Solution of the Schrodinger Equation for Four Specific Problems (Fundamentals of Electron Theory) Part 2

Finite Potential Barrier (Tunnel Effect)

Let us assume that a free electron, propagating in the positive x-direction, encounters a potential barrier whose potential energy, V0, ("height" of the barrier) is larger than the total energy, E, of the electron, but is still finite (Fig. 4.6). For this case we have to write two Schrodinger equations, which take into account the two different areas. In region I (x < 0) the electron is assumed to be free, and we can write

tmp6C-95_thumb[2] Finite potential barrier. Inside the potential barrier (x > 0) the Schrodinger equation reads

Figure 4.6. Finite potential barrier. Inside the potential barrier (x > 0) the Schrodinger equation reads


tmp6C-97_thumb[2]

The solutions to these equations are as before (see topic 1):

tmp6C-98_thumb[2]

where

tmp6C-99_thumb[2]

and

tmp6C-100_thumb[2]

with

tmp6C-101_thumb[2]

A word of caution has to be inserted here. We stipulated above that V0 is larger than E. As a consequence of this (E — V0) is negative and b becomes imaginary. To prevent this, we define a new parameter:

tmp6C-102_thumb[2]

This yields, for (4.32),

tmp6C-103_thumb[2]

The parameter g is now prevented under the stated conditions from becoming imaginary. Rearranging (4.33) to obtain

tmp6C-104_thumb[2]

and inserting (4.35) into (4.31) yields

tmp6C-105_thumb[2]

Next, one of the constants C or D needs to be determined by means of a boundary condition:

Fortmp6C-106_thumb[2]it follows from (4.36) that tmp6C-108_thumb[2]

The consequence of (4.37) could be thattmp6C-109_thumb[2]and thereforetmp6C-110_thumb[2]are infinity. Since the probabilitytmp6C-111_thumb[2]can never be larger than one (certainty), tmp6C-112_thumb[2]is no solution. To avoid this, C has to go to zero:

tmp6C-117_thumb[2]

Then, (4.36) reduces to

tmp6C-118_thumb[2]

which reveals that the C-function decreases in Region II exponentially, as shown in Fig. 4.7 (solid line). The decrease is stronger the larger g is chosen, i.e., for a large potential barrier, V0.

The electron wave C(x, t) is then given, using (A.27) and (4.39), by

tmp6C-119_thumb[2]

(damped wave) as shown by the dashed curve in Fig. 4.7. In other words, (4.39) provides the envelope (or decreasing amplitude) for the electron wave that propagates in the finite potential barrier. If the potential barrier is only moderately high and relatively narrow, the electron wave may continue on the opposite side of the barrier. This behavior is analogous to that for a light wave, which likewise penetrates to a certain degree into a material and whose amplitude also decreases exponentially, as we shall see in the optics part of this topic, specifically in Fig. 10.4. The penetration of a potential barrier by an electron wave is called "tunneling" and has important applications in solid state physics (tunnel diode, tunnel electron microscope, field ion microscope). Tunneling is a quantum mechanical effect. In classical physics, the electron (particle) would be described to be entirely reflected back from the barrier (at x = 0) if its kinetic energy is smaller than V0.

*For the complete solution of the behavior of an electron wave that penetrates a finite potential barrier (Fig. 4.6), some additional boundary conditions need to be taken into consideration:

C-function (solid line) and electron wave (dashed line) meeting a finite potential barrier.

Figure 4.7. C-function (solid line) and electron wave (dashed line) meeting a finite potential barrier.

(1) The functions CI and Cn are continuous at x = 0. As a consequence, CI = Cn at x = 0. This yields, with (4.29), (4.36), and (4.38),


tmp6C-121_thumb[2]

With x = 0, we obtain

tmp6C-122_thumb[2]

(2) The slopes of the wave functions in Regions I and II are continuous at x = 0, i.e.,tmp6C-123_thumb[2]This yields tmp6C-125_thumb[2]

With x = 0, one obtains

tmp6C-126_thumb[2]

Inserting (4.40) into (4.42) yields

tmp6C-127_thumb[2]

and

tmp6C-128_thumb[2]

From this, the C-functions can be expressed in terms of a constant D. Figure 4.8 illustrates the modification of Fig. 4.4(a) when tunneling is taken into consideration. A penetration of the C-function into the potential barriers is depicted.

Square well with finite potential barriers. (The zero points on the vertical axis have been shifted for clarity.)

Figure 4.8. Square well with finite potential barriers. (The zero points on the vertical axis have been shifted for clarity.)

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