Haug H., Koch S. Quantum theory of the optical and electronic properties of semiconductors
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Magneto-Optics 379
tionary polarization equation is in the rotating wave approximation
$
ω−e
e
k,ν
−e
h
k,ν
+iγ
%
p
k,ν
=
&
f
e
k,ν
+f
h
k,ν
− 1
'
d
c
vE
0
+
q,ν
V
eh
νν
(q)p
k−q,ν
,
(19.33)
where p
k,ν
= P
k,ν
e
−iωt
. From the scattering term, we took only a phe-
nomenological dephasing or damping rate γ into account. The electron and
hole distributions are thermal Fermi functions, e.g.,
f
e
k,ν
=
1
e
$
ω
eff
&
ν+
1
2
'
+
2
k
2
2m
e
eff
−µ
e
%
β
+1
. (19.34)
With this ansatz, we not only assume that the electrons of a given Landau
subband are in thermal equilibrium, but that all electron subbands are in
equilibrium among each other.
It is advantageous to eliminate the amplitude of the light field by intro-
ducing a susceptibility component
χ
k,ν
(ω)=
P
k,ν
(t)
E(t)
, (19.35)
which depends on the carrier frequency of the light. The total susceptibility
is then obtained as
χ(ω)=d
cv
k,ν
P
k,ν
(ω) . (19.36)
The equation for the susceptibility component can be written in the form
χ
k,ν
(ω)=χ
0
nk
(ω) − χ
0
nk
(ω)
1
d
cv
k
,ν
V
eh
νν
(k − k
)χ
0
nk
(ω) ,
(19.37)
susceptibility integral equation
where we again used the spectral representation according to Eq. (15.18).
From the susceptibility, one gets the luminescence spectrum using the equi-
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
380 Quantum Theory of the Optical and Electronic Properties of Semiconductors
a)
b)
Intensity (a.u.)
Energy [eV]
Energy [eV]
1.43
1.43
1.44
1.44
1.45
1.451.46
1.46
1.47
1.47
Fig. 19.1 (a) Calculated and (b) measured luminescence spectra. The experiments were
done using a modulated barrier In
0.13
Ga
0.87
As/GaAs quantum wire at B =10.5 T. The
plasma densities and temperatures used in the theoretical spectra have been determined
to fit the experimental spectra with increasing excitation power. [After M. Bayer et al.
(1997).]
librium relation
I(ω) ∝
Im{χ(ω)}
e
β(ω−µ)
− 1
, (19.38)
where µ = µ
e
+ µ
h
is the electron-hole chemical potential with respect to
the band gap.
As an example of the resulting spectra, we show in Fig. 19.1 calcu-
lated luminescence spectra for a field B =10.5 T, where the magnetic
confinement dominates already over the lateral barrier confinement (see
problem 19.3). In this situation, the wire resembles already in some re-
spects a 2D system in a strong magnetic field. The spectra are calculated
for various carrier densities n and plasma temperatures T . The densities
and the temperatures are chosen to give the best fit to the corresponding
experimental spectra shown in the same figure. The investigated sample is
based on a In
0.13
Ga
0.87
As/GaAs quantum well. The top GaAs layer has
been removed by selective etching, except for a 29 nm wide stripe. This
remaining stripe modulates the ground-state energy of the quantum well
and causes a lateral confinement. The resulting quantum wire had a width
of L
x
=29nm. The electron–hole concentration has been generated by a
cw Ar
+
laser (λ = 514.5 nm) with power densities of up to 3kWcm
−2
.
The sample has been immersed in liquid He (T = 1.8 K). The appropriate
material parameters are: the subband spacing Ω 2E
0
8.4meV,where
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Magneto-Optics 381
E
0
is the 3D exciton Rydberg of GaAs. The broadening has been taken
to be γ =1E
0
. At the lowest density of n =0.12 × 10
6
cm
−1
and the
corresponding lowest excitation power, only the emission from the lowest
wire subband is seen. Note that the excess energy of the exciting laser
causes heating of the plasma (but not of the lattice), so that the plasma
temperature increases with the excitation power from 43 K at the lowest
excitation power to 117 K at the highest power. At higher powers the n =2
wire subband starts to be filled so that also luminescence from this next
higher level takes place, while the n =1line saturates. In the excitation
region, where also the n =2subband line is present, both peaks of the two
luminescence lines show a slight shift to lower energies at higher excitation
powers, which is more pronounced for the n =2transitions.
In particular, one can see that the n =1line does not shift at all as
long as the the n =2subband is not filled (the three lowest densities). The
calculations show that here already a strong band-gap shift exists, but it
is completely compensated by the reduction of the exciton correlations. It
is known that for a 2D system the compensation is exact in the limit of a
strong magnetic field essentially due to the neutrality of the electron–hole
pair, which explains the absence of any shift with only one occupied sub-
band. Due to the interaction between carriers in two different subbands, the
energy renormalization becomes stronger than the effect of the attractive
interactions.
REFERENCES
For the literature of the 2D electron in a perpendicular magnetic field see:
L.D. Landau and L.M. Lifshitz, Quantum Mechanics, Pergamon, New York
1958
I.V. Lerner and Yu.E. Lozovik, Sov. Phys. JETP 53, 763 (1981)
D. Paquet, T.M. Rice, and K. Ueda, Phys. Rev. B32, 5208 (1985)
2D magneto excitons are treated in:
O. Akimoto and H. Hasegawa, J. Phys. Soc. Jpn. 22, 181 (1967)
A.H. McDonald and D.S. Ritchie, Phys. Rev. B33, 8336 (1986)
C. Stafford, S. Schmitt-Rink, and W. Schäfer, Phys. Rev. B41, 10000
(1990)
G.E.W. Bauer, in Optics of Excitons in Confined Systems,eds.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
382 Quantum Theory of the Optical and Electronic Properties of Semiconductors
A. D’Andrea, R. Del Sole R. Girlanda, and A. Quattropani, Inst. of Phys.
Conf. Series No. 123, 283 (1992)
2D magneto-plasmas are treated in:
T. Uenoyama and L.J. Sham, Phys. Rev. B39, 11044 (1989)
G.E.W. Bauer, Phys. Rev. Lett. 64, 60 (1990) and Phys. Rev. B45, 9153
(1992)
M. Bayer, Ch. Schlier, Ch. Gréus, S. Benner, and H. Haug, Phys. Rev.
B55, 13180 (1997)
M.W. Wu and H. Haug, Phys. Rev. B58, 13060 (1998)
PROBLEMS
Problem 19.1: Derive the single-particle Hamiltonian (19.8) for the states
φ
k,ν
.
Problem 19.2: Show that the Coulomb matrix element can be written in
the form (19.24), by calculating the integral
dxdy
dx
y
ψ
∗
k
n
(x, y)ψ
∗
k,ν
(x
,y
)V (x − x
,y− y
)ψ
k
,ν
(x
y
)ψ
k,ν
(x, y) .
Use the Fourier transform of the 2D Coulomb potential
V (x − x
,y− y
)=
q,q
x
2πe
2
0
q
2
+ q
2
x
e
iq(y−y
)
e
iq
x
(x−x
)
,
and the eigenfunctions (19.14).
Problem 19.3: (a)Calculate the cyclotron energy ω
c
= eB/m for B =
10.5 TintheInGaAs/Ga quantum wire in units of meV. Use a reduced
electron–hole mass of m =0.06m
0
,wherem
0
is the free electron mass. In
SI units, the specific electron charge e/m
0
=1.759 ×10
11
Askg
−1
, while a
Tesla is T=kgs
−2
A
−1
. Here, A denotes Ampere.
(b) Calculate for comparison the energy of the subband spacing Ω in meV
with the same reduced mass.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Chapter 20
Quantum Dots
The ultimate quantum-confinement effects occur in very small semicon-
ductor structures, called quantum dots, which confine the laser-excited
electron–hole pairs in all three space dimensions. Such systems can be re-
alized in many different ways, including glasses doped with semiconductor
microcrystallites, microstructures fabricated by lithographic techniques, in-
terface fluctuations in quantum wells, strain-induced quantum dots and
many more. A common aspect of all of these systems is that their geometry
provides a confinement potential with a spatial extension comparable to the
exciton Bohr radius in the respective semiconductor material. The detailed
shape of the potential depends on the respective system, including more
or less spherical confinement in microcrystallites, parabolic confinement in
interface dots, or more complicated shapes in other systems. For more de-
tailed discussion of experimental realizations of quantum dots and of the
resulting confinement potentials, we refer the interested reader to the lit-
erature cited at the end of this chapter. Here, we only want to discuss the
basic principles of optical excitations in quantum-dot systems.
20.1 Effective Mass Approximation
For most of the discussion in this chapter, we consider a spherical hard
wall confinement potential. More precisely, we assume a sphere of radius
R and background dielectric constant
2
embedded in another material
with background dielectric constant
1
. It is reasonably straightforward to
modify the results for other (simple) geometries, such as microcubes, boxes,
or parabolic confinement potentials. We concentrate on systems where
R ≤ a
0
, (20.1)
383
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
384 Quantum Theory of the Optical and Electronic Properties of Semiconductors
but still much larger than the semiconductor lattice constant. Hence, these
quantum dots are mesoscopic structures in the sense of Chap. 4.
Using the envelope function approximation, we assume again that the
electron wave function can be written as
ψ(r)=ζ(r) u
λ
(k 0, r) , (20.2)
compare Eq. (4.1). Here, u
λ
(k 0, r) is the Bloch function of the bulk
material, and ζ(r) is the envelope function. The wave function ψ(r) has
to satisfy the boundary conditions of the quantum dot. For simplicity, we
analyze the case of ideal quantum confinement, i.e.,
ψ(r ≥ R)=0 . (20.3)
Furthermore, we assume that the energy eigenvalues of the electron in the
periodic lattice, i.e., the energy bands, are not appreciably modified through
the quantum confinement. Therefore, we use the effective mass approxima-
tion to describe the free motion of electrons and holes.
The Hamiltonian for one electron–hole pair is
H = H
e
+ H
h
+ V
ee
+ V
hh
+ V
eh
, (20.4)
where the kinetic terms are
H
e
= −
2
2m
e
d
3
r
ˆ
ψ
†
e
(r) ∇
2
ˆ
ψ
e
(r)+E
g
d
3
r
ˆ
ψ
†
e
(r)
ˆ
ψ
e
(r) , (20.5)
and
H
h
= −
2
2m
h
d
3
r
ˆ
ψ
†
h
(r)∇
2
ˆ
ψ
h
(r) , (20.6)
and the Coulomb interaction is described by
V
ee
=
1
2
d
3
rd
3
r
ˆ
ψ
†
e
(r)
ˆ
ψ
†
e
(r
) V (r, r
)
ˆ
ψ
e
(r
)
ˆ
ψ
e
(r) , (20.7)
V
eh
= −
d
3
rd
3
r
ˆ
ψ
†
e
(r)
ˆ
ψ
†
h
(r
) V (r, r
)
ˆ
ψ
h
(r
)
ˆ
ψ
e
(r) , (20.8)
and
V
hh
= V
ee
(e → h) , (20.9)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Quantum Dots 385
where V (r) is the effective Coulomb interaction potential inside the quan-
tum dot.
The Coulomb interaction between two point charges in a spherical dot,
which is embedded in a material with different background dielectric con-
stant, is
V (r
1
, r
2
)|
R
= V (r
1
, r
2
)|
R=∞
+ δV (r
1
, r
2
) , (20.10)
where V (r
1
, r
2
)|
R=∞
is the usual bulk Coulomb interaction, and the addi-
tional term is caused by the induced surface charge of the sphere,
δV (r
1
, r
2
)=Q
1
(r
1
)+Q
1
(r
2
) ∓ Q
2
(r
1
, r
2
) , (20.11)
with -(+) for charges with opposite (equal) sign, respectively (Brus, 1984).
The different contributions in Eq. (20.11) are
Q
1
(r)=
∞
n=0
Q
1,n
(r) , (20.12)
with
Q
1,n
(r)=
e
2
2R
α
n
(r/R)
2n
, (20.13)
and
Q
2
(r
1
, r
2
)=
∞
n=0
Q
2,n
(r
1
, r
2
) , (20.14)
with
Q
2,n
(r
1
, r
2
)=α
n
e
2
R
$
r
1
r
2
R
2
%
n
P
n
cos(θ)
, (20.15)
where θ is the angle between r
1
and r
2
, P
n
is the n-th order Legendre
polynomial and
α
n
=
(
2
/
1
− 1)(n +1)
2
(n
2
/
1
+ n +1)
. (20.16)
Obviously, the surface polarization term δV vanishes for
1
=
2
.
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
386 Quantum Theory of the Optical and Electronic Properties of Semiconductors
20.2 Single Particle Properties
The eigenstates and energy eigenvalues for a single electron in the quantum
dot are determined by the Schrödinger equation
H|ψ
e
= E
e
|ψ
e
. (20.17)
The eigenstate is of the form
|ψ
e
=
d
3
rζ
e
(r)
ˆ
ψ
†
e
(r) |0 , (20.18)
where |0 is the crystal ground state, i.e., the state without excited electrons
or holes. The coefficients ζ(r) in Eq. (20.18) have to be determined from
Eq. (20.17). Using the Hamiltonians (20.6) – (20.9) in Eq. (20.17), we find
V
ee
|ψ
e
= V
eh
|ψ
e
= V
hh
|ψ
e
= H
h
|ψ
e
=0 . (20.19)
However, we have
H
e
|ψ
e
= −
2
2m
e
d
3
r
[∇
2
ˆ
ψ
†
e
(r
)]
ˆ
ψ
e
(r
)
d
3
rζ
e
(r)
ˆ
ψ
†
e
(r) |0
+E
g
d
3
r
d
3
r
ˆ
ψ
†
e
(r
) ψ
e
(r
) ζ
e
(r)
ˆ
ψ
†
e
(r) |0 ,
which can be written as
H
e
|ψ
e
= −
2
2m
e
d
3
r
d
3
rζ
e
(r) δ(r − r
)[∇
2
ˆ
ψ
†
e
(r
)] |0
+E
g
d
3
r
d
3
rζ
e
(r)
ˆ
ψ
†
e
(r) δ(r − r
) |0
= −
2
2m
e
d
3
r [∇
2
ζ
e
(r)]
ˆ
ψ
†
e
(r)|0 + E
g
d
3
rζ
e
(r)
ˆ
ψ
†
e
(r) |0
= E
e
d
3
rζ
e
(r)
ˆ
ψ
†
e
(r) |0 . (20.20)
This equation is satisfied if
−
2
2m
e
∇
2
ζ
e
(r)=(E
e
− E
g
) ζ
e
(r) , (20.21)
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
Quantum Dots 387
which is the one-electron eigenvalue equation. Similarly, we find for the
one-hole state
−
2
2m
h
∇
2
ζ
h
(r)=E
h
ζ
h
(r) . (20.22)
The problem is completely defined with the boundary conditions
ζ
e
(r)=ζ
h
(r)=0 for |r|≥R. (20.23)
The solution is
ζ
e,nlm
(r) =
2
R
3
j
l
(α
nl
r/R)
j
l+1
(α
nl
)
Y
l,m
(Ω) , (20.24)
single-particle wave function in quantum dot
where j
l
is the spherical Bessel function of order l and Y
l,m
denotes the
spherical harmonics. The boundary condition (20.23) is satisfied if
j
l
(α
nl
)=0 for n =1, 2, ··· (20.25)
and
α
10
= π, α
11
=4.4934,α
12
=5.7635,α
20
=6.2832 ,
α
21
=7.7253,α
22
=9.0950,α
30
=9.4248 ··· .
(20.26)
Since the wave function (20.24) depends only on R and not on any physi-
cal parameters which are specific for the electron, the corresponding wave
function for the hole must have the same form, and we can drop the index
e/h of the functions ζ(r) from now on. Inserting (20.24) into Eq. (20.22),
we obtain the discrete energies
E
e,nlm
= E
g
+
2
2m
e
α
2
nl
R
2
(20.27)
and
E
h,nlm
=
2
2m
h
α
2
nl
R
2
. (20.28)
The nl eigenstates are usually referred to as 1s, 1p,etc. Thelowesttwo
energy levels given by Eqs. (20.27) and (20.28) are shown schematically
in Fig. 20.1. In practice, however, the single-particle spectrum is rather
January 26, 2004 16:26 WSPC/Book Trim Size for 9in x 6in b ook2
388 Quantum Theory of the Optical and Electronic Properties of Semiconductors
uninteresting since it is not observed in optical absorption measurements.
As discussed before, the absorption is always given by the electron–hole–
pair excitation spectrum.
E
k
E
e, 1p
e, 1s
h, 1s
h, 1p
Fig. 20.1 Schematic plot of the single-particle energy spectrum in bulk semiconductors
(left). The single-particle energies for electrons (e) and holes (h) in small quantum dots
are shown in the right part of the figure.
20.3 Pair States
For the electron-hole-pair eigenstates, we make the ansatz
|ψ
eh
=
d
3
r
e
d
3
r
h
ψ
eh
(r
e
, r
h
)
ˆ
ψ
†
e
(r
e
)
ˆ
ψ
†
h
(r
h
) |0 (20.29)
and obtain the Schrödinger equation for the pair
−
2
2m
e
∇
2
e
−
2
2m
h
∇
2
h
−V (r
e
, r
h
)
ψ
eh
(r
e
, r
h
)=(E −E
g
)ψ
eh
(r
e
, r
h
) .
(20.30)
Because of the boundary conditions
ψ
eh
(r
e
, r
h
)=0if |r
e
| >R or |r
h
| >R , (20.31)
it is not useful to introduce relative and center-of-mass coordinates in con-
trast to the case of bulk or quantum-well semiconductor materials.
If the quantum dot radius is smaller than the bulk-exciton Bohr ra-
dius, R<a
0
, electron and hole are closer together than they would be in