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  4. Quantum mechanics
  5. The quantum tunneling of particles

7.6 The quantum tunneling of particles through potential barriers  (Page 9/22)

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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.
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 E electron in aluminum arsenide are not aligned with their energy levels E dot 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.

Summary

  • A quantum particle that is incident on a potential barrier of a finite width and height may cross the barrier and appear on its other side. This phenomenon is called ‘quantum tunneling.’ It does not have a classical analog.
  • To find the probability of quantum tunneling, we assume the energy of an incident particle and solve the stationary Schrӧdinger equation to find wave functions inside and outside the barrier. The tunneling probability is a ratio of squared amplitudes of the wave past the barrier to the incident wave.
  • The tunneling probability depends on the energy of the incident particle relative to the height of the barrier and on the width of the barrier. It is strongly affected by the width of the barrier in a nonlinear, exponential way so that a small change in the barrier width causes a disproportionately large change in the transmission probability.
  • Quantum-tunneling phenomena govern radioactive nuclear decays. They are utilized in many modern technologies such as STM and nano-electronics. STM allows us to see individual atoms on metal surfaces. Electron-tunneling devices have revolutionized electronics and allow us to build fast electronic devices of miniature sizes.

Key equations

Normalization condition in one dimension P ( x = − ∞ , + ∞ ) = ∫ − ∞ ∞ | Ψ ( x , t ) | 2 d x = 1
Probability of finding a particle in a narrow interval of position in one dimension ( x , x + d x ) P ( x , x + d x ) = Ψ * ( x , t ) Ψ ( x , t ) d x
Expectation value of position in one dimension ⟨ x ⟩ = ∫ − ∞ ∞ Ψ * ( x , t ) x Ψ ( x , t ) d x
Heisenberg’s position-momentum uncertainty principle Δ x Δ p ≥ ℏ 2
Heisenberg’s energy-time uncertainty principle Δ E Δ t ≥ ℏ 2
Schrӧdinger’s time-dependent equation − ℏ 2 2 m ∂ 2 Ψ ( x , t ) ∂ x 2 + U ( x , t ) Ψ ( x , t ) = i ℏ ∂ 2 Ψ ( x , t ) ∂ t
General form of the wave function for a time-independent potential in one dimension Ψ ( x , t ) = ψ ( x ) e − i ω t
Schrӧdinger’s time-independent equation − ℏ 2 2 m d 2 ψ ( x ) d x 2 + U ( x ) ψ ( x ) = E ψ ( x )
Schrӧdinger’s equation (free particle) − ℏ 2 2 m ∂ 2 ψ ( x ) ∂ x 2 = E ψ ( x )
Allowed energies (particle in box of length L ) E n = n 2 π 2 ℏ 2 2 m L 2 , n = 1 , 2 , 3 , . . .
Stationary states (particle in a box of length L ) ψ n ( x ) = 2 L sin n π x L , n = 1 , 2 , 3 , . . .
Potential-energy function of a harmonic oscillator U ( x ) = 1 2 m ω 2 x 2
Stationary Schrӧdinger equation − ℏ 2 m d 2 ψ ( x ) d x 2 + 1 2 m ω 2 x 2 ψ ( x ) = E ψ ( x )
The energy spectrum E n = ( n + 1 2 ) ℏ ω , n = 0 , 1 , 2 , 3 , . . .
The energy wave functions ψ n ( x ) = N n e − β 2 x 2 / 2 H n ( β x ) , n = 0 , 1 , 2 , 3 , . . .
Potential barrier U ( x ) = { 0 , when x < 0 U 0 , when 0 ≤ x ≤ L 0 , when x > L
Definition of the transmission coefficient T ( L , E ) = | ψ tra ( x ) | 2 | ψ in ( x ) | 2
A parameter in the transmission coefficient β 2 = 2 m ℏ 2 ( U 0 − E )
Transmission coefficient, exact T ( L , E ) = 1 cosh 2 β L + ( γ / 2 ) 2 sinh 2 β L
Transmission coefficient, approximate T ( L , E ) = 16 E U 0 ( 1 − E U 0 ) e − 2 β L
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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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