ELEC 315 - Assignment 2
Term 2 Winter 2021-22
Due Feb. 14, 2022
1. We want to use a 3D infinite potential well to model a quantum well, an extremely thin
film of semiconductor, used for tuning the emission wavelengths and bandwidth of a
semiconductor laser. Assume that the potential is zero inside the film ( Lx = L y = 10 nm
and Lz = 0.1 nm) and infinity outside the film.
(a) Similar to what was described in class for 2D case, the solution ψ n ,n
x
y ,n z
(x, y, z,t) for
Schrödinger’s equation has three quantum numbers and the energy for each state can be
2
2
2
! 2π 2 ⎛⎜ nx n y nz ⎞⎟
+
+
written as En ,n ,n =
. ( m0 = 9.1 ´ 10 -31 kg and
2
2
2 ⎟
⎜
x y z
2m0 ⎝ Lx Ly Lz ⎠
h = 2p ! = 6.63 ´10 -34 Js ).
0.1 nm
(b) Calculate the six lowest energy levels for electrons in this
10 nm
quantum well? Are there any degenerate states among these states?
10 nm
Comment on how do the different quantum numbers n x and n z
change the energy and relate this to the asymmetry in the size of this potential well in x
and z directions.
(c) Find the energy of the photons emitted from transition of an electron: y 1,1, 2 ® y 1,1,1
and y 1, 2,1 ® y 1,1,1 .
2. Assume an electron is traveling in positive x direction
toward a barrier with the height Ubarrier and the width W.
The electron has a total energy E and assume its wave
function has an amplitude A1. The transmission
probability T for electron tunneling through the barrier
is given by
A
T= 3
A1
V(x)
U
1
Ubarrier
2
3
2
−1
=⎡⎣1+ Dsinh 2 ( k2W )⎤⎦ , where
E, p
2
2m (U barrier − E )
U barrier
2
D=
and k2 =
.
2
4 E (U barrier - E )
V=0
0 W
Assume that for a gate of transistor, gate metal is separated from semiconductor with an
oxide film with a thickness of 2 nm. The oxide layer acts as a square potential barrier
(like the picture) with a height of 2 eV for free electrons (kinetic energy of 1 eV) in the
gate. Find the transmission probability for the electron through this oxide layer. If there is
an electron current of 1 µA in the gate, estimate the tunneling current through the gate
insulator. If there are 1 billion transistors and the power supply of the chip is 1.5 V,
approximate the power consumption due to the tunneling current through the gate.
x
3. We want to compare two semiconductors in this problem: Si and GaAs.
(a) At room temperature ( T = 300 K), the band gaps of Si and GaAs are 1.12 eV and
1.42 eV, respectively. Given that the density of intrinsic electrons at room
temperature are 1´ 1010 cm-3 and 2.1 ´ 10 6 cm-3 for Si and GaAs, respectively, find
the density of intrinsic electrons and holes at 350 K for Si and GaAs, assuming
the bandgap does not change with temperature.
(b) Calculate conductivities of Si and GaAs at room temperature and 350 K, if
µ n = 1400 and µ p = 450 cm2/Vs for Si and µ n = 8500 and µ p = 400 cm2/Vs
for GaAs. Assume mobility values do not change with temperature.
(c) Given that for Si mn* = 0.49m0 and m*p = 0.98m0 , and for GaAs mn* = 0.063m0
and m*p = 0.53m0 , find N C and NV (cm-3) at room temperature for these
semiconductors.