Physics of Engineered and Nanostructured Materials (SS2011)
Sheet1 - 6.5.2011
Properties of the 2d-electron gas of a quantum well system
In lectures 1 and 2 we learnt about methods to fabricate low dimensional nanostructures.
Molecular beam epitaxy (MBE) makes it possible to grow semiconductor layers with a thickness of only a few Angström. We consider a so-called quantum well system for which a thin
layer of GaAs (with a width - d) is sandwiched by Alx Ga(1−x) As cladding layers with an Alcontent x < 0.42. In todays exercise we’ll consider the properties of the electron gas that can
be created e.g. by doping the GaAs region. In the quantum well the free electron movement
is confined to two dimensions (xy-plane of the GaAs layer) but the motion is quantized along
the z-direction. The effective mass of the electrons in the GaAs layer is me .
a) Approximate the quantum well system by a rectangular potential well in z-direction with
infinitely high barriers. Derive expressions for the wavefunctions of the first two quantized
states (n = 1, n = 2) in this potential.
b) The 2-dimensional movement of an electron in the free electron gas in the GaAs layer (xyplane) can be described by the following single-particle Schrödinger Equation (the interaction
of the electrons with each other and with the crystal lattice as well as surface effects shall be
neglected):
2
~2
∂
∂2
−
·
ψ(x,y)
+
ψ(x,y)
= E · ψ(x,y)
2me
∂x2
∂y 2
A solution of this equation is given by the function ψ(x,y) = A · ei · (kx · x+ky · y) with the wave
vectors kx , ky . Determine the energies E(kx , ky ).
c) The quantum well system has the dimensions L × L in the xy-direction. Assume periodic
boundary conditions such that
ψ(x + L,y) = ψ(x,y) and ψ(x,y + L) = ψ(x,y)
Determine the allowed values for kx and ky . Derive the mean area of one single electron state
in wavevector (k) space (kx ky -plane).
d) Prove that the number of allowed states N , that lie within a circle of the Fermi-radius
2 ·π
kF
kF in the wavevector space, is given by N = 2 · (2π/L)
2 . Take into account that each state in
the wavevector space is twofold spin-degenerate.
e) The energy of the 2-dimensional free movement and the quantised energy in the potential
well contribute to the total energy E(kx ,ky ,n) of an electron in the GaAs layer. Derive an
expression for E(kx ,ky ,n). What are the contours of constant energy in the wavevector space?
f) Prove that the expresssion for the density of states D(E) = dN/dE of the 2-dimensional
electron gas results in
L2 · me
D(E) =
π · ~2
Use the dispersion relation E(kx ,ky ) and the result of d.
*g) The excitation of states in the 2-dimensional electron gas is decribed by the Fermi-Dirac
distribution f (E,T ). The density of electrons in the GaAs layer shall be small enough that
only the ground state with respect to the quantisation in z-direction has to be considered.
Schematically sketch the number of occupied electron states per energy interval D(E) · f (E,T )
as a function of their energy for the cases T = 0 and T > 0.