A Quantum Box
We study quantum boxes (also called quantum dots) of nanometric dimensions, inside which the conduction electrons of a solid are confined at low temperatures. The possibility to control the energy levels of such devices leads to very interesting applicati
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A Quantum Box
In recent years, it has become possible to devise quantum boxes (also called quantum dots) of nanometric dimensions, inside which the conduction electrons of a solid are confined at low temperatures. The ensuing possibility to control the energy levels of such devices leads to very interesting applications in micro-electronics and opto-electronics. A quantum box is made of a material A on which another material B is deposited. A set of quantum boxes is shown on Fig. 26.1. The dots of In As (material B) are deposited on a substrate of Ga As (material A). In this chapter, we are interested in the motion of an electron in a twodimensional box. We note −q the electric charge of the electron, and we neglect spin effects. We shall assume that in a solid, the dynamics of an electron is described by the usual Schrödinger equation where: (i) the mass of the electron is replaced by an effective mass μ, (ii) the atoms of the materials A and B create an effective potential V (x, y) which is slowly varying on the atomic scale.
26.1
Results on the One-Dimensional Harmonic Oscillator
Consider a particle of mass μ placed in the one-dimensional potential V (x) = μω2 x 2 /2. We recall the definition of the annihilation and creation operators aˆ x and aˆ x† of the oscillator in terms of the position and momentum operators xˆ and pˆ x pˆx μω 1 +i√ aˆ x = √ xˆ h¯ h¯ μω 2
pˆx μω 1 aˆ x† = √ xˆ . −i√ h¯ h¯ μω 2
© Springer Nature Switzerland AG 2019 J.-L. Basdevant, J. Dalibard, The Quantum Mechanics Solver, https://doi.org/10.1007/978-3-030-13724-3_26
(26.1)
273
274
26 A Quantum Box
1 0.8 0.6 0
0.4 0.2
0.4
0.2 0.6
0
0.8
1
Fig. 26.1 Picture of a set of quantum boxes obtained with a tunneling microscope. The side of the square is 1 µm long and the vertical scale will be studied below
The Hamiltonian of the system can be written as 1 2 2 pˆ x2 1 ˆ + μω xˆ = hω nˆ x + Hx = ¯ 2μ 2 2
where nˆ x = aˆ x† aˆ x .
(26.2)
We also recall that the eigenvalues of the number operator nˆ x are the non-negative integers. Noting |nx the eigenvector corresponding to the eigenvalue nx , we have aˆ x† |nx =
nx + 1|nx + 1 aˆ x |nx =
√ nx |nx − 1
(26.3)
26.1.1 We recall that the ground state wave function is ψ0 (x) =
μω π h¯
1/4
μωx 2 exp − . 2h
(26.4)
What is the characteristic extension 0 of the electron’s position distribution in this state? 26.1.2 The effective mass μ of the electron in the quantum box is μ = 0.07 m0 , where m0 is the electron mass in vacuum. We assume that h¯ ω = 0.060 eV, i.e. ω/(2π) = 1.45 × 1013 Hz. (a) Evaluate 0 numerically. (b) At a temperature of 10 Kelvin, how many levels of the oscillator are populated significantly? (c) What is the absorption wavelength of radiation in a transition between two consecutive levels?
26.2 The Quantum Box
26.2
275
The Quantum Box
We assume that the effective two-dimensional potential seen by an electron in the quantum box is: V (x, y) =
1 2 2 μω (x + y 2 ). 2
(26.5)
We note Hˆ 0 = (pˆ x2 + pˆy2 )/2μ + V (x, y) t
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