{"id":205,"date":"2019-04-09T00:57:21","date_gmt":"2019-04-09T04:57:21","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&#038;p=205"},"modified":"2019-04-12T19:00:50","modified_gmt":"2019-04-12T23:00:50","slug":"3-6-the-quantum-tunneling-of-particles-through-potential-barriers","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/3-6-the-quantum-tunneling-of-particles-through-potential-barriers\/","title":{"raw":"3.6 The Quantum Tunneling of Particles through Potential Barriers","rendered":"3.6 The Quantum Tunneling of Particles through Potential Barriers"},"content":{"raw":"<div data-type=\"abstract\" id=\"19923\" class=\"ui-has-child-title\"><header>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Describe how a quantum particle may tunnel across a potential barrier<\/li>\r\n \t<li>Identify important physical parameters that affect the tunneling probability<\/li>\r\n \t<li>Identify the physical phenomena where quantum tunneling is observed<\/li>\r\n \t<li>Explain how quantum tunneling is utilized in modern technologies<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<strong style=\"font-size: 14pt\" data-effect=\"bold\">Quantum tunneling<\/strong><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a phenomenon in which particles penetrate a potential energy barrier with a height greater than the total energy of the particles. The phenomenon is interesting and important because it violates the principles of classical mechanics. Quantum tunneling is important in models of the Sun and has a wide range of applications, such as the scanning tunneling microscope and the tunnel diode.<\/span>\r\n\r\n<\/header><\/div>\r\n<section id=\"fs-id1170899453628\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Tunneling and Potential Energy<\/h3>\r\n<p id=\"fs-id1170901605351\">To illustrate<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term314\">quantum tunneling<\/span>, consider a ball rolling along a surface with a kinetic energy of 100 J. As the ball rolls, it encounters a hill. The potential energy of the ball placed atop the hill is 10 J. Therefore, the ball (with 100 J of kinetic energy) easily rolls over the hill and continues on. In classical mechanics, the probability that the ball passes over the hill is exactly 1\u2014it makes it over every time. If, however, the height of the hill is increased\u2014a ball placed atop the hill has a potential energy of 200 J\u2014the ball proceeds only part of the way up the hill, stops, and returns in the direction it came. The total energy of the ball is converted entirely into potential energy before it can reach the top of the hill. We do not expect, even after repeated attempts, for the 100-J ball to ever be found beyond the hill. Therefore, the probability that the ball passes over the hill is exactly 0, and probability it is turned back or \u201creflected\u201d by the hill is exactly 1. The ball<span>\u00a0<\/span><em data-effect=\"italics\">never<\/em><span>\u00a0<\/span>makes it over the hill. The existence of the ball beyond the hill is an impossibility or \u201cenergetically forbidden.\u201d<\/p>\r\n<p id=\"fs-id1170901510354\">However, according to quantum mechanics, the ball has a wave function and this function is defined over all space. The wave function may be highly localized, but there is always a chance that as the ball encounters the hill, the ball will suddenly be found beyond it. Indeed, this probability is appreciable if the \u201cwave packet\u201d of the ball is wider than the barrier.<\/p>\r\n\r\n<div class=\"media-2 ui-has-child-title\" data-type=\"note\" id=\"fs-id1170902008280\"><header>\r\n<h3 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\">INTERACTIVE<\/span><\/h3>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1170902917781\">View this<span>\u00a0<\/span><a href=\"https:\/\/openstax.org\/l\/21intquatanvid\" rel=\"nofollow\">interactive simulation<\/a><span>\u00a0<\/span>for a simulation of tunneling.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1170901600117\">In the language of quantum mechanics, the hill is characterized by a<span>\u00a0<\/span><span data-type=\"term\" id=\"term315\">potential barrier<\/span>. A finite-height square barrier is described by the following potential-energy function:<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-324-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)={0,when x&lt;0U0,when 0\u2264x\u2264L0,when x&gt;L.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.59]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901928797\">The potential barrier is illustrated in<span>\u00a0<\/span>Figure 3.16. When the height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-325-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span><span>\u00a0<\/span>of the barrier is infinite, the wave packet representing an incident quantum particle is unable to penetrate it, and the quantum particle bounces back from the barrier boundary, just like a classical particle. When the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the barrier is infinite and its height is finite, a part of the wave packet representing an incident quantum particle can filter through the barrier boundary and eventually perish after traveling some distance inside the barrier.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_barrier\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"516\"]<img alt=\"The potential U of x is plotted as a function of x. U is zero for x less than 0 and for x greater than L. It is equal to U sub 0 between x =0 and x=L. The constant energy E is indicated as a dotted horizontal line at a value less than U sub 0. The region x less than 0 is labeled as region I and has both incident and reflected waves, going to the right and left respectively. The region between x=0 and x=L is labeled as region II. The region x greater than L is labeled as region III and has only transmitted waves going to the right.\" data-media-type=\"image\/jpeg\" id=\"70394\" src=\"https:\/\/cnx.org\/resources\/f934a9164c237f284add8c0eccb2d0c9b03a8468\" width=\"516\" height=\"421\" \/> Figure 3.16 A potential energy barrier of height U0 creates three physical regions with three different wave behaviors. In region I where x&lt;0, an incident wave packet (incident particle) moves in a potential-free zone and coexists with a reflected wave packet (reflected particle). In region II, a part of the incident wave that has not been reflected at x=0 moves as a transmitted wave in a constant potential U(x)=+U0 and tunnels through to region III at x=L. In region III for x&gt;L, a wave packet (transmitted particle) that has tunneled through the potential barrier moves as a free particle in potential-free zone. The energy E of the incident particle is indicated by the horizontal line.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170901769645\">When both the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>and the height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-332-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span><span>\u00a0<\/span>are finite, a part of the quantum wave packet incident on one side of the barrier can penetrate the barrier boundary and continue its motion inside the barrier, where it is gradually attenuated on its way to the other side. A part of the incident quantum wave packet eventually emerges on the other side of the barrier in the form of the transmitted wave packet that tunneled through the barrier. How much of the incident wave can tunnel through a barrier depends on the barrier width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>and its height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-333-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span>, and on the energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>of the quantum particle incident on the barrier. This is the physics of tunneling.<\/p>\r\n<p id=\"fs-id1170903049760\">Barrier penetration by quantum wave functions was first analyzed theoretically by Friedrich<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term316\">Hund<\/span><span>\u00a0<\/span>in 1927, shortly after Schr\u04e7dinger published the equation that bears his name. A year later, George<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term317\">Gamow<\/span><span>\u00a0<\/span>used the formalism of quantum mechanics to explain the radioactive<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-334-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-decay of atomic nuclei as a quantum-tunneling phenomenon. The invention of the tunnel diode in 1957 made it clear that quantum tunneling is important to the semiconductor industry. In modern nanotechnologies, individual atoms are manipulated using a knowledge of quantum tunneling.<\/p>\r\n\r\n<\/section><section id=\"fs-id1170902863895\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Tunneling and the Wave Function<\/h3>\r\n<p id=\"fs-id1170901904236\">Suppose a uniform and time-independent beam of electrons or other quantum particles with energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>traveling along the<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>-axis (in the positive direction to the right) encounters a potential barrier described by<span>\u00a0<\/span>Equation 3.59. The question is: What is the probability that an individual particle in the beam will tunnel through the potential barrier? The answer can be found by solving the boundary-value problem for the time-independent Schr\u04e7dinger equation for a particle in the beam. The general form of this equation is given by<span>\u00a0<\/span>Equation 3.60, which we reproduce here:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170901793747\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-335-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-335-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8(x)dx2+U(x)\u03c8(x)=E\u03c8(x),where\u2212\u221e&lt;x&lt;+\u221e.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.60]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">In<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.60, the potential function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">U<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">) is defined by<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.59. We assume that the given energy<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">of the incoming particle is smaller than the height<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-336-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U0<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">of the potential barrier,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-337-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">E&lt;U0<\/span><\/span><span style=\"font-size: 14pt\">, because this is the interesting physical case. Knowing the energy<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">of the incoming particle, our task is to solve<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.60<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">for a function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-338-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8(x)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">that is continuous and has continuous first derivatives for all<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">. In other words, we are looking for a \u201csmooth-looking\u201d solution (because this is how wave functions look) that can be given a probabilistic interpretation so that<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-339-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03c8(x)|2=\u03c8*(x)\u03c8(x)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the probability density.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901916178\">We divide the real axis into three regions with the boundaries defined by the potential function in<span>\u00a0<\/span>Equation 3.59<span>\u00a0<\/span>(illustrated in<span>\u00a0<\/span>Figure 3.16) and transcribe<span>\u00a0<\/span>Equation 3.60<span>\u00a0<\/span>for each region. Denoting by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-340-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8I(x)<\/span><\/span><\/span><span>\u00a0<\/span>the solution in region I for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-341-Frame\"><span class=\"MathJax_MathContainer\"><span>x&lt;0<\/span><\/span><\/span>, by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-342-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8II(x)<\/span><\/span><\/span><span>\u00a0<\/span>the solution in region II for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-343-Frame\"><span class=\"MathJax_MathContainer\"><span>0\u2264x\u2264L<\/span><\/span><\/span>, and by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-344-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8III(x)<\/span><\/span><\/span><span>\u00a0<\/span>the solution in region III for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-345-Frame\"><span class=\"MathJax_MathContainer\"><span>x&gt;L<\/span><\/span><\/span>, the stationary Schr\u04e7dinger equation has the following forms in these three regions:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170903064710\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-346-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-346-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8I(x)dx2=E\u03c8I(x),in region I:\u2212\u221e&lt;x&lt;0,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.61]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901948077\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-347-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-347-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8II(x)dx2+U0\u03c8II(x)=E\u03c8II(x),in region II:0\u2264x\u2264L,<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.62]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170903070603\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-348-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-348-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8III(x)dx2=E\u03c8III(x),in region III:L&lt;x&lt;+\u221e.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.63]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The continuity condition at region boundaries requires that:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901741204\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-349-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-349-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8I(0)=\u03c8II(0),at the boundary between regions I and II and<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.64]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">and<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170899245855\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-350-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-350-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8II(L)=\u03c8III(L),at the boundary between regions II and III.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.65]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The \u201csmoothness\u201d condition requires the first derivative of the solution be continuous at region boundaries:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901533726\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-351-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-351-Frame\"><span class=\"MathJax_MathContainer\"><span>d\u03c8I(x)dx|x=0=d\u03c8II(x)dx|x=0,at the boundary between regions I and II;<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.66]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">and<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902790371\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-352-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-352-Frame\"><span class=\"MathJax_MathContainer\"><span>d\u03c8II(x)dx|x=L=d\u03c8III(x)dx|x=L,at the boundary between regions II and III.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.67]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">In what follows, we find the functions<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-353-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8I(x)<\/span><\/span><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-354-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8II(x)<\/span><\/span><span style=\"font-size: 14pt\">, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-355-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8III(x)<\/span><\/span><span style=\"font-size: 14pt\">.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901980803\">We can easily verify (by substituting into the original equation and differentiating) that in regions I and III, the solutions must be in the following general forms:<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170903010927\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-356-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-356-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8I(x)=Ae+ikx+Be\u2212ikx<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\">\r\n\r\n<span class=\"os-number\">[3.68]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox\">\r\n<div data-type=\"equation\" id=\"fs-id1170903010927\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-356-Frame\">\r\n\r\n<span style=\"font-size: 14pt\">\u03c8III(x)=Fe+ikx+Ge\u2212ikx<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170903043787\">\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.69]<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-358-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">k=2mE\/\u210f<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a wave number and the complex exponent denotes oscillations,<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901539686\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-359-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-359-Frame\"><span class=\"MathJax_MathContainer\"><span>e\u00b1ikx=coskx\u00b1isinkx.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.70]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The constants<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">B<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">G<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.68<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.69<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">may be complex. These solutions are illustrated in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.16. In region I, there are two waves\u2014one is incident (moving to the right) and one is reflected (moving to the left)\u2014so none of the constants<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A\u00a0<\/em><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">B<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.68<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">may vanish. In region III, there is only one wave (moving to the right), which is the transmitted wave, so the constant<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">G<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">must be zero in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.69,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-360-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">G=0<\/span><\/span><span style=\"font-size: 14pt\">. We can write explicitly that the incident wave is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-361-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8in(x)=Ae+ikx<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and that the reflected wave is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-362-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8ref(x)=Be\u2212ikx<\/span><\/span><span style=\"font-size: 14pt\">, and that the transmitted wave is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-363-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8tra(x)=Fe+ikx<\/span><\/span><span style=\"font-size: 14pt\">. The amplitude of the incident wave is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903084073\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-364-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8in(x)|2=\u03c8in*(x)\u03c8in(x)=(Ae+ikx)*Ae+ikx=A*e\u2212ikxAe+ikx=A*A=|A|2.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901634975\">Similarly, the amplitude of the reflected wave is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-365-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8ref(x)|2=|B|2<\/span><\/span><\/span><span>\u00a0<\/span>and the amplitude of the transmitted wave is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-366-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8tra(x)|2=|F|2<\/span><\/span><\/span>. We know from the theory of waves that the square of the wave amplitude is directly proportional to the wave intensity. If we want to know how much of the incident wave tunnels through the barrier, we need to compute the square of the amplitude of the transmitted wave. The<span>\u00a0<\/span><span data-type=\"term\" id=\"term318\">transmission probability<\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span data-type=\"term\" id=\"term319\">tunneling probability<\/span><span>\u00a0<\/span>is the ratio of the transmitted intensity<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-367-Frame\"><span class=\"MathJax_MathContainer\"><span>(|F|2)<\/span><\/span><\/span><span>\u00a0<\/span>to the incident intensity<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-368-Frame\"><span class=\"MathJax_MathContainer\"><span>(|A|2)<\/span><\/span><\/span>, written as<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-369-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=|\u03c8tra(x)|2|\u03c8in(x)|2=|F|2|A|2=|FA|2<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.71]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170903037273\">where<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>is the width of the barrier and<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is the total energy of the particle. This is the probability an individual particle in the incident beam will tunnel through the potential barrier. Intuitively, we understand that this probability must depend on the barrier height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-370-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span>.<\/p>\r\n<p id=\"fs-id1170903111657\">In region II, the terms in equation<span>\u00a0<\/span>Equation 3.62<span>\u00a0<\/span>can be rearranged to<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170901903999\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-371-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-371-Frame\"><span class=\"MathJax_MathContainer\"><span>d2\u03c8II(x)dx2=\u03b22\u03c8II(x)<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.72]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-372-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b22<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is positive because<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-373-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U0&gt;E<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and the parameter<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-374-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b2<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a real number,<\/span>\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-375-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b22=2m\u210f2(U0\u2212E).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.73]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The general solution to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.72<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is not oscillatory (unlike in the other regions) and is in the form of exponentials that describe a gradual attenuation of<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-376-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8II(x)<\/span><\/span><span style=\"font-size: 14pt\">,<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902869487\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-377-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-377-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8II(x)=Ce\u2212\u03b2x+De+\u03b2x.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.74]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">The two types of solutions in the three regions are illustrated in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.17.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_tunneling\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"516\"]<img alt=\"A solution to the barrier potential U of x is plotted as a function of x. U is zero for x less than 0 and for x greater than L. It is equal to U sub 0 between x =0 and x=L. The wave function oscillates in the region x less than zero. The wave function is labeled psi sub I in this region. It decays exponentially in the region between x=0 and x=L, and is labeled psi sub I I in this region. It oscillates again in the x greater than L region, where it is labeled psi sub I I I. The amplitude of the oscillations is smaller in region I I I than in region I but the wavelength is the same. The wave function and its derivative are continuous at x=0 and x=L.\" data-media-type=\"image\/jpeg\" id=\"35274\" src=\"https:\/\/cnx.org\/resources\/b95198c890dbf4f2538a07307bb26ad959617f7f\" width=\"516\" height=\"419\" \/> Figure 3.17 Three types of solutions to the stationary Schr\u04e7dinger equation for the quantum-tunneling problem: Oscillatory behavior in regions I and III where a quantum particle moves freely, and exponential-decay behavior in region II (the barrier region) where the particle moves in the potential U0.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170901711932\">Now we use the boundary conditions to find equations for the unknown constants.<span>\u00a0<\/span>Equation 3.68<span>\u00a0<\/span>and<span>\u00a0<\/span>Equation 3.74<span>\u00a0<\/span>are substituted into<span>\u00a0<\/span>Equation 3.64<span>\u00a0<\/span>to give<\/p>\r\n\r\n<div data-type=\"equation\" id=\"fs-id1170901672030\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-379-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-379-Frame\"><span class=\"MathJax_MathContainer\"><span>A+B=C+D.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.75]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Equation 3.74<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.69<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">are substituted into<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.65<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to give<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902011772\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-380-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-380-Frame\"><span class=\"MathJax_MathContainer\"><span>Ce\u2212\u03b2L+De+\u03b2L=Fe+ikL.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.76]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Similarly, we substitute<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.68<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.74<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">into<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.66, differentiate, and obtain<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170901583450\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-381-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-381-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212ik(A\u2212B)=\u03b2(D\u2212C).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.77]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">Similarly, the boundary condition<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.67<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">reads explicitly<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170903074449\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-382-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-382-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b2(De+\u03b2L\u2212Ce\u2212\u03b2L)=\u2212ikFe+ikL.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.78]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">We now have four equations for five unknown constants. However, because the quantity we are after is the transmission coefficient, defined in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.71<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">by the fraction<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, the number of equations is exactly right because when we divide each of the above equations by<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, we end up having only four unknown fractions:<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">B<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">C<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">D<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, three of which can be eliminated to find<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">. The actual algebra that leads to expression for<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is pretty lengthy, but it can be done either by hand or with a help of computer software. The end result is<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"equation\" id=\"fs-id1170902951016\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-383-Frame\">\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-383-Frame\"><span class=\"MathJax_MathContainer\"><span>FA=e\u2212ikLcosh(\u03b2L)+i(\u03b3\/2)sinh(\u03b2L).<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.79]<\/span><\/div>\r\n<\/div>\r\n<span style=\"font-size: 14pt\">In deriving<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.79, to avoid the clutter, we use the substitutions<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-384-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b3\u2261\u03b2\/k\u2212k\/\u03b2<\/span><\/span><span style=\"font-size: 14pt\">,<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902707922\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-385-Frame\"><span class=\"MathJax_MathContainer\"><span>coshy=ey+e\u2212y2,andsinhy=ey\u2212e\u2212y2.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902738264\">We substitute<span>\u00a0<\/span>Equation 3.79<span>\u00a0<\/span>into<span>\u00a0<\/span>Equation 3.71<span>\u00a0<\/span>and obtain the exact expression for the transmission coefficient for the barrier,<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903079519\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-386-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=(FA)*FA=e+ikLcosh(\u03b2L)\u2212i(\u03b3\/2)sinh(\u03b2L)\u00b7e\u2212ikLcosh(\u03b2L)+i(\u03b3\/2)sinh(\u03b2L)<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901533679\">or<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-387-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=1cosh2(\u03b2L)+(\u03b3\/2)2sinh2(\u03b2L)<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.80]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901609283\">where<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901492355\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-388-Frame\"><span class=\"MathJax_MathContainer\"><span>(\u03b32)2=14(1\u2212E\/U0E\/U0+E\/U01\u2212E\/U0\u22122).<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170899453560\">For a wide and high barrier that transmits poorly,<span>\u00a0<\/span>Equation 3.80<span>\u00a0<\/span>can be approximated by<\/p>\r\n\r\n<div class=\"textbox\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-389-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=16EU0(1\u2212EU0)e\u22122\u03b2L.<\/span><\/span><\/div>\r\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.81]<\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170901525408\">Whether it is the exact expression<span>\u00a0<\/span>Equation 3.80<span>\u00a0<\/span>or the approximate expression<span>\u00a0<\/span>Equation 3.81, we see that the tunneling effect very strongly depends on the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the potential barrier. In the laboratory, we can adjust both the potential height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-390-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span><span>\u00a0<\/span>and the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>to design nano-devices with desirable transmission coefficients.<\/p>\r\n\r\n<div data-type=\"example\" id=\"fs-id1170902720394\" class=\"ui-has-child-title\"><header><\/header><section>\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.12<\/span><span class=\"os-divider\"><\/span><\/h3>\r\n<\/header><section>\r\n<p id=\"fs-id1170903140982\"><span data-type=\"title\"><strong>Transmission Coefficient<\/strong><\/span><\/p>\r\nTwo copper nanowires are insulated by a copper oxide nano-layer that provides a 10.0-eV potential barrier. Estimate the tunneling probability between the nanowires by 7.00-eV electrons through a 5.00-nm thick oxide layer. What if the thickness of the layer were reduced to just 1.00 nm? What if the energy of electrons were increased to 9.00 eV?\r\n\r\n<span data-type=\"title\"><strong>Strategy<\/strong><\/span>\r\n\r\nTreating the insulating oxide layer as a finite-height potential barrier, we use<span>\u00a0<\/span>Equation 3.81. We identify<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-391-Frame\"><span class=\"MathJax_MathContainer\"><span>U0=10.0eV<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-392-Frame\"><span class=\"MathJax_MathContainer\"><span>E1=7.00eV<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-393-Frame\"><span class=\"MathJax_MathContainer\"><span>E2=9.00eV<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-394-Frame\"><span class=\"MathJax_MathContainer\"><span>L1=5.00nm<\/span><\/span><\/span>, and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-395-Frame\"><span class=\"MathJax_MathContainer\"><span>L2=1.00nm<\/span><\/span><\/span>. We use<span>\u00a0<\/span>Equation 3.73<span>\u00a0<\/span>to compute the exponent. Also, we need the rest mass of the electron<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-396-Frame\"><span class=\"MathJax_MathContainer\"><span>m=511keV\/c2<\/span><\/span><\/span><span>\u00a0<\/span>and Planck\u2019s constant<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-397-Frame\"><span class=\"MathJax_MathContainer\"><span>\u210f=0.1973keV\u00b7nm\/c<\/span><\/span><\/span>. It is typical for this type of estimate to deal with very small quantities that are often not suitable for handheld calculators. To make correct estimates of orders, we make the conversion<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-398-Frame\"><span class=\"MathJax_MathContainer\"><span>ey=10y\/ln10<\/span><\/span><\/span>.\r\n\r\n<span data-type=\"title\"><strong>Solution<\/strong><\/span>\r\n\r\nConstants:\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901744088\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-399-Frame\"><span class=\"MathJax_MathContainer\"><span>2m\u210f2=2(511keV\/c2)(0.1973keV\u00b7nm\/c)2=26,2541keV\u00b7(nm)2,<\/span><\/span><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902597918\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-400-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>\u03b2=2m\u210f2(U0\u2212E)=26,254(10.0eV\u2212E)keV\u00b7(nm)2=26.254(10.0eV\u2212E)\/eV1nm.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">For a lower-energy electron with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-401-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">E1=7.00eV<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901492418\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-402-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b21=26.254(10.00eV\u2212E1)\/eV1nm=26.254(10.00\u22127.00)1nm=8.875nm,<\/span><\/span><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901510689\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-403-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>T(L,E1)=16E1U0(1\u2212E1U0)e\u22122\u03b21L=16710(1\u2212710)e\u221217.75L\/nm=3.36e\u221217.75L\/nm.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">For a higher-energy electron with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-404-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">E2=9.00eV<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902598093\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-405-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b22=26.254(10.00eV\u2212E2)\/eV1nm=26.254(10.00\u22129.00)1nm=5.124nm,<\/span><\/span><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902795334\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-406-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>T(L,E2)=16E2U0(1\u2212E2U0)e\u22122\u03b22L=16910(1\u2212910)e\u22125.12L\/nm=1.44e\u22125.12L\/nm.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">For a broad barrier with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-407-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L1=5.00nm<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170899293709\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-408-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L1,E1)=3.36e\u221217.75L1\/nm=3.36e\u221217.75\u00b75.00nm\/nm=3.36e\u221288=3.36(6.2\u00d710\u221239)=2.1%\u00d710\u221236,<\/span><\/span><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903032292\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-409-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>T(L1,E2)=1.44e\u22125.12L1\/nm=1.44e\u22125.12\u00b75.00nm\/nm=1.44e\u221225.6=1.44(7.62\u00d710\u221212)=1.1%\u00d710\u22129.<\/span><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">For a narrower barrier with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-410-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L2=1.00nm<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901924352\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-411-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L2,E1)=3.36e\u221217.75L2\/nm=3.36e\u221217.75\u00b71.00nm\/nm=3.36e\u221217.75=3.36(5.1\u00d710\u22127)=1.7%\u00d710\u22124,<\/span><\/span><\/div>\r\n<\/div>\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901478827\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-412-Frame\">\r\n\r\n<span class=\"MathJax_MathContainer\"><span>T(L2,E2)=1.44e\u22125.12L2\/nm=1.44e\u22125.12\u00b71.00nm\/nm=1.44e\u22125.12=1.44(5.98\u00d710\u22123)=0.86%.<\/span><\/span>\r\n\r\n<span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span>\r\n\r\n<span style=\"text-indent: 1em;font-size: 1rem\">We see from these estimates that the probability of tunneling is affected more by the width of the potential barrier than by the energy of an incident particle. In today\u2019s technologies, we can manipulate individual atoms on metal surfaces to create potential barriers that are fractions of a nanometer, giving rise to measurable tunneling currents. One of many applications of this technology is the scanning tunneling microscope (STM), which we discuss later in this section.<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1170901541920\" data-depth=\"1\">\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a03<\/span><\/span><span class=\"os-number\">.10<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\"><header><span style=\"font-size: 1rem\">A proton with kinetic energy 1.00 eV is incident on a square potential barrier with height 10.00 eV. If the proton is to have the same transmission probability as an electron of the same energy, what must the width of the barrier be relative to the barrier width encountered by an electron?<\/span><\/header><\/div>\r\n<\/div>\r\n<h3 data-type=\"title\">Radioactive Decay<\/h3>\r\n<p id=\"fs-id1170901541925\">In 1928, Gamow identified quantum tunneling as the mechanism responsible for the<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term320\">radioactive decay<\/span><span>\u00a0<\/span>of atomic nuclei. He observed that some isotopes of thorium, uranium, and bismuth disintegrate by emitting<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-413-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles (which are doubly ionized helium atoms or, simply speaking, helium nuclei). In the process of emitting an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-414-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle, the original nucleus is transformed into a new nucleus that has two fewer neutrons and two fewer protons than the original nucleus. The<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-415-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles emitted by one isotope have approximately the same kinetic energies. When we look at variations of these energies among isotopes of various elements, the lowest kinetic energy is about 4 MeV and the highest is about 9 MeV, so these energies are of the same order of magnitude. This is about where the similarities between various isotopes end.<\/p>\r\n<p id=\"fs-id1170899266027\">When we inspect half-lives (a half-life is the time in which a radioactive sample loses half of its nuclei due to decay), different isotopes differ widely. For example, the half-life of polonium-214 is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-416-Frame\"><span class=\"MathJax_MathContainer\"><span>160\u00b5s<\/span><\/span><\/span><span>\u00a0<\/span>and the half-life of uranium is 4.5 billion years. Gamow explained this variation by considering a \u2018spherical-box\u2019 model of the nucleus, where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-417-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles can bounce back and forth between the walls as free particles. The confinement is provided by a strong nuclear potential at a spherical wall of the box. The thickness of this wall, however, is not infinite but finite, so in principle, a nuclear particle has a chance to escape this nuclear confinement. On the inside wall of the confining barrier is a high nuclear potential that keeps the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-418-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle in a small confinement. But when an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-419-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle gets out to the other side of this wall, it is subject to electrostatic Coulomb repulsion and moves away from the nucleus. This idea is illustrated in<span>\u00a0<\/span>Figure 3.18. The width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the potential barrier that separates an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-420-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle from the outside world depends on the particle\u2019s kinetic energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em>. This width is the distance between the point marked by the nuclear radius<span>\u00a0<\/span><em data-effect=\"italics\">R<\/em><span>\u00a0<\/span>and the point<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-421-Frame\"><span class=\"MathJax_MathContainer\"><span>R0<\/span><\/span><\/span><span>\u00a0<\/span>where an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-422-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle emerges on the other side of the barrier,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-423-Frame\"><span class=\"MathJax_MathContainer\"><span>L=R0\u2212R<\/span><\/span><\/span>. At the distance<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-424-Frame\"><span class=\"MathJax_MathContainer\"><span>R0<\/span><\/span><\/span>, its kinetic energy must at least match the electrostatic energy of repulsion,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-425-Frame\"><span class=\"MathJax_MathContainer\"><span>E=(4\u03c0\u03b50)\u22121Ze2\/R0<\/span><\/span><\/span><span>\u00a0<\/span>(where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-426-Frame\"><span class=\"MathJax_MathContainer\"><span>+Ze<\/span><\/span><\/span><span>\u00a0<\/span>is the charge of the nucleus). In this way we can estimate the width of the nuclear barrier,<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902793216\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-427-Frame\"><span class=\"MathJax_MathContainer\"><span>L=e24\u03c0\u03b50ZE\u2212R.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170899368006\">We see from this estimate that the higher the energy of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-428-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle, the narrower the width of the barrier that it is to tunnel through. We also know that the width of the potential barrier is the most important parameter in tunneling probability. Thus, highly energetic<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-429-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles have a good chance to escape the nucleus, and, for such nuclei, the nuclear disintegration half-life is short. Notice that this process is highly nonlinear, meaning a small increase in the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-430-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle energy has a disproportionately large enhancing effect on the tunneling probability and, consequently, on shortening the half-life. This explains why the half-life of polonium that emits 8-MeV<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-431-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles is only hundreds of milliseconds and the half-life of uranium that emits 4-MeV<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-432-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles is billions of years.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_nucleus\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"373\"]<img alt=\"The potential U of r is plotted as a function of r. For r less than R, U of r is constant and negative. At r = R, the potential rises vertically to some maximum positive value, then decays toward zero. The area under the curve is shaded. U of r equals E at r equal to R sub 0. A horizontal dashed line at E=E and a vertical dashed line at r=R sub 0 are shown.\" data-media-type=\"image\/jpeg\" id=\"47863\" src=\"https:\/\/cnx.org\/resources\/1d92a84dbfb6f34dbf2e0ed5e4509f2e24d70726\" width=\"373\" height=\"642\" \/> Figure 3.18 The potential energy barrier for an \u03b1-particle bound in the nucleus: To escape from the nucleus, an \u03b1-particle with energy Emust tunnel across the barrier from distance R to distance R0 away from the center.[\/caption]<\/figure>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170899265382\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Field Emission<\/h3>\r\n<p id=\"fs-id1170899265387\"><strong data-effect=\"bold\">Field emission<\/strong><span>\u00a0<\/span>is a process of emitting electrons from conducting surfaces due to a strong external electric field that is applied in the direction normal to the surface (Figure 3.19). As we know from our study of electric fields in earlier chapters, an applied external electric field causes the electrons in a conductor to move to its surface and stay there as long as the present external field is not excessively strong. In this situation, we have a constant electric potential throughout the inside of the conductor, including its surface. In the language of potential energy, we say that an electron inside the conductor has a constant potential energy<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-436-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=\u2212U0<\/span><\/span><\/span><span>\u00a0<\/span>(here, the<span>\u00a0<\/span><em data-effect=\"italics\">x\u00a0<\/em>means inside the conductor). In the situation represented in<span>\u00a0<\/span>Figure 3.19, where the external electric field is uniform and has magnitude<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-437-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span>, if an electron happens to be outside the conductor at a distance<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em><span>\u00a0<\/span>away from its surface, its potential energy would have to be<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-438-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=\u2212eEgx<\/span><\/span><\/span><span>\u00a0<\/span>(here,<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em><span>\u00a0<\/span>denotes distance to the surface). Taking the origin at the surface, so that<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-439-Frame\"><span class=\"MathJax_MathContainer\"><span>x=0<\/span><\/span><\/span><span>\u00a0<\/span>is the location of the surface, we can represent the potential energy of conduction electrons in a metal as the potential energy barrier shown in<span>\u00a0<\/span>Figure 3.20. In the absence of the external field, the potential energy becomes a step barrier defined by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-440-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x\u22640)=\u2212U0<\/span><\/span><\/span><span>\u00a0<\/span>and by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-441-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x&gt;0)=0<\/span><\/span><\/span>.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_field\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"450\"]<img alt=\"The potential U of r is plotted as a function of r. For r less than R, U of r is constant and negative. At r = R, the potential rises vertically to some maximum positive value, then decays toward zero. The area under the curve is shaded. U of r equals E at r equal to R sub 0. A horizontal dashed line at E=E and a vertical dashed line at r=R sub 0 are shown.\" data-media-type=\"image\/jpeg\" id=\"49581\" src=\"https:\/\/cnx.org\/resources\/333cb466c65c1ffc2ea59a313a3b742eb648a6f3\" width=\"450\" height=\"512\" \/> Figure 3.19 A normal-direction external electric field at the surface of a conductor: In a strong field, the electrons on a conducting surface may get detached from it and accelerate against the external electric field away from the surface.[\/caption]<\/figure>\r\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\"><\/span><span class=\"os-caption\">\r\n<\/span><\/em><\/div>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_metal\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"672\"]<img alt=\"U of x is plotted as a function of x. For x less than zero, U of x has a constant value of minus U sub zero. At x=0, U of x jumps to a value of zero. For x larger than zero, U of x equals minus e times E sub g times x. The area under the curve is shaded. The energy is a negative constant, shown as a dashed line, at a value of minus phi. U of x equals E at x equal phi divided by the quantity e times E sub g.\" data-media-type=\"image\/jpeg\" id=\"71000\" src=\"https:\/\/cnx.org\/resources\/0cfe7bcb891cef9a31464af3f1002afa56bda2a1\" width=\"672\" height=\"495\" \/> Figure 3.20 The potential energy barrier at the surface of a metallic conductor in the presence of an external uniform electric field Eg normal to the surface: It becomes a step-function barrier when the external field is removed. The work function of the metal is indicated by \u03d5.[\/caption]<\/figure>\r\n<\/div>\r\n<p id=\"fs-id1170901704550\">When an external electric field is strong, conduction electrons at the surface may get detached from it and accelerate along electric field lines in a direction antiparallel to the external field, away from the surface. In short, conduction electrons may escape from the surface. The<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term321\">field emission<\/span><span>\u00a0<\/span>can be understood as the quantum tunneling of conduction electrons through the potential barrier at the conductor\u2019s surface. The physical principle at work here is very similar to the mechanism of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-444-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-emission from a radioactive nucleus.<\/p>\r\n<p id=\"fs-id1170899245185\">Suppose a conduction electron has a kinetic energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>(the average kinetic energy of an electron in a metal is the work function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-445-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>for the metal and can be measured, as discussed for the photoelectric effect in<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:fdd4b413-6910-44d5-801d-0f4223bc7a31@5\" data-page=\"48\">Photons and Matter Waves<\/a>), and an external electric field can be locally approximated by a uniform electric field of strength<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-446-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span>. The width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the potential barrier that the electron must cross is the distance from the conductor\u2019s surface to the point outside the surface where its kinetic energy matches the value of its potential energy in the external field. In<span>\u00a0<\/span>Figure 3.20, this distance is measured along the dashed horizontal line<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-447-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=E<\/span><\/span><\/span><span>\u00a0<\/span>from<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-448-Frame\"><span class=\"MathJax_MathContainer\"><span>x=0<\/span><\/span><\/span><span>\u00a0<\/span>to the intercept with<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-449-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=\u2212eEgx<\/span><\/span><\/span>, so the barrier width is<\/p>\r\n\r\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170899266044\">\r\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-450-Frame\"><span class=\"MathJax_MathContainer\"><span>L=e\u22121EEg=e\u22121\u03d5Eg.<\/span><\/span><\/div>\r\n<\/div>\r\n<p id=\"fs-id1170902851400\">We see that<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>is inversely proportional to the strength<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-451-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span><span>\u00a0<\/span>of an external field. When we increase the strength of the external field, the potential barrier outside the conductor becomes steeper and its width decreases for an electron with a given kinetic energy. In turn, the probability that an electron will tunnel across the barrier (conductor surface) becomes exponentially larger. The electrons that emerge on the other side of this barrier form a current (tunneling-electron current) that can be detected above the surface. The tunneling-electron current is proportional to the tunneling probability. The tunneling probability depends nonlinearly on the barrier width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em>, and<span>\u00a0<\/span><em data-effect=\"italics\">L\u00a0<\/em>can be changed by adjusting<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-452-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span>. Therefore, the tunneling-electron current can be tuned by adjusting the strength of an external electric field at the surface. When the strength of an external electric field is constant, the tunneling-electron current has different values at different elevations<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>above the surface.<\/p>\r\n<p id=\"fs-id1170901616105\">The quantum tunneling phenomenon at metallic surfaces, which we have just described, is the physical principle behind the operation of the<span>\u00a0<\/span><span data-type=\"term\" id=\"term322\">scanning tunneling microscope (STM)<\/span>, invented in 1981 by Gerd Binnig and Heinrich Rohrer. The STM device consists of a scanning tip (a needle, usually made of tungsten, platinum-iridium, or gold); a piezoelectric device that controls the tip\u2019s elevation in a typical range of 0.4 to 0.7 nm above the surface to be scanned; some device that controls the motion of the tip along the surface; and a computer to display images. While the sample is kept at a suitable voltage bias, the scanning tip moves along the surface (Figure 3.21), and the tunneling-electron current between the tip and the surface is registered at each position. The amount of the current depends on the probability of electron tunneling from the surface to the tip, which, in turn, depends on the elevation of the tip above the surface. Hence, at each tip position, the distance from the tip to the surface is measured by measuring how many electrons tunnel out from the surface to the tip. This method can give an unprecedented resolution of about 0.001 nm, which is about 1% of the average diameter of an atom. In this way, we can see individual atoms on the surface, as in the image of a carbon nanotube in<span>\u00a0<\/span>Figure 3.22.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_stm\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"693\"]<img alt=\"An illustration of a scanning tunneling microscope. The atoms in the tip and sample are represented by spheres, orange for the S T M tip and purple for the sample. The atoms on the surface of the atoms being scanned are arranged in this illustration in a grid of four atoms by five atoms. The tip is above one of the atoms, and a tunneling electron current is shown between the tip and the surface atom. The image on the computer monitor is a 4 by 5 grid of spots.\" data-media-type=\"image\/jpeg\" id=\"74886\" src=\"https:\/\/cnx.org\/resources\/e419d79bf6204b92dfe14e6b97f581a05c20ff00\" width=\"693\" height=\"509\" \/> Figure 3.21 In STM, a surface at a constant potential is being scanned by a narrow tip moving along the surface. When the STM tip moves close to surface atoms, electrons can tunnel from the surface to the tip. This tunneling-electron current is continually monitored while the tip is in motion. The amount of current at location (x,y) gives information about the elevation of the tip above the surface at this location. In this way, a detailed topographical map of the surface is created and displayed on a computer monitor.[\/caption]<\/figure>\r\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\"><\/span><span class=\"os-caption\">\r\n<\/span><\/em><\/div>\r\n<\/div>\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_nanotube\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"488\"]<img alt=\"An STM image of a carbon nanotube showing the atoms as red points in a grid like pattern.\" data-media-type=\"image\/jpeg\" id=\"57721\" src=\"https:\/\/cnx.org\/resources\/57753c12751920a2283483c54c90d7109d3fe614\" width=\"488\" height=\"156\" \/> Figure 3.22 An STM image of a carbon nanotube: Atomic-scale resolution allows us to see individual atoms on the surface. STM images are in gray scale, and coloring is added to bring up details to the human eye. (credit: Taner Yildirim, NIST)[\/caption]<\/figure>\r\n<\/div>\r\n<\/section><section id=\"fs-id1170903136127\" data-depth=\"1\">\r\n<h3 data-type=\"title\">Resonant Quantum Tunneling<\/h3>\r\n<p id=\"fs-id1170901840333\">Quantum tunneling has numerous applications in semiconductor devices such as electronic circuit components or integrated circuits that are designed at nanoscales; hence, the term \u2018<span data-type=\"term\" id=\"term323\">nanotechnology<\/span>.\u2019 For example, a diode (an electric-circuit element that causes an electron current in one direction to be different from the current in the opposite direction, when the polarity of the bias voltage is reversed) can be realized by a tunneling junction between two different types of semiconducting materials. In such a<span>\u00a0<\/span><span data-type=\"term\" id=\"term324\">tunnel diode<\/span>, electrons tunnel through a single potential barrier at a contact between two different semiconductors. At the junction, tunneling-electron current changes nonlinearly with the applied potential difference across the junction and may rapidly decrease as the bias voltage is increased. This is unlike the Ohm\u2019s law behavior that we are familiar with in household circuits. This kind of rapid behavior (caused by quantum tunneling) is desirable in high-speed electronic devices.<\/p>\r\n<p id=\"fs-id1170901840342\">Another kind of electronic nano-device utilizes<span>\u00a0<\/span><span data-type=\"term\" id=\"term325\">resonant tunneling<\/span><span>\u00a0<\/span>of electrons through potential barriers that occur in quantum dots. A<span>\u00a0<\/span><span data-type=\"term\" id=\"term326\">quantum dot<\/span><span>\u00a0<\/span>is a small region of a semiconductor nanocrystal that is grown, for example, in a silicon or aluminum arsenide crystal.<span>\u00a0<\/span>Figure 3.23(a) shows a quantum dot of gallium arsenide embedded in an aluminum arsenide wafer. The quantum-dot region acts as a potential well of a finite height (shown in<span>\u00a0<\/span>Figure 3.23(b)) that has two finite-height potential barriers at dot boundaries. Similarly, as for a quantum particle in a box (that is, an infinite potential well), lower-lying energies of a quantum particle trapped in a finite-height potential well are quantized. The difference between the box and the well potentials is that a quantum particle in a box has an infinite number of quantized energies and is trapped in the box indefinitely, whereas a quantum particle trapped in a potential well has a finite number of quantized energy levels and can tunnel through potential barriers at well boundaries to the outside of the well. Thus, a quantum dot of gallium arsenide sitting in aluminum arsenide is a potential well where low-lying energies of an electron are quantized, indicated as<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-453-Frame\"><span class=\"MathJax_MathContainer\"><span>Edot<\/span><\/span><\/span>in part (b) in the figure. When the energy<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-454-Frame\"><span class=\"MathJax_MathContainer\"><span>Eelectron<\/span><\/span><\/span><span>\u00a0<\/span>of an electron in the outside region of the dot does not match its energy<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-455-Frame\"><span class=\"MathJax_MathContainer\"><span>Edot<\/span><\/span><\/span><span>\u00a0<\/span>that it would have in the dot, the electron does not tunnel through the region of the dot and there is no current through such a circuit element, even if it were kept at an electric voltage difference (bias). However, when this voltage bias is changed in such a way that one of the barriers is lowered, so that<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-456-Frame\"><span class=\"MathJax_MathContainer\"><span>Edot<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-457-Frame\"><span class=\"MathJax_MathContainer\"><span>Eelectron<\/span><\/span><\/span><span>\u00a0<\/span>become aligned, as seen in part (c) of the figure, an electron current flows through the dot. When the voltage bias is now increased, this alignment is lost and the current stops flowing. When the voltage bias is increased further, the electron tunneling becomes improbable until the bias voltage reaches a value for which the outside electron energy matches the next electron energy level in the dot. The word \u2018resonance\u2019 in the device name means that the tunneling-electron current occurs only when a selected energy level is matched by tuning an applied voltage bias, such as in the operation mechanism of the<span>\u00a0<\/span><span data-type=\"term\" id=\"term327\">resonant-tunneling diode<\/span><span>\u00a0<\/span>just described. Resonant-tunneling diodes are used as super-fast nano-switches.<\/p>\r\n\r\n<div class=\"os-figure\">\r\n<figure id=\"CNX_UPhysics_40_06_dot\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"859\"]<img alt=\"Figure a is an illustration of a tunneling diode. The quantum dot is a small region of gallium arsenide embedded in aluminum arsenide. Additional small regions of gallium arsenide are also embedded on either side of the quantum dot, separated from it by a small barrier of aluminum arsenide. The left end of the structure is attached to a negative electrode, and the right to a positive electrode. Figure b is a graph of the potential U as a function of x with no bias. The potential is constant except in two narrow regions where it has a larger constant value. The electron energy, represented by a dashed line, is between the lower and higher values of U, closer to the lower one. Two allowed energy levels, labeled as E sub dot, are shown. Both are higher than the electron energy and less than the maximum value of U. Figure c shows the potential U of x with a voltage bias across the device. The potential has the same constant value to the left of the barriers as in figure a, but decreases linearly between the barriers. U is constant again to the right of the barriers but at a lower value than before. The allowed energies are also pulled down, and the lower one now coincides with the energy of the electron.\" data-media-type=\"image\/jpeg\" id=\"1737\" src=\"https:\/\/cnx.org\/resources\/392116708b13f551148c46e4c676efba1e5138b1\" width=\"859\" height=\"289\" \/> Figure 3.23 Resonant-tunneling diode: (a) A quantum dot of gallium arsenide embedded in aluminum arsenide. (b) Potential well consisting of two potential barriers of a quantum dot with no voltage bias. Electron energies Eelectron in aluminum arsenide are not aligned with their energy levels Edot in the quantum dot, so electrons do not tunnel through the dot. (c) Potential well of the dot with a voltage bias across the device. A suitably tuned voltage difference distorts the well so that electron-energy levels in the dot are aligned with their energies in aluminum arsenide, causing the electrons to tunnel through the dot.[\/caption]<\/figure>\r\n<div>\r\n\r\n&nbsp;\r\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section>","rendered":"<div data-type=\"abstract\" id=\"19923\" class=\"ui-has-child-title\">\n<header>\n<div class=\"textbox textbox--learning-objectives\"><\/div>\n<\/header>\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Describe how a quantum particle may tunnel across a potential barrier<\/li>\n<li>Identify important physical parameters that affect the tunneling probability<\/li>\n<li>Identify the physical phenomena where quantum tunneling is observed<\/li>\n<li>Explain how quantum tunneling is utilized in modern technologies<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p><strong style=\"font-size: 14pt\" data-effect=\"bold\">Quantum tunneling<\/strong><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a phenomenon in which particles penetrate a potential energy barrier with a height greater than the total energy of the particles. The phenomenon is interesting and important because it violates the principles of classical mechanics. Quantum tunneling is important in models of the Sun and has a wide range of applications, such as the scanning tunneling microscope and the tunnel diode.<\/span><\/p>\n<section id=\"fs-id1170899453628\" data-depth=\"1\">\n<h3 data-type=\"title\">Tunneling and Potential Energy<\/h3>\n<p id=\"fs-id1170901605351\">To illustrate<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term314\">quantum tunneling<\/span>, consider a ball rolling along a surface with a kinetic energy of 100 J. As the ball rolls, it encounters a hill. The potential energy of the ball placed atop the hill is 10 J. Therefore, the ball (with 100 J of kinetic energy) easily rolls over the hill and continues on. In classical mechanics, the probability that the ball passes over the hill is exactly 1\u2014it makes it over every time. If, however, the height of the hill is increased\u2014a ball placed atop the hill has a potential energy of 200 J\u2014the ball proceeds only part of the way up the hill, stops, and returns in the direction it came. The total energy of the ball is converted entirely into potential energy before it can reach the top of the hill. We do not expect, even after repeated attempts, for the 100-J ball to ever be found beyond the hill. Therefore, the probability that the ball passes over the hill is exactly 0, and probability it is turned back or \u201creflected\u201d by the hill is exactly 1. The ball<span>\u00a0<\/span><em data-effect=\"italics\">never<\/em><span>\u00a0<\/span>makes it over the hill. The existence of the ball beyond the hill is an impossibility or \u201cenergetically forbidden.\u201d<\/p>\n<p id=\"fs-id1170901510354\">However, according to quantum mechanics, the ball has a wave function and this function is defined over all space. The wave function may be highly localized, but there is always a chance that as the ball encounters the hill, the ball will suddenly be found beyond it. Indeed, this probability is appreciable if the \u201cwave packet\u201d of the ball is wider than the barrier.<\/p>\n<div class=\"media-2 ui-has-child-title\" data-type=\"note\" id=\"fs-id1170902008280\">\n<header>\n<h3 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\">INTERACTIVE<\/span><\/h3>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1170902917781\">View this<span>\u00a0<\/span><a href=\"https:\/\/openstax.org\/l\/21intquatanvid\" rel=\"nofollow\">interactive simulation<\/a><span>\u00a0<\/span>for a simulation of tunneling.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1170901600117\">In the language of quantum mechanics, the hill is characterized by a<span>\u00a0<\/span><span data-type=\"term\" id=\"term315\">potential barrier<\/span>. A finite-height square barrier is described by the following potential-energy function:<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-324-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)={0,when x&lt;0U0,when 0\u2264x\u2264L0,when x&gt;L.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.59]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901928797\">The potential barrier is illustrated in<span>\u00a0<\/span>Figure 3.16. When the height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-325-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span><span>\u00a0<\/span>of the barrier is infinite, the wave packet representing an incident quantum particle is unable to penetrate it, and the quantum particle bounces back from the barrier boundary, just like a classical particle. When the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the barrier is infinite and its height is finite, a part of the wave packet representing an incident quantum particle can filter through the barrier boundary and eventually perish after traveling some distance inside the barrier.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_barrier\">\n<figure style=\"width: 516px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The potential U of x is plotted as a function of x. U is zero for x less than 0 and for x greater than L. It is equal to U sub 0 between x =0 and x=L. The constant energy E is indicated as a dotted horizontal line at a value less than U sub 0. The region x less than 0 is labeled as region I and has both incident and reflected waves, going to the right and left respectively. The region between x=0 and x=L is labeled as region II. The region x greater than L is labeled as region III and has only transmitted waves going to the right.\" data-media-type=\"image\/jpeg\" id=\"70394\" src=\"https:\/\/cnx.org\/resources\/f934a9164c237f284add8c0eccb2d0c9b03a8468\" width=\"516\" height=\"421\" \/><figcaption class=\"wp-caption-text\">Figure 3.16 A potential energy barrier of height U0 creates three physical regions with three different wave behaviors. In region I where x&lt;0, an incident wave packet (incident particle) moves in a potential-free zone and coexists with a reflected wave packet (reflected particle). In region II, a part of the incident wave that has not been reflected at x=0 moves as a transmitted wave in a constant potential U(x)=+U0 and tunnels through to region III at x=L. In region III for x&gt;L, a wave packet (transmitted particle) that has tunneled through the potential barrier moves as a free particle in potential-free zone. The energy E of the incident particle is indicated by the horizontal line.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170901769645\">When both the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>and the height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-332-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span><span>\u00a0<\/span>are finite, a part of the quantum wave packet incident on one side of the barrier can penetrate the barrier boundary and continue its motion inside the barrier, where it is gradually attenuated on its way to the other side. A part of the incident quantum wave packet eventually emerges on the other side of the barrier in the form of the transmitted wave packet that tunneled through the barrier. How much of the incident wave can tunnel through a barrier depends on the barrier width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>and its height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-333-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span>, and on the energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>of the quantum particle incident on the barrier. This is the physics of tunneling.<\/p>\n<p id=\"fs-id1170903049760\">Barrier penetration by quantum wave functions was first analyzed theoretically by Friedrich<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term316\">Hund<\/span><span>\u00a0<\/span>in 1927, shortly after Schr\u04e7dinger published the equation that bears his name. A year later, George<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term317\">Gamow<\/span><span>\u00a0<\/span>used the formalism of quantum mechanics to explain the radioactive<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-334-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-decay of atomic nuclei as a quantum-tunneling phenomenon. The invention of the tunnel diode in 1957 made it clear that quantum tunneling is important to the semiconductor industry. In modern nanotechnologies, individual atoms are manipulated using a knowledge of quantum tunneling.<\/p>\n<\/section>\n<section id=\"fs-id1170902863895\" data-depth=\"1\">\n<h3 data-type=\"title\">Tunneling and the Wave Function<\/h3>\n<p id=\"fs-id1170901904236\">Suppose a uniform and time-independent beam of electrons or other quantum particles with energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>traveling along the<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em>-axis (in the positive direction to the right) encounters a potential barrier described by<span>\u00a0<\/span>Equation 3.59. The question is: What is the probability that an individual particle in the beam will tunnel through the potential barrier? The answer can be found by solving the boundary-value problem for the time-independent Schr\u04e7dinger equation for a particle in the beam. The general form of this equation is given by<span>\u00a0<\/span>Equation 3.60, which we reproduce here:<\/p>\n<div data-type=\"equation\" id=\"fs-id1170901793747\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-335-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-335-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8(x)dx2+U(x)\u03c8(x)=E\u03c8(x),where\u2212\u221e&lt;x&lt;+\u221e.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.60]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">In<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.60, the potential function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">U<\/em><span style=\"font-size: 14pt\">(<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">) is defined by<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.59. We assume that the given energy<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">of the incoming particle is smaller than the height<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-336-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U0<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">of the potential barrier,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-337-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">E&lt;U0<\/span><\/span><span style=\"font-size: 14pt\">, because this is the interesting physical case. Knowing the energy<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">E<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">of the incoming particle, our task is to solve<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.60<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">for a function<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-338-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8(x)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">that is continuous and has continuous first derivatives for all<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">x<\/em><span style=\"font-size: 14pt\">. In other words, we are looking for a \u201csmooth-looking\u201d solution (because this is how wave functions look) that can be given a probabilistic interpretation so that<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-339-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">|\u03c8(x)|2=\u03c8*(x)\u03c8(x)<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is the probability density.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170901916178\">We divide the real axis into three regions with the boundaries defined by the potential function in<span>\u00a0<\/span>Equation 3.59<span>\u00a0<\/span>(illustrated in<span>\u00a0<\/span>Figure 3.16) and transcribe<span>\u00a0<\/span>Equation 3.60<span>\u00a0<\/span>for each region. Denoting by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-340-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8I(x)<\/span><\/span><\/span><span>\u00a0<\/span>the solution in region I for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-341-Frame\"><span class=\"MathJax_MathContainer\"><span>x&lt;0<\/span><\/span><\/span>, by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-342-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8II(x)<\/span><\/span><\/span><span>\u00a0<\/span>the solution in region II for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-343-Frame\"><span class=\"MathJax_MathContainer\"><span>0\u2264x\u2264L<\/span><\/span><\/span>, and by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-344-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8III(x)<\/span><\/span><\/span><span>\u00a0<\/span>the solution in region III for<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-345-Frame\"><span class=\"MathJax_MathContainer\"><span>x&gt;L<\/span><\/span><\/span>, the stationary Schr\u04e7dinger equation has the following forms in these three regions:<\/p>\n<div data-type=\"equation\" id=\"fs-id1170903064710\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-346-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-346-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8I(x)dx2=E\u03c8I(x),in region I:\u2212\u221e&lt;x&lt;0,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.61]<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901948077\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-347-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-347-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8II(x)dx2+U0\u03c8II(x)=E\u03c8II(x),in region II:0\u2264x\u2264L,<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.62]<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170903070603\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-348-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-348-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212\u210f22md2\u03c8III(x)dx2=E\u03c8III(x),in region III:L&lt;x&lt;+\u221e.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.63]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The continuity condition at region boundaries requires that:<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901741204\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-349-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-349-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8I(0)=\u03c8II(0),at the boundary between regions I and II and<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.64]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">and<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170899245855\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-350-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-350-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8II(L)=\u03c8III(L),at the boundary between regions II and III.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.65]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The \u201csmoothness\u201d condition requires the first derivative of the solution be continuous at region boundaries:<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901533726\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-351-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-351-Frame\"><span class=\"MathJax_MathContainer\"><span>d\u03c8I(x)dx|x=0=d\u03c8II(x)dx|x=0,at the boundary between regions I and II;<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.66]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">and<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902790371\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-352-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-352-Frame\"><span class=\"MathJax_MathContainer\"><span>d\u03c8II(x)dx|x=L=d\u03c8III(x)dx|x=L,at the boundary between regions II and III.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.67]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">In what follows, we find the functions<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-353-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8I(x)<\/span><\/span><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-354-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8II(x)<\/span><\/span><span style=\"font-size: 14pt\">, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-355-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8III(x)<\/span><\/span><span style=\"font-size: 14pt\">.<\/span><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1170901980803\">We can easily verify (by substituting into the original equation and differentiating) that in regions I and III, the solutions must be in the following general forms:<\/p>\n<div data-type=\"equation\" id=\"fs-id1170903010927\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-356-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-356-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8I(x)=Ae+ikx+Be\u2212ikx<\/span><\/span><\/div>\n<div class=\"os-equation-number\">\n<p><span class=\"os-number\">[3.68]<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"textbox\">\n<div data-type=\"equation\" id=\"fs-id1170903010927\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-356-Frame\">\n<p><span style=\"font-size: 14pt\">\u03c8III(x)=Fe+ikx+Ge\u2212ikx<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170903043787\">\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.69]<\/span><\/div>\n<\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-358-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">k=2mE\/\u210f<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a wave number and the complex exponent denotes oscillations,<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901539686\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-359-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-359-Frame\"><span class=\"MathJax_MathContainer\"><span>e\u00b1ikx=coskx\u00b1isinkx.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.70]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The constants<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">B<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">G<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.68<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.69<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">may be complex. These solutions are illustrated in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.16. In region I, there are two waves\u2014one is incident (moving to the right) and one is reflected (moving to the left)\u2014so none of the constants<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A\u00a0<\/em><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">B<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.68<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">may vanish. In region III, there is only one wave (moving to the right), which is the transmitted wave, so the constant<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">G<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">must be zero in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.69,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-360-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">G=0<\/span><\/span><span style=\"font-size: 14pt\">. We can write explicitly that the incident wave is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-361-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8in(x)=Ae+ikx<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and that the reflected wave is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-362-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8ref(x)=Be\u2212ikx<\/span><\/span><span style=\"font-size: 14pt\">, and that the transmitted wave is<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-363-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8tra(x)=Fe+ikx<\/span><\/span><span style=\"font-size: 14pt\">. The amplitude of the incident wave is<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903084073\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-364-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8in(x)|2=\u03c8in*(x)\u03c8in(x)=(Ae+ikx)*Ae+ikx=A*e\u2212ikxAe+ikx=A*A=|A|2.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901634975\">Similarly, the amplitude of the reflected wave is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-365-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8ref(x)|2=|B|2<\/span><\/span><\/span><span>\u00a0<\/span>and the amplitude of the transmitted wave is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-366-Frame\"><span class=\"MathJax_MathContainer\"><span>|\u03c8tra(x)|2=|F|2<\/span><\/span><\/span>. We know from the theory of waves that the square of the wave amplitude is directly proportional to the wave intensity. If we want to know how much of the incident wave tunnels through the barrier, we need to compute the square of the amplitude of the transmitted wave. The<span>\u00a0<\/span><span data-type=\"term\" id=\"term318\">transmission probability<\/span><span>\u00a0<\/span>or<span>\u00a0<\/span><span data-type=\"term\" id=\"term319\">tunneling probability<\/span><span>\u00a0<\/span>is the ratio of the transmitted intensity<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-367-Frame\"><span class=\"MathJax_MathContainer\"><span>(|F|2)<\/span><\/span><\/span><span>\u00a0<\/span>to the incident intensity<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-368-Frame\"><span class=\"MathJax_MathContainer\"><span>(|A|2)<\/span><\/span><\/span>, written as<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-369-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=|\u03c8tra(x)|2|\u03c8in(x)|2=|F|2|A|2=|FA|2<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.71]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170903037273\">where<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>is the width of the barrier and<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>is the total energy of the particle. This is the probability an individual particle in the incident beam will tunnel through the potential barrier. Intuitively, we understand that this probability must depend on the barrier height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-370-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span>.<\/p>\n<p id=\"fs-id1170903111657\">In region II, the terms in equation<span>\u00a0<\/span>Equation 3.62<span>\u00a0<\/span>can be rearranged to<\/p>\n<div data-type=\"equation\" id=\"fs-id1170901903999\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-371-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-371-Frame\"><span class=\"MathJax_MathContainer\"><span>d2\u03c8II(x)dx2=\u03b22\u03c8II(x)<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.72]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">where<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-372-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b22<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is positive because<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-373-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">U0&gt;E<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and the parameter<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-374-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b2<\/span><\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is a real number,<\/span><\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-375-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b22=2m\u210f2(U0\u2212E).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.73]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The general solution to<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.72<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is not oscillatory (unlike in the other regions) and is in the form of exponentials that describe a gradual attenuation of<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-376-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03c8II(x)<\/span><\/span><span style=\"font-size: 14pt\">,<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902869487\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-377-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-377-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03c8II(x)=Ce\u2212\u03b2x+De+\u03b2x.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.74]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">The two types of solutions in the three regions are illustrated in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Figure 3.17.<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_tunneling\">\n<figure style=\"width: 516px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"A solution to the barrier potential U of x is plotted as a function of x. U is zero for x less than 0 and for x greater than L. It is equal to U sub 0 between x =0 and x=L. The wave function oscillates in the region x less than zero. The wave function is labeled psi sub I in this region. It decays exponentially in the region between x=0 and x=L, and is labeled psi sub I I in this region. It oscillates again in the x greater than L region, where it is labeled psi sub I I I. The amplitude of the oscillations is smaller in region I I I than in region I but the wavelength is the same. The wave function and its derivative are continuous at x=0 and x=L.\" data-media-type=\"image\/jpeg\" id=\"35274\" src=\"https:\/\/cnx.org\/resources\/b95198c890dbf4f2538a07307bb26ad959617f7f\" width=\"516\" height=\"419\" \/><figcaption class=\"wp-caption-text\">Figure 3.17 Three types of solutions to the stationary Schr\u04e7dinger equation for the quantum-tunneling problem: Oscillatory behavior in regions I and III where a quantum particle moves freely, and exponential-decay behavior in region II (the barrier region) where the particle moves in the potential U0.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170901711932\">Now we use the boundary conditions to find equations for the unknown constants.<span>\u00a0<\/span>Equation 3.68<span>\u00a0<\/span>and<span>\u00a0<\/span>Equation 3.74<span>\u00a0<\/span>are substituted into<span>\u00a0<\/span>Equation 3.64<span>\u00a0<\/span>to give<\/p>\n<div data-type=\"equation\" id=\"fs-id1170901672030\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-379-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-379-Frame\"><span class=\"MathJax_MathContainer\"><span>A+B=C+D.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.75]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Equation 3.74<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.69<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">are substituted into<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.65<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">to give<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902011772\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-380-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-380-Frame\"><span class=\"MathJax_MathContainer\"><span>Ce\u2212\u03b2L+De+\u03b2L=Fe+ikL.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.76]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Similarly, we substitute<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.68<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.74<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">into<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.66, differentiate, and obtain<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170901583450\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-381-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-381-Frame\"><span class=\"MathJax_MathContainer\"><span>\u2212ik(A\u2212B)=\u03b2(D\u2212C).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.77]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">Similarly, the boundary condition<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.67<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">reads explicitly<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170903074449\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-382-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-382-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b2(De+\u03b2L\u2212Ce\u2212\u03b2L)=\u2212ikFe+ikL.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.78]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">We now have four equations for five unknown constants. However, because the quantity we are after is the transmission coefficient, defined in<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.71<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">by the fraction<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, the number of equations is exactly right because when we divide each of the above equations by<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, we end up having only four unknown fractions:<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">B<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">C<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">,<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">D<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, and<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">, three of which can be eliminated to find<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">. The actual algebra that leads to expression for<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">F<\/em><span style=\"font-size: 14pt\">\/<\/span><em style=\"font-size: 14pt\" data-effect=\"italics\">A<\/em><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">is pretty lengthy, but it can be done either by hand or with a help of computer software. The end result is<\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"equation\" id=\"fs-id1170902951016\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-383-Frame\">\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-383-Frame\"><span class=\"MathJax_MathContainer\"><span>FA=e\u2212ikLcosh(\u03b2L)+i(\u03b3\/2)sinh(\u03b2L).<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.79]<\/span><\/div>\n<\/div>\n<p><span style=\"font-size: 14pt\">In deriving<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span style=\"font-size: 14pt\">Equation 3.79, to avoid the clutter, we use the substitutions<\/span><span style=\"font-size: 14pt\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-384-Frame\" style=\"font-size: 14pt\"><span class=\"MathJax_MathContainer\">\u03b3\u2261\u03b2\/k\u2212k\/\u03b2<\/span><\/span><span style=\"font-size: 14pt\">,<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902707922\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-385-Frame\"><span class=\"MathJax_MathContainer\"><span>coshy=ey+e\u2212y2,andsinhy=ey\u2212e\u2212y2.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902738264\">We substitute<span>\u00a0<\/span>Equation 3.79<span>\u00a0<\/span>into<span>\u00a0<\/span>Equation 3.71<span>\u00a0<\/span>and obtain the exact expression for the transmission coefficient for the barrier,<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903079519\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-386-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=(FA)*FA=e+ikLcosh(\u03b2L)\u2212i(\u03b3\/2)sinh(\u03b2L)\u00b7e\u2212ikLcosh(\u03b2L)+i(\u03b3\/2)sinh(\u03b2L)<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901533679\">or<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-387-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=1cosh2(\u03b2L)+(\u03b3\/2)2sinh2(\u03b2L)<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.80]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901609283\">where<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901492355\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-388-Frame\"><span class=\"MathJax_MathContainer\"><span>(\u03b32)2=14(1\u2212E\/U0E\/U0+E\/U01\u2212E\/U0\u22122).<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170899453560\">For a wide and high barrier that transmits poorly,<span>\u00a0<\/span>Equation 3.80<span>\u00a0<\/span>can be approximated by<\/p>\n<div class=\"textbox\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-389-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L,E)=16EU0(1\u2212EU0)e\u22122\u03b2L.<\/span><\/span><\/div>\n<div class=\"os-equation-number\"><span class=\"os-number\">[3.81]<\/span><\/div>\n<\/div>\n<p id=\"fs-id1170901525408\">Whether it is the exact expression<span>\u00a0<\/span>Equation 3.80<span>\u00a0<\/span>or the approximate expression<span>\u00a0<\/span>Equation 3.81, we see that the tunneling effect very strongly depends on the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the potential barrier. In the laboratory, we can adjust both the potential height<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-390-Frame\"><span class=\"MathJax_MathContainer\"><span>U0<\/span><\/span><\/span><span>\u00a0<\/span>and the width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>to design nano-devices with desirable transmission coefficients.<\/p>\n<div data-type=\"example\" id=\"fs-id1170902720394\" class=\"ui-has-child-title\">\n<header><\/header>\n<section>\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\"><span class=\"os-title-label\">EXAMPLE<span>\u00a03<\/span><\/span><span class=\"os-number\">.12<\/span><span class=\"os-divider\"><\/span><\/h3>\n<\/header>\n<section>\n<p id=\"fs-id1170903140982\"><span data-type=\"title\"><strong>Transmission Coefficient<\/strong><\/span><\/p>\n<p>Two copper nanowires are insulated by a copper oxide nano-layer that provides a 10.0-eV potential barrier. Estimate the tunneling probability between the nanowires by 7.00-eV electrons through a 5.00-nm thick oxide layer. What if the thickness of the layer were reduced to just 1.00 nm? What if the energy of electrons were increased to 9.00 eV?<\/p>\n<p><span data-type=\"title\"><strong>Strategy<\/strong><\/span><\/p>\n<p>Treating the insulating oxide layer as a finite-height potential barrier, we use<span>\u00a0<\/span>Equation 3.81. We identify<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-391-Frame\"><span class=\"MathJax_MathContainer\"><span>U0=10.0eV<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-392-Frame\"><span class=\"MathJax_MathContainer\"><span>E1=7.00eV<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-393-Frame\"><span class=\"MathJax_MathContainer\"><span>E2=9.00eV<\/span><\/span><\/span>,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-394-Frame\"><span class=\"MathJax_MathContainer\"><span>L1=5.00nm<\/span><\/span><\/span>, and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-395-Frame\"><span class=\"MathJax_MathContainer\"><span>L2=1.00nm<\/span><\/span><\/span>. We use<span>\u00a0<\/span>Equation 3.73<span>\u00a0<\/span>to compute the exponent. Also, we need the rest mass of the electron<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-396-Frame\"><span class=\"MathJax_MathContainer\"><span>m=511keV\/c2<\/span><\/span><\/span><span>\u00a0<\/span>and Planck\u2019s constant<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-397-Frame\"><span class=\"MathJax_MathContainer\"><span>\u210f=0.1973keV\u00b7nm\/c<\/span><\/span><\/span>. It is typical for this type of estimate to deal with very small quantities that are often not suitable for handheld calculators. To make correct estimates of orders, we make the conversion<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-398-Frame\"><span class=\"MathJax_MathContainer\"><span>ey=10y\/ln10<\/span><\/span><\/span>.<\/p>\n<p><span data-type=\"title\"><strong>Solution<\/strong><\/span><\/p>\n<p>Constants:<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901744088\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-399-Frame\"><span class=\"MathJax_MathContainer\"><span>2m\u210f2=2(511keV\/c2)(0.1973keV\u00b7nm\/c)2=26,2541keV\u00b7(nm)2,<\/span><\/span><\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902597918\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-400-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>\u03b2=2m\u210f2(U0\u2212E)=26,254(10.0eV\u2212E)keV\u00b7(nm)2=26.254(10.0eV\u2212E)\/eV1nm.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">For a lower-energy electron with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-401-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">E1=7.00eV<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901492418\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-402-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b21=26.254(10.00eV\u2212E1)\/eV1nm=26.254(10.00\u22127.00)1nm=8.875nm,<\/span><\/span><\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901510689\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-403-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>T(L,E1)=16E1U0(1\u2212E1U0)e\u22122\u03b21L=16710(1\u2212710)e\u221217.75L\/nm=3.36e\u221217.75L\/nm.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">For a higher-energy electron with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-404-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">E2=9.00eV<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902598093\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-405-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b22=26.254(10.00eV\u2212E2)\/eV1nm=26.254(10.00\u22129.00)1nm=5.124nm,<\/span><\/span><\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902795334\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-406-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>T(L,E2)=16E2U0(1\u2212E2U0)e\u22122\u03b22L=16910(1\u2212910)e\u22125.12L\/nm=1.44e\u22125.12L\/nm.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">For a broad barrier with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-407-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L1=5.00nm<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170899293709\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-408-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L1,E1)=3.36e\u221217.75L1\/nm=3.36e\u221217.75\u00b75.00nm\/nm=3.36e\u221288=3.36(6.2\u00d710\u221239)=2.1%\u00d710\u221236,<\/span><\/span><\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170903032292\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-409-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>T(L1,E2)=1.44e\u22125.12L1\/nm=1.44e\u22125.12\u00b75.00nm\/nm=1.44e\u221225.6=1.44(7.62\u00d710\u221212)=1.1%\u00d710\u22129.<\/span><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">For a narrower barrier with<\/span><span style=\"text-indent: 1em;font-size: 1rem\">\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-410-Frame\" style=\"text-indent: 1em;font-size: 1rem\"><span class=\"MathJax_MathContainer\">L2=1.00nm<\/span><\/span><span style=\"text-indent: 1em;font-size: 1rem\">:<\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901924352\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-411-Frame\"><span class=\"MathJax_MathContainer\"><span>T(L2,E1)=3.36e\u221217.75L2\/nm=3.36e\u221217.75\u00b71.00nm\/nm=3.36e\u221217.75=3.36(5.1\u00d710\u22127)=1.7%\u00d710\u22124,<\/span><\/span><\/div>\n<\/div>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170901478827\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-412-Frame\">\n<p><span class=\"MathJax_MathContainer\"><span>T(L2,E2)=1.44e\u22125.12L2\/nm=1.44e\u22125.12\u00b71.00nm\/nm=1.44e\u22125.12=1.44(5.98\u00d710\u22123)=0.86%.<\/span><\/span><\/p>\n<p><span data-type=\"title\" style=\"text-indent: 1em;font-size: 1rem\"><strong>Significance<\/strong><\/span><\/p>\n<p><span style=\"text-indent: 1em;font-size: 1rem\">We see from these estimates that the probability of tunneling is affected more by the width of the potential barrier than by the energy of an incident particle. In today\u2019s technologies, we can manipulate individual atoms on metal surfaces to create potential barriers that are fractions of a nanometer, giving rise to measurable tunneling currents. One of many applications of this technology is the scanning tunneling microscope (STM), which we discuss later in this section.<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1170901541920\" data-depth=\"1\">\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\"><span class=\"os-title-label\">CHECK YOUR UNDERSTANDING<span>\u00a03<\/span><\/span><span class=\"os-number\">.10<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<header><span style=\"font-size: 1rem\">A proton with kinetic energy 1.00 eV is incident on a square potential barrier with height 10.00 eV. If the proton is to have the same transmission probability as an electron of the same energy, what must the width of the barrier be relative to the barrier width encountered by an electron?<\/span><\/header>\n<\/div>\n<\/div>\n<h3 data-type=\"title\">Radioactive Decay<\/h3>\n<p id=\"fs-id1170901541925\">In 1928, Gamow identified quantum tunneling as the mechanism responsible for the<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term320\">radioactive decay<\/span><span>\u00a0<\/span>of atomic nuclei. He observed that some isotopes of thorium, uranium, and bismuth disintegrate by emitting<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-413-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles (which are doubly ionized helium atoms or, simply speaking, helium nuclei). In the process of emitting an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-414-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle, the original nucleus is transformed into a new nucleus that has two fewer neutrons and two fewer protons than the original nucleus. The<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-415-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles emitted by one isotope have approximately the same kinetic energies. When we look at variations of these energies among isotopes of various elements, the lowest kinetic energy is about 4 MeV and the highest is about 9 MeV, so these energies are of the same order of magnitude. This is about where the similarities between various isotopes end.<\/p>\n<p id=\"fs-id1170899266027\">When we inspect half-lives (a half-life is the time in which a radioactive sample loses half of its nuclei due to decay), different isotopes differ widely. For example, the half-life of polonium-214 is<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-416-Frame\"><span class=\"MathJax_MathContainer\"><span>160\u00b5s<\/span><\/span><\/span><span>\u00a0<\/span>and the half-life of uranium is 4.5 billion years. Gamow explained this variation by considering a \u2018spherical-box\u2019 model of the nucleus, where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-417-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles can bounce back and forth between the walls as free particles. The confinement is provided by a strong nuclear potential at a spherical wall of the box. The thickness of this wall, however, is not infinite but finite, so in principle, a nuclear particle has a chance to escape this nuclear confinement. On the inside wall of the confining barrier is a high nuclear potential that keeps the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-418-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle in a small confinement. But when an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-419-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle gets out to the other side of this wall, it is subject to electrostatic Coulomb repulsion and moves away from the nucleus. This idea is illustrated in<span>\u00a0<\/span>Figure 3.18. The width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the potential barrier that separates an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-420-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle from the outside world depends on the particle\u2019s kinetic energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em>. This width is the distance between the point marked by the nuclear radius<span>\u00a0<\/span><em data-effect=\"italics\">R<\/em><span>\u00a0<\/span>and the point<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-421-Frame\"><span class=\"MathJax_MathContainer\"><span>R0<\/span><\/span><\/span><span>\u00a0<\/span>where an<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-422-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle emerges on the other side of the barrier,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-423-Frame\"><span class=\"MathJax_MathContainer\"><span>L=R0\u2212R<\/span><\/span><\/span>. At the distance<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-424-Frame\"><span class=\"MathJax_MathContainer\"><span>R0<\/span><\/span><\/span>, its kinetic energy must at least match the electrostatic energy of repulsion,<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-425-Frame\"><span class=\"MathJax_MathContainer\"><span>E=(4\u03c0\u03b50)\u22121Ze2\/R0<\/span><\/span><\/span><span>\u00a0<\/span>(where<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-426-Frame\"><span class=\"MathJax_MathContainer\"><span>+Ze<\/span><\/span><\/span><span>\u00a0<\/span>is the charge of the nucleus). In this way we can estimate the width of the nuclear barrier,<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170902793216\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-427-Frame\"><span class=\"MathJax_MathContainer\"><span>L=e24\u03c0\u03b50ZE\u2212R.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170899368006\">We see from this estimate that the higher the energy of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-428-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle, the narrower the width of the barrier that it is to tunnel through. We also know that the width of the potential barrier is the most important parameter in tunneling probability. Thus, highly energetic<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-429-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles have a good chance to escape the nucleus, and, for such nuclei, the nuclear disintegration half-life is short. Notice that this process is highly nonlinear, meaning a small increase in the<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-430-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particle energy has a disproportionately large enhancing effect on the tunneling probability and, consequently, on shortening the half-life. This explains why the half-life of polonium that emits 8-MeV<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-431-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles is only hundreds of milliseconds and the half-life of uranium that emits 4-MeV<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-432-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-particles is billions of years.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_nucleus\">\n<figure style=\"width: 373px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The potential U of r is plotted as a function of r. For r less than R, U of r is constant and negative. At r = R, the potential rises vertically to some maximum positive value, then decays toward zero. The area under the curve is shaded. U of r equals E at r equal to R sub 0. A horizontal dashed line at E=E and a vertical dashed line at r=R sub 0 are shown.\" data-media-type=\"image\/jpeg\" id=\"47863\" src=\"https:\/\/cnx.org\/resources\/1d92a84dbfb6f34dbf2e0ed5e4509f2e24d70726\" width=\"373\" height=\"642\" \/><figcaption class=\"wp-caption-text\">Figure 3.18 The potential energy barrier for an \u03b1-particle bound in the nucleus: To escape from the nucleus, an \u03b1-particle with energy Emust tunnel across the barrier from distance R to distance R0 away from the center.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<\/section>\n<section id=\"fs-id1170899265382\" data-depth=\"1\">\n<h3 data-type=\"title\">Field Emission<\/h3>\n<p id=\"fs-id1170899265387\"><strong data-effect=\"bold\">Field emission<\/strong><span>\u00a0<\/span>is a process of emitting electrons from conducting surfaces due to a strong external electric field that is applied in the direction normal to the surface (Figure 3.19). As we know from our study of electric fields in earlier chapters, an applied external electric field causes the electrons in a conductor to move to its surface and stay there as long as the present external field is not excessively strong. In this situation, we have a constant electric potential throughout the inside of the conductor, including its surface. In the language of potential energy, we say that an electron inside the conductor has a constant potential energy<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-436-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=\u2212U0<\/span><\/span><\/span><span>\u00a0<\/span>(here, the<span>\u00a0<\/span><em data-effect=\"italics\">x\u00a0<\/em>means inside the conductor). In the situation represented in<span>\u00a0<\/span>Figure 3.19, where the external electric field is uniform and has magnitude<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-437-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span>, if an electron happens to be outside the conductor at a distance<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em><span>\u00a0<\/span>away from its surface, its potential energy would have to be<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-438-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=\u2212eEgx<\/span><\/span><\/span><span>\u00a0<\/span>(here,<span>\u00a0<\/span><em data-effect=\"italics\">x<\/em><span>\u00a0<\/span>denotes distance to the surface). Taking the origin at the surface, so that<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-439-Frame\"><span class=\"MathJax_MathContainer\"><span>x=0<\/span><\/span><\/span><span>\u00a0<\/span>is the location of the surface, we can represent the potential energy of conduction electrons in a metal as the potential energy barrier shown in<span>\u00a0<\/span>Figure 3.20. In the absence of the external field, the potential energy becomes a step barrier defined by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-440-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x\u22640)=\u2212U0<\/span><\/span><\/span><span>\u00a0<\/span>and by<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-441-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x&gt;0)=0<\/span><\/span><\/span>.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_field\">\n<figure style=\"width: 450px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"The potential U of r is plotted as a function of r. For r less than R, U of r is constant and negative. At r = R, the potential rises vertically to some maximum positive value, then decays toward zero. The area under the curve is shaded. U of r equals E at r equal to R sub 0. A horizontal dashed line at E=E and a vertical dashed line at r=R sub 0 are shown.\" data-media-type=\"image\/jpeg\" id=\"49581\" src=\"https:\/\/cnx.org\/resources\/333cb466c65c1ffc2ea59a313a3b742eb648a6f3\" width=\"450\" height=\"512\" \/><figcaption class=\"wp-caption-text\">Figure 3.19 A normal-direction external electric field at the surface of a conductor: In a strong field, the electrons on a conducting surface may get detached from it and accelerate against the external electric field away from the surface.<\/figcaption><\/figure>\n<\/figure>\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\"><\/span><span class=\"os-caption\"><br \/>\n<\/span><\/em><\/div>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_metal\">\n<figure style=\"width: 672px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"U of x is plotted as a function of x. For x less than zero, U of x has a constant value of minus U sub zero. At x=0, U of x jumps to a value of zero. For x larger than zero, U of x equals minus e times E sub g times x. The area under the curve is shaded. The energy is a negative constant, shown as a dashed line, at a value of minus phi. U of x equals E at x equal phi divided by the quantity e times E sub g.\" data-media-type=\"image\/jpeg\" id=\"71000\" src=\"https:\/\/cnx.org\/resources\/0cfe7bcb891cef9a31464af3f1002afa56bda2a1\" width=\"672\" height=\"495\" \/><figcaption class=\"wp-caption-text\">Figure 3.20 The potential energy barrier at the surface of a metallic conductor in the presence of an external uniform electric field Eg normal to the surface: It becomes a step-function barrier when the external field is removed. The work function of the metal is indicated by \u03d5.<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<p id=\"fs-id1170901704550\">When an external electric field is strong, conduction electrons at the surface may get detached from it and accelerate along electric field lines in a direction antiparallel to the external field, away from the surface. In short, conduction electrons may escape from the surface. The<span>\u00a0<\/span><span class=\"no-emphasis\" data-type=\"term\" id=\"term321\">field emission<\/span><span>\u00a0<\/span>can be understood as the quantum tunneling of conduction electrons through the potential barrier at the conductor\u2019s surface. The physical principle at work here is very similar to the mechanism of<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-444-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03b1<\/span><\/span><\/span>-emission from a radioactive nucleus.<\/p>\n<p id=\"fs-id1170899245185\">Suppose a conduction electron has a kinetic energy<span>\u00a0<\/span><em data-effect=\"italics\">E<\/em><span>\u00a0<\/span>(the average kinetic energy of an electron in a metal is the work function<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-445-Frame\"><span class=\"MathJax_MathContainer\"><span>\u03d5<\/span><\/span><\/span><span>\u00a0<\/span>for the metal and can be measured, as discussed for the photoelectric effect in<span>\u00a0<\/span><a class=\"target-chapter\" href=\"https:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1:fdd4b413-6910-44d5-801d-0f4223bc7a31@5\" data-page=\"48\">Photons and Matter Waves<\/a>), and an external electric field can be locally approximated by a uniform electric field of strength<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-446-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span>. The width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>of the potential barrier that the electron must cross is the distance from the conductor\u2019s surface to the point outside the surface where its kinetic energy matches the value of its potential energy in the external field. In<span>\u00a0<\/span>Figure 3.20, this distance is measured along the dashed horizontal line<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-447-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=E<\/span><\/span><\/span><span>\u00a0<\/span>from<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-448-Frame\"><span class=\"MathJax_MathContainer\"><span>x=0<\/span><\/span><\/span><span>\u00a0<\/span>to the intercept with<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-449-Frame\"><span class=\"MathJax_MathContainer\"><span>U(x)=\u2212eEgx<\/span><\/span><\/span>, so the barrier width is<\/p>\n<div class=\"unnumbered\" data-label=\"\" data-type=\"equation\" id=\"fs-id1170899266044\">\n<div class=\"MathJax_MathML\" id=\"MathJax-Element-450-Frame\"><span class=\"MathJax_MathContainer\"><span>L=e\u22121EEg=e\u22121\u03d5Eg.<\/span><\/span><\/div>\n<\/div>\n<p id=\"fs-id1170902851400\">We see that<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>is inversely proportional to the strength<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-451-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span><span>\u00a0<\/span>of an external field. When we increase the strength of the external field, the potential barrier outside the conductor becomes steeper and its width decreases for an electron with a given kinetic energy. In turn, the probability that an electron will tunnel across the barrier (conductor surface) becomes exponentially larger. The electrons that emerge on the other side of this barrier form a current (tunneling-electron current) that can be detected above the surface. The tunneling-electron current is proportional to the tunneling probability. The tunneling probability depends nonlinearly on the barrier width<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em>, and<span>\u00a0<\/span><em data-effect=\"italics\">L\u00a0<\/em>can be changed by adjusting<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-452-Frame\"><span class=\"MathJax_MathContainer\"><span>Eg<\/span><\/span><\/span>. Therefore, the tunneling-electron current can be tuned by adjusting the strength of an external electric field at the surface. When the strength of an external electric field is constant, the tunneling-electron current has different values at different elevations<span>\u00a0<\/span><em data-effect=\"italics\">L<\/em><span>\u00a0<\/span>above the surface.<\/p>\n<p id=\"fs-id1170901616105\">The quantum tunneling phenomenon at metallic surfaces, which we have just described, is the physical principle behind the operation of the<span>\u00a0<\/span><span data-type=\"term\" id=\"term322\">scanning tunneling microscope (STM)<\/span>, invented in 1981 by Gerd Binnig and Heinrich Rohrer. The STM device consists of a scanning tip (a needle, usually made of tungsten, platinum-iridium, or gold); a piezoelectric device that controls the tip\u2019s elevation in a typical range of 0.4 to 0.7 nm above the surface to be scanned; some device that controls the motion of the tip along the surface; and a computer to display images. While the sample is kept at a suitable voltage bias, the scanning tip moves along the surface (Figure 3.21), and the tunneling-electron current between the tip and the surface is registered at each position. The amount of the current depends on the probability of electron tunneling from the surface to the tip, which, in turn, depends on the elevation of the tip above the surface. Hence, at each tip position, the distance from the tip to the surface is measured by measuring how many electrons tunnel out from the surface to the tip. This method can give an unprecedented resolution of about 0.001 nm, which is about 1% of the average diameter of an atom. In this way, we can see individual atoms on the surface, as in the image of a carbon nanotube in<span>\u00a0<\/span>Figure 3.22.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_stm\">\n<figure style=\"width: 693px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"An illustration of a scanning tunneling microscope. The atoms in the tip and sample are represented by spheres, orange for the S T M tip and purple for the sample. The atoms on the surface of the atoms being scanned are arranged in this illustration in a grid of four atoms by five atoms. The tip is above one of the atoms, and a tunneling electron current is shown between the tip and the surface atom. The image on the computer monitor is a 4 by 5 grid of spots.\" data-media-type=\"image\/jpeg\" id=\"74886\" src=\"https:\/\/cnx.org\/resources\/e419d79bf6204b92dfe14e6b97f581a05c20ff00\" width=\"693\" height=\"509\" \/><figcaption class=\"wp-caption-text\">Figure 3.21 In STM, a surface at a constant potential is being scanned by a narrow tip moving along the surface. When the STM tip moves close to surface atoms, electrons can tunnel from the surface to the tip. This tunneling-electron current is continually monitored while the tip is in motion. The amount of current at location (x,y) gives information about the elevation of the tip above the surface at this location. In this way, a detailed topographical map of the surface is created and displayed on a computer monitor.<\/figcaption><\/figure>\n<\/figure>\n<div class=\"os-caption-container\"><em><span class=\"os-title-label\"><\/span><span class=\"os-caption\"><br \/>\n<\/span><\/em><\/div>\n<\/div>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_nanotube\">\n<figure style=\"width: 488px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"An STM image of a carbon nanotube showing the atoms as red points in a grid like pattern.\" data-media-type=\"image\/jpeg\" id=\"57721\" src=\"https:\/\/cnx.org\/resources\/57753c12751920a2283483c54c90d7109d3fe614\" width=\"488\" height=\"156\" \/><figcaption class=\"wp-caption-text\">Figure 3.22 An STM image of a carbon nanotube: Atomic-scale resolution allows us to see individual atoms on the surface. STM images are in gray scale, and coloring is added to bring up details to the human eye. (credit: Taner Yildirim, NIST)<\/figcaption><\/figure>\n<\/figure>\n<\/div>\n<\/section>\n<section id=\"fs-id1170903136127\" data-depth=\"1\">\n<h3 data-type=\"title\">Resonant Quantum Tunneling<\/h3>\n<p id=\"fs-id1170901840333\">Quantum tunneling has numerous applications in semiconductor devices such as electronic circuit components or integrated circuits that are designed at nanoscales; hence, the term \u2018<span data-type=\"term\" id=\"term323\">nanotechnology<\/span>.\u2019 For example, a diode (an electric-circuit element that causes an electron current in one direction to be different from the current in the opposite direction, when the polarity of the bias voltage is reversed) can be realized by a tunneling junction between two different types of semiconducting materials. In such a<span>\u00a0<\/span><span data-type=\"term\" id=\"term324\">tunnel diode<\/span>, electrons tunnel through a single potential barrier at a contact between two different semiconductors. At the junction, tunneling-electron current changes nonlinearly with the applied potential difference across the junction and may rapidly decrease as the bias voltage is increased. This is unlike the Ohm\u2019s law behavior that we are familiar with in household circuits. This kind of rapid behavior (caused by quantum tunneling) is desirable in high-speed electronic devices.<\/p>\n<p id=\"fs-id1170901840342\">Another kind of electronic nano-device utilizes<span>\u00a0<\/span><span data-type=\"term\" id=\"term325\">resonant tunneling<\/span><span>\u00a0<\/span>of electrons through potential barriers that occur in quantum dots. A<span>\u00a0<\/span><span data-type=\"term\" id=\"term326\">quantum dot<\/span><span>\u00a0<\/span>is a small region of a semiconductor nanocrystal that is grown, for example, in a silicon or aluminum arsenide crystal.<span>\u00a0<\/span>Figure 3.23(a) shows a quantum dot of gallium arsenide embedded in an aluminum arsenide wafer. The quantum-dot region acts as a potential well of a finite height (shown in<span>\u00a0<\/span>Figure 3.23(b)) that has two finite-height potential barriers at dot boundaries. Similarly, as for a quantum particle in a box (that is, an infinite potential well), lower-lying energies of a quantum particle trapped in a finite-height potential well are quantized. The difference between the box and the well potentials is that a quantum particle in a box has an infinite number of quantized energies and is trapped in the box indefinitely, whereas a quantum particle trapped in a potential well has a finite number of quantized energy levels and can tunnel through potential barriers at well boundaries to the outside of the well. Thus, a quantum dot of gallium arsenide sitting in aluminum arsenide is a potential well where low-lying energies of an electron are quantized, indicated as<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-453-Frame\"><span class=\"MathJax_MathContainer\"><span>Edot<\/span><\/span><\/span>in part (b) in the figure. When the energy<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-454-Frame\"><span class=\"MathJax_MathContainer\"><span>Eelectron<\/span><\/span><\/span><span>\u00a0<\/span>of an electron in the outside region of the dot does not match its energy<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-455-Frame\"><span class=\"MathJax_MathContainer\"><span>Edot<\/span><\/span><\/span><span>\u00a0<\/span>that it would have in the dot, the electron does not tunnel through the region of the dot and there is no current through such a circuit element, even if it were kept at an electric voltage difference (bias). However, when this voltage bias is changed in such a way that one of the barriers is lowered, so that<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-456-Frame\"><span class=\"MathJax_MathContainer\"><span>Edot<\/span><\/span><\/span><span>\u00a0<\/span>and<span>\u00a0<\/span><span class=\"MathJax_MathML\" id=\"MathJax-Element-457-Frame\"><span class=\"MathJax_MathContainer\"><span>Eelectron<\/span><\/span><\/span><span>\u00a0<\/span>become aligned, as seen in part (c) of the figure, an electron current flows through the dot. When the voltage bias is now increased, this alignment is lost and the current stops flowing. When the voltage bias is increased further, the electron tunneling becomes improbable until the bias voltage reaches a value for which the outside electron energy matches the next electron energy level in the dot. The word \u2018resonance\u2019 in the device name means that the tunneling-electron current occurs only when a selected energy level is matched by tuning an applied voltage bias, such as in the operation mechanism of the<span>\u00a0<\/span><span data-type=\"term\" id=\"term327\">resonant-tunneling diode<\/span><span>\u00a0<\/span>just described. Resonant-tunneling diodes are used as super-fast nano-switches.<\/p>\n<div class=\"os-figure\">\n<figure id=\"CNX_UPhysics_40_06_dot\">\n<figure style=\"width: 859px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" alt=\"Figure a is an illustration of a tunneling diode. The quantum dot is a small region of gallium arsenide embedded in aluminum arsenide. Additional small regions of gallium arsenide are also embedded on either side of the quantum dot, separated from it by a small barrier of aluminum arsenide. The left end of the structure is attached to a negative electrode, and the right to a positive electrode. Figure b is a graph of the potential U as a function of x with no bias. The potential is constant except in two narrow regions where it has a larger constant value. The electron energy, represented by a dashed line, is between the lower and higher values of U, closer to the lower one. Two allowed energy levels, labeled as E sub dot, are shown. Both are higher than the electron energy and less than the maximum value of U. Figure c shows the potential U of x with a voltage bias across the device. The potential has the same constant value to the left of the barriers as in figure a, but decreases linearly between the barriers. U is constant again to the right of the barriers but at a lower value than before. The allowed energies are also pulled down, and the lower one now coincides with the energy of the electron.\" data-media-type=\"image\/jpeg\" id=\"1737\" src=\"https:\/\/cnx.org\/resources\/392116708b13f551148c46e4c676efba1e5138b1\" width=\"859\" height=\"289\" \/><figcaption class=\"wp-caption-text\">Figure 3.23 Resonant-tunneling diode: (a) A quantum dot of gallium arsenide embedded in aluminum arsenide. (b) Potential well consisting of two potential barriers of a quantum dot with no voltage bias. Electron energies Eelectron in aluminum arsenide are not aligned with their energy levels Edot in the quantum dot, so electrons do not tunnel through the dot. (c) Potential well of the dot with a voltage bias across the device. A suitably tuned voltage difference distorts the well so that electron-energy levels in the dot are aligned with their energies in aluminum arsenide, causing the electrons to tunnel through the dot.<\/figcaption><\/figure>\n<\/figure>\n<div>\n<p>&nbsp;<\/p>\n<div class=\"textbox\"><em>Download for free at http:\/\/cnx.org\/contents\/af275420-6050-4707-995c-57b9cc13c358@11.1<\/em><\/div>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":615,"menu_order":7,"template":"","meta":{"pb_show_title":"on","pb_short_title":"3. 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