# Advanced Semiconductor Materials Lecture 8, Quantum Wells, Quantum

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"Advanced Semiconductor Materials Lecture 8, Quantum Wells, Quantum"

```					                     Advanced Semiconductor Materials
Lecture 8, Quantum Wells, Quantum Wires and
Quantum Dots

Need for low dimensional structures
Carrier confinement
Ballistic transport        => High performance transistors and lasers
Elastic scattering: Energy does not change between collisions
Inelastic scattering: Energy changes with collision
distance
Ballistic transport: At low enough dimensions (< average distance between two elastic scattering),
electrons travel in straight lines => Light beams in geometrical optics

Outline
Quantum wells (Well with finite potential)
Quantum wires
Quantum dots

Sebastian Lourdudoss
PARTICLE IN AN
INFINITE WELL

Consider first the particle trapped in
an infinitely deep one- dimensional
potential well with a specific dimension
Observations
• Energy is quantized, Even the lowest
energy level has a positive value and
not zero
• The probability of finding the particle
is restricted to the respective energy
levels only and not in-between
• Classical E-p curve is continuous. In
quantum mechanics, p = hk with k =
nπ/l where n = ±1, ±2, ±3 etc.
π
En = h2k2/2m
= n2 π2 h2/2ml2
In fact the negative values are not
counted since the probability of finding
the electrons in n=1 and n=-1 is the
same and also E is the same at these
values
• When l is large, energies at En and
En+1 move closer to each other =>
classical systems, energy is continuous.
Sebastian Lourdudoss
PARTICLE IN A FINITE WELL

Region II:
A cos kx (symmetric solutions)           (1)
ΨII =
One dimensional
A sin kx (antisymmetric solutions)        (2)                                 finite well
where k2 = 2mE/h2
h                                  (3)
Region III:
ΨIII = Be-γx where γ2 = 2m(V0-E)/h2
γ                      h          (4) and (5)
Region I:
ΨI = Beγx (eq. 6) but by symmetry we use only the
single boundary condition at x = l/2 between II and III
At x = l/2: Ψ = Ψ and Ψ ´ = Ψ ´
II   III        II      III  (7)
For the symmetric solutions, i.e., for (1),                     • (10) and (11) have electron energy
A cos (kl/2) = Be-γ l/2
γ                         (8)                E on both sides via k and γ => only
Ak sin (kl/2) = Bγe-γ l/2
γ γ                        (9)                discrete E values satisfy boundary
(9)/(8): k tan (kl/2) = γ                    (10)               conditions (7)
For antisymmetric solutions, i.e., for (2),
k tan (kl/2 - π/2) = γ                 (11)                   From Coldren and Corzine,
(In (11), cot x = - tan (x- π/2) has been used to show the          Diode lasers and photonic
similarity between symmetric and antisymmeteric                     integrated circuits, Wiley,
characteristic equations                   Sebastian Lourdudoss  1995
PARTICLE IN A FINITE WELL

Observations
• The wave functions are not
zero at the boundaries as in
the infinite potential well
• Allowed particle energies
depend on the well depth
Infinite well
•Finite well energy levels <
Finite well
corresponding infinite well
energy levels
Energy levels and wave functions
• The deeper the finite well,    in a one dimensional finite well.
the better the infinite well        Three bound solutions are
approximation for the low-                  illustrated
lying energy values
• Quantum mechanical
tunnelling possible                                                  a) Shallow well with single allowed
• Quantum mechanical                                                 level kl = π/4
reflection possible at E>V0                                          N.B: k2 = 2mE/h2h
b) Increase of allowed levels as kl
exceeds π; here kl = 3π + π/4
π
c) Comparison of the finite-well (solid
line) and infinite well (dashed line)
energies; here kl = 8π + π/4
π
Sebastian Lourdudoss
ENERGY LEVELS IN A FINITE WELL IN TERMS OF
THE FIRST LEVEL OF INFINITE WELL

• For infinite well case, En = n2 E1∞        (12)
where   E1∞   = h2k12/2m                  (13)
= π2 h2/2ml2                 (14)
• Can we arrive at a similar relation for the
finite well case? YES
How?
Solve (10) and (11) using (12) with (3) & (5):

Quantum number in the quantum well:
nQW = (En/E1∞)½                             (15)
Maximum number of bound states:
nmax = (V0/ E1∞)½                            (16)

Example:
V0 = 25E1∞ => From (16), nmax = 5
(If necessary, round up to the nearest integer)
nQW = 0.886, 1.77, 2.65, 3.51, 4.33 from figure

From Coldren and Corzine,
Diode lasers and photonic
Plot of quantum numbers as a function of the maximum allowed
integrated circuits, Wiley,                              quantum number which is determined by the potential height V0
1995
Sebastian Lourdudoss
RELATION BETWEEN ENERGY LEVELS IN A FINITE
WELL WITH THE FIRST LEVEL OF INFINITE WELL

Example:
V0 = 25E1∞ => From (16), nmax = 5
nQW = 0.886, 1.77, 2.65, 3.51, 4.33 from the
figure

Some figures:
• The energy spacing between the energy levels for
the quantum wells with thickness ~10 nm is a few
10’           100’
10’s to a few 100’s meV
meV.
• At room temperature kT ~ 26 meV. This means
only the first energy levels can be occupied by
electrons under typical device operational
conditions

From Coldren and Corzine,
Diode lasers and photonic                           Plot of quantum numbers as a function of the maximum allowed
quantum number which is determined by the potential height V0
integrated circuits, Wiley,
1995                                     Sebastian Lourdudoss
Bound states as a function of well thickness

 2m V l 
*       2

n = 1 + Int     e   0
max

  π h 
2

2

Sebastian Lourdudoss
Optical absorption/emission in the quantum wells

    hπ n    2
hπ n 
2       2
hπ n
2        2       2       2   2   2
1   1
E − E = E +
C

i    i
V

C
−E −           =E +
i
V
i
g
i

m + m 
     2m l      2m l                                                                    
2         *   *
*
e
2
2l         *
h
2
e   h

1 1     1
= +          m = optical effective mass         *

 m m
*             *               *
m   eh             e               h
eh

Sebastian Lourdudoss
Density of states in the low dimensional
structures

Lower the dimension greater
the density of states near the
band edge
=> Greater proportion of the
injected carriers contribute to
the band edge population
inversion and gain (in lasers)

Sebastian Lourdudoss
Quantum wires

Sebastian Lourdudoss
Quantum dots

• Quantization in all the three directions
• With a finite potential, the problem can be treated as a spherical
dot like an atom of radius R with a surrounding potential
V (r) = 0 for r ≤ R and
= Vb for r ≥ R    Here r is the co-ordinate
• The solutions resemble those for the spectra of atoms
• Total number of states
*       3/ 2
( 2m V ) L L L
N =t
e   b              x   y   z

3π h        2      3

Sebastian Lourdudoss
Courtesy: W.Seifert
Sebastian Lourdudoss
Growth modes

Epitaxial layer (e)
γe/v
Complete wetting:
γs/e                                      Layer-by-layer or
γe/v + γs/e < γs/v
Frank - van der Merwe
Substrate (s)

2D+3D or
Stranski - Krastanow

γe/v
Non-complete wetting:
γs/v            γs/e                               3D or
γe/v + γs/e > γs/v                                                                    Volmer - Weber

γe/v and γs/v: surface energies of epimaterial and substrate, γs/e: interface energy substrate/epimaterial

Courtesy: W.Seifert
Sebastian Lourdudoss
Quantum wire and dot fabrication

Coupled QWRs -Evidence for
tunneling and electronic coupling
shown - Wire is GaAs, barrirer is
Formed from reorganisation of a
AlGaAs
sequence of AlGaAs and strained
InGaAs epitaxial films grown on
From                                                                                 GaAs (311)B substrates by MOCVD.
http://www.ifm.liu.se/Matephys/AAnew/research/iii_v/qwr.htm#S1.7                     The size of the quantum dots are as
small as 20 nm
Sebastian Lourdudoss
Nanorods

Courtesy: W.Seifert
Sebastian Lourdudoss
Sebastian Lourdudoss
Etched Quantum Dots By E-Beam Lithography

GaAs
AlGaAs   QW
AlGaAs
GaAs

• E-beam lithography used for
Au-liftoff etch mask                   • SiCl4/SiF4 RIE etch

• Mask size =15-22 nm                  • Dot Size= 15-25 nm
• Dot Density = 3x1010cm-2
• Etched dots have poor optical quality
• Dot density is low
• Device applications require regrowth                       Courtesy: P.Bhattacharya,
University of Michigan
Sebastian Lourdudoss
Researchers Vie to Achieve a Quantum-Dot laser
(Physics Today, May 1996)

Room temperature quantum dot laser

Courtesy:
P.Bhattacharya,
University of
Michigan

K. Kamath, P. Bhattacharya, T. Sosnowksi, J. Phillips, and T. Norris, Electron. Lett., 30, 1374, 1996.
Sebastian Lourdudoss
Tunnel Injection QD Lasers Grown by MBE
Active region
Single mode ridge waveguide lasers
1.5µm n- Al0.55Ga0.45As                         20Å Al0.55Ga0.45As 1.5µm p- Al0.55Ga0.45As          W=3µm
L=200-1300µm
18Å GaAs
650Å                      barriers      750Å GaAs
GaAs

p-AlGaAs      Quantum dots
95Å                                                           hωLO
ω
Courtesy:                          In0.25Ga0.75As
P.Bhattacharya,                    Injector well
University of
In0.4Ga0.6As                                           n-AlGaAs
Michigan
quantum dots

2.5
T=12K          Quantum Dot(~980nm)               • The laser heterostructures are grown by solid
2                                                          source molecular beam epitaxy
PL Intensity (a.u.)

C,
• The quantum dots are grown at 530° the quantum
1.5               Injector
(~950nm)                                                       C,
well is grown at 490° and the rest of the structure
at 630°  C
1
• The high strain due to the In0.25Ga0.75As QW limits the
number of dot layers to less than 4.
0.5
• The energy separation between the quantum well
injector layer ground state and quantum dot ground
0
850       900          950          1000        1050     state is tuned by adjusting the In and Ga charge in
Sebastian Lourdudoss
QD
Wavelength (nm)
History of Heterostructure Lasers

1000000
DHS - Diode Heterostructure
Threshold Current Density (A/cm2)
QW - Quantum Well
GaAs pn                                 QD - Quantum Dot
100000                        QW Miller et. al.

T=300K
10000
DHS
QW Dupuis et. al.
Alferov
et. al.                                       QD Kamath et. al.
1000
DHS QW                                        Mirin et. al.
Alferov et. al.                               Shoji et. al.
QW Tsang
Hayashi et. al.
100
QD Ledenstov et. al.
QW Alferov et. al.                 QD Liu et. al.
Chand et. al.
10
1960       1970          1980        1990            2000        2010
Courtesy:
Year
P.Bhattacharya,
University of
Michigan
Sebastian Lourdudoss

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