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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.


  • 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 2
Heisenberg’s energy-time uncertainty principle Δ E Δ t 2
Schrӧdinger’s time-dependent equation 2 2 m 2 Ψ ( x , t ) x 2 + U ( x , t ) Ψ ( x , t ) = i 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 2 2 m d 2 ψ ( x ) d x 2 + U ( x ) ψ ( x ) = E ψ ( x )
Schrӧdinger’s equation (free particle) 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 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 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 ) 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 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

Questions & Answers

A Pb wire wound in a tight solenoid of diameter of 4.0 mm is cooled to a temperature of 5.0 K. The wire is connected in series with a 50-Ωresistor and a variable source of emf. As the emf is increased, what value does it have when the superconductivity of the wire is destroyed?
Rupal Reply
how does colour appear in thin films
Nwjwr Reply
in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
Kimani Reply
derivation of lateral shieft
James Reply
total binding energy of ionic crystal at equilibrium is
All Reply
How does, ray of light coming form focus, behaves in concave mirror after refraction?
Bishesh Reply
Refraction does not occur in concave mirror. If refraction occurs then I don't know about this.
What is motion
Izevbogie Reply
Anything which changes itself with respect to time or surrounding
and what's time? is time everywhere same
how can u say that
do u know about black hole
Not so more
Radioactive substance
These substance create harmful radiation like alpha particle radiation, beta particle radiation, gamma particle radiation
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
explain cavendish experiment to determine the value of gravitational concept.
Celine Reply
 Cavendish Experiment to Measure Gravitational Constant. ... This experiment used a torsion balance device to attract lead balls together, measuring the torque on a wire and equating it to the gravitational force between the balls. Then by a complex derivation, the value of G was determined.
For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
Evan Reply
as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
acceleration also inc
which paper will be subjective and which one objective
normal distributiin of errors report
normal distribution of errors
acceleration also increases
there are two correct answers depending on whether air resistance is considered. none of those answers have acceleration increasing.
Acceleration is the change in velocity over time, hence it's the derivative of the velocity with respect to time. So this case would depend on the velocity. More specifically the change in velocity in the system.
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
this msg is out of mistake. sorry friends​.
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
physics is the study of non living things if we added it with biology it becomes biophysics and bio is the study of living things tell me please what is this?
physics is the study of matter,energy and their interactions
all living things are matter
why rolling friction is less than sliding friction
thanks buvanas

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