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STM Physical Backgrounds hhhttttttppp::://////wwwwwwwww...nnntttmmmdddttt...cccooommm STM Physical Backgrounds Tunneling Effect The idea of particles tunneling appeared almost simultaneously with quantum mechanics. In classical mechanics, to describe a system of material points at a certain moment of time, it is enough to set every point coordinates and momentum components. In quantum mechanics it is in principle impossible to determine Fig. 1. Rectangular potential barrier and particle wave simultaneously coordinates and momentum function Ψ components of even single point according to the He-isenberg uncertainty principle. To describe the The stationary Schrodinger equations can be written system completely, an associated complex function as follows is introduced in quantum mechanics (the wavefunction). The wavefunction Ψ , which 2 ⎧Ψ&& +κ Ψ = ,0 при z < ;0 is a function of time and all system particles ⎪ 1 ⎪ 2 position, is a solution of the wave Schrodinger ⎨Ψ&& −κ 2 Ψ = ,0 при z∈ 0[ 0; L] ; (2) equation. In order to use the system wavefunction, ⎪ Ψ&& +κ 2 Ψ = ,0 при z> L one should determine Ψ 2 rather than Ψ . Then, the ⎩⎪ 1 probability for finding particles in an elementary volume dxdydz is given by Ψ 2 dxdydz . 2mE 2m ( U0 − E If particles impinge on a potential barrier of a where κ1 = , κ 2 = – wave h h limited width, the quantum mechanics predicts the −34 effect of particles penetration through the potential vectors, h =1,05 ⋅ 10 Дж ⋅ с – Planck's constant. barrier even if particle total energy is less than the The solution to the wave equation at z<0 can be barrier height which is unknown in classical expressed as a sum of incident and reflected physics. waves Ψ = exp(ik1 z )+ a exp( − ik1 z ) , while solution Lets calculate the transparency of the rectangular at z>L – as a transmitted wave Ψ = bexp( ik z ) . barrier [1], [2]. Suppose that electrons of potential 1 A general solution inside the potential barrier energy 0<z<L is written as Ψ = cexp( k2 z )+ d exp( − k2 z ) . ⎧ ,0 при z < ;0 Constants a, b, c, d are determined from the ⎪ wavefunction Ψ and Ψ& continuity condition at U() z = ⎨U 0 при 0≤z ≤ L ; (1) z=0 and z=L. ⎪ ⎪ ,0 при z> L ⎩ The barrier transmission coefficient can be naturally considered as a ratio of the transmitted electrons impinge on the rectangular potential barrier and the probability flux density to that one of the incident total energy E is less than U0 (Fig. 1). electrons. In the case under consideration this ratio 1.1 STM Physical Backgrounds is just equal to the squared wavefunction module at References z>L because the incident wave amplitude is assumed to be 1 and wave vectors of both incident 1. Sivuhin D.V. A General course of physics. and transmitted waves coincide. Nauka, volume 5, chapter 1, 1988 (In Russian) 2. Goldin L.L., Novikova H.I. The introduction in 2 −1 quantum physics. Nauka, 1988 (In Russian). ⎛ 1 ⎛ k k ⎞ ⎞ D= bb* = ⎜ ch2 () k L +⎜ 2 − 1 ⎟ sh2 () k L ⎟ (3) ⎜ 2 ⎜ ⎟ 2 ⎟ 4 ⎝ k1 k2 ⎠ ⎝ ⎠ Tunneling Effect in Quasiclassical Approximation If k2 L >>1, then both ch() k2 L and sh() k2 L can be The quasiclassical qualitative condition imply that approximated to exp(k2 L )/ 2 and (3) will be written as de Broglie wavelength of the particle λ is less than characteristic length L determining the conditions of ⎧ 2L ⎫ the problem. This condition means that the particle DED( )=0 exp⎨ − 2m ( U0 − E ⎬ (4) wavelength should not change considerably within ⎩ h ⎭ the length of the wavelength order −1 dD 2 <<1 (1) ⎡ 1 ⎛ k k ⎞ ⎤ dz ⎜ 2 1 ⎟ where D0 =4⎢ 1 +⎜ − ⎟ ⎥ . ⎢ 4 ⎝ k1 k2 ⎠ ⎥ ⎣ ⎦ where = λ 2/ π , λ(z )= 2π / p ( z ) – de Broglie D h wavelength of the particle expressed by way of the Thus, analytical calculation of the rectangular particle classical momentum p(z) [1]. barrier transmission coefficient is rather a simple task. However, in many quantum mechanical Condition (1) can be expressed in another form problems it is necessary to find the transmission taking into account that coefficient of the more complicated shape barrier. In this case, there is no common analytical solution dp d m dU mF for the D coefficient. Nevertheless, if the problem = 2m ( W− U = − = (2) parameters satisfy the quasiclassical condition, the dz dz p dz p transmission coefficient can be calculated in a general form. (see chapter Tunneling Effect in where F= − dU/ dz means classical force acting Quasiclassical Approximation). upon the particle in the external field. Introducing this force, we get Summary m F h <<1 (3) − In quantum mechanics tunneling effect is p3 particles penetration through the potential barrier even if particle total energy is less than From (3) it is clear that the quasiclassical the barrier height. approximation is not valid at too small momentum − To calculate the transparency of the potential of the particle. In particular, it is deliberately invalid barrier, one should solve Shrodinger equation at near positions in which the particle, according to continuity condition of wavefunction and its classical mechanics, should stop, then start moving first derivative. in the opposite direction. These points are the so − The transparency coefficient of the rectangular called "turning points". Their coordinates z1and z2 barrier decreases exponentially with the barrier are determined from the condition E=U(z). width, when wave vectors of both incident and It should be emphasized that condition (3) itself can transmitted waves coincide. be insufficient for the permissibility of the quasiclassical approach. One more condition should be met: the barrier height should not change much over the length L. 1.1 STM Physical Backgrounds Let us consider the particles move in the field References shown in Fig. 1 which is characterized by the presence of the potential barrier with potential 1. Landau L.D., Lifshitz E. M. Quantum energy U(z) exceeding particle total energy E and mechanics. Nauka, 1989 (In Russian) meeting all the quasiclassics conditions. In this case 2. Mott N., Sneddon I. Wave mechanics and its points z1and z2 are the turning points. application. Nauka, 1966 (In Russian) Metal Energy-Band Structure To derive the formula of tunneling current in the metal-insulator-metal (MIM) system (John G. Simmons formula, see chapter "John G. Simmons Formula"), we must make some assumptions and remind of metal electronic theory fundamentals. Fig. 1. Potential barrier of arbitrary shape − Firstly, let us consider that the solid (metal) is a The approximation technique of the Schrodinger three-dimensional potential well with a plain equation solution when quasiclassical conditions bottom (the so called Sommerfeld model) and are met was first used by Wentzel, Kramers and electrons do not interact with each other. The Brillouin. This technique is known as WKB energy of the electron resting at the bottom of approximation or quasiclassical quantization such potential well is less than vacuum level – method. In this textbook we do not present the the energy of electron resting infinitely far from Schrodinger equations solution for the given case. the solid surface. The well depth U is However, it can be found in [1], [2] and the barrier determined by the averaged field of all positive transparency in this case is ions of the lattice and of all electrons. − Secondly, because all electrons are considered z ⎪⎧ 2 2 ⎪⎫ to be noninteracting, the solution to the DE( )∞ exp⎨ − ∫ 2m ( U ( z )− E ) dz ⎬ (4) Schrodinger equation for the system of electrons ⎪ h z ⎪ ⎩ 1 ⎭ is turned to the solution to the Schrodinger equation for single electron moving in the averaged field which allows to use formulas (3) Comparing expressions (3) in chapter in chapter Tunneling Effect or (4) Tunneling Effect for transmission coefficients of in chapter Tunneling Effect in Quasiclassical rectangular barrier (precise solution of Shrodinger Approximation. equation) and (4) for quasiclassical approximation, − Thirdly, in the potential box with the plain we can notice that there is no qualitative difference bottom, the dependence of energy allowed between them. In both cases the transparency values on wave vector allowed values looks like decreases exponentially with the barrier width. points on the parabolic dependence of the energy on the wave vector components Summary 2 2 E= p/ 2 m = (h k ) / 2 m for the free electron in − If the problem parameters satisfy quasiclassical void. If electrons mass m is isotropic in space, conditions, then transmission coefficient can be then the following expression is valid 2 2 2 2 calculated in a general form using (4). E= px / 2 m + py / 2 m + pz / 2 m = p / 2 m , − In case of the square barrier there is no where Px, Py, Pz,– axial electron momentum qualitative difference between transmission components. coefficients calculated using quantum mechanics and quasiclassical approximation. In From the statistical physics [1] it is known that in a both cases the transparency decreases system in thermodynamic equilibrium all quantum exponentially with the barrier width. states with the same energy level E are occupied with electrons equally. The mean number of electrons in one quantum state with energy E at 1.1 STM Physical Backgrounds temperature T is given by the Fermi-Dirac distribution: 1 f() E = (1) ⎛ E − μ ⎞ 1+ exp⎜ ⎟ ⎝ kB T ⎠ –1 where kB=B (11600) eV/К – Boltzmann's constant, μ – parameter having the dimension of energy and called chemical potential. The chemical potential μ is defined by the normalization condition ∞ Fig. 1. Diagram of MIM system in equilibrium. n()() E f E dE= n (2) j1 and j2 – work function of the left and right metals, respectively ∫ e 0 where ne – number of conduction band electrons per volume unit (concentration), 3 1 ⎛ 2m ⎞ n( E ) dE = 2(2π )3 d3 k = ⎜ ⎟ EdE – ∫∫∫ 2 ⎜ ⎟ []E, E+ dE 2π ⎝ h ⎠ number of elecron states per volume unit in the energy range from E to E + dE.
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