Download - Quantum Wells Wires Dots- Lecture 8-2005
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2B 1700/2B1823, Advanced Semiconductor
Materials
Lecture 8, Quantum Wells, Quantum Wires and
Quantum Dots
Need for low dimensional structuresNeed for low dimensional structures
Carrier confinementCarrier confinement
Ballistic transportBallistic transportElastic scattering: Energy does not change between collisionsElastic scattering: Energy does not change between collisions
Inelastic scattering: Energy changes with collisionInelastic scattering: Energy changes with collision
Ballistic transport: At low enough dimensions (< average distancBallistic transport: At low enough dimensions (< average distanc e between two elastic scattering),e between two elastic scattering),
electrons travel inelectrons travel in straghtstraght lines => Light beams in geometrical opticslines => Light beams in geometrical optics
OutlineOutline
Quantum wells (Well with finite potential)Quantum wells (Well with finite potential)
Quantum wiresQuantum wires
Quantum dotsQuantum dots
=> High performance transistors and lasers
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Consider first the particle trapped in
an infinitely deep one- dimensional
potential well with a specific dimensionObservations
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 energylevels only and not in-between
Classical E-p curve is continuous. In
quantum mechanics, p = hk with k =
n/lwhere n = 1, 2, 3 etc.
En = h2k2/2m
= n22 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.
PARTICLE IN AN
INFINITE WELL
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One dimensionalfinite well
Region II:
A cos kx (symmetric solutions) (1)
II =A sin kx (antisymmetric solutions) (2)
where k2 = 2mE/h2 (3)
Region III:
III = Be-x where 2 = 2m(V0-E)/h
2 (4) and (5)
Region I:
I = Bex (eq. 6) but by symmetry we use only the
single boundary condition at x = l/2 between II and III
At x = l/2: II = III and II =
III (7)
For the symmetric solutions, i.e., for (1),
A cos (kl/2) = Be-l/2 (8)
Ak sin (kl/2) = Be- l/2 (9)
(9)/(8): k tan kl/2 = (10)For antisymmetric solutions, i.e., for (2),
k tan (kl/2 - /2) = (11)(In (11), cot x = - tan (x- /2) has been used to show the
similarity between symmetric and antisymmeteric
characteristic equations
(10) and (11) have electron energy
E on both sides via k and => onlydiscrete E values satisfy boundary
conditions (7)
PARTICLE IN A FINITE WELL
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PARTICLE IN A FINITE WELL
Observations
The wave functions are not
zero at the boundaries as inthe infinite potential well
Allowed particle energies
depend on the well depth
Finite well energy levels V0
Energy levels and wave functions
in a one dimensional finite well.
Three bound solutions are
illustrated
a) Shallow well with single allowedlevel kl= /4
b) Increase of allowed levels as kl
exceeds (kl= 3 + /4)
c) Comparison of the finite-well (solid
line) and infinite well (dashed line)
energies (kl = 8 + /4);
Infinite well
Finite well
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For infinite well case, En = n2 E1
(12)
where E1 = h2k1
2 /2m (13)
= 2h2/2ml2 (14)
Can we arrive at a similar relation for the
finite well case? YES
How?
Solve (10) and (11) using (12) with (3) & (5):
ENERGY LEVELS IN A FINITE WELL IN TERMS OF
THE FIRST LEVEL OF INFINITE WELL
Plot of quantum numbers as a function of the maximum allowed
quantum number which is determined by the potential height V0
Quantum number in the quantum well:nQW = (En/E1) (15)
Maximum number of bound states:
nmax = (V0/ E1) (16)
Example:
V0 = 25E1 => From (16), nmax = 5nQW = 0.886, 1.77, 2.65, 3.51, 4.33 (from (15) or from
figure
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RELATION BETWEEN ENERGY LEVELS IN A FINITE
WELL WITH THE FIRST LEVEL OF INFINITE WELL
Some figures:Some figures:
The energy spacing between the energy levels forThe energy spacing between the energy levels for
the quantum wells with thickness ~10 nm is a fewthe quantum wells with thickness ~10 nm is a few
1010s to a few 100s to a few 100ss meVmeV
At room temperatureAt room temperature kTkT ~ 26~ 26 meVmeV. This means. This means
only the first energy levels can be occupied byonly the first energy levels can be occupied by
electrons under typical device operationalelectrons under typical device operational
conditionsconditions
Example:Example:
VV00 = 25= 25EE11 => From (16),=> From (16), nnmaxmax = 5= 5
nnQWQW = 0.886, 1.77, 2.65, 3.51, 4.33 from (15)= 0.886, 1.77, 2.65, 3.51, 4.33 from (15)
or from the figureor from the figure
Plot of quantum numbers as a function of the maximum allowed
quantum number which is determined by the potential height V0
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Bound states as a function of well thickness
+=22
2
0
*
max
21
h
lVmIntn e
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Optical absorption/emission in the quantum wells
+=
2*
222
2*
222
22 lm
nE
lm
nEEE
h
i
V
e
i
C
V
i
C
i
hh
++=**2
222 11
2 he
i
g
mml
nE
h
+=
***
111
heehmmm meh
* = optical effective mass
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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)
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Quantum wires
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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
32
2/3*
3
)2(
h
zyxbe
t
LLLVmN =
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e/v
e/v+ s/e> s/v
Non-complete wetting:
s/v s/e
e/v
s/ee/v+ s/e< s/v
Epitaxial layer (e)
Complete wetting:
Substrate (s)
Growth modes
Layer-by-layer or
Frank - van der Merwe
2D+3D or
Stranski - Krastanow
3Dor
Volmer - Weber
e/v ands/v: surface energies of epimaterial and substrate, s/e: interface energy substrate/epimaterial
Courtesy: W.Seifert
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Quantum wire and dot fabrication
From
http://www.ifm.liu.se/Matephys/AAnew/resear
ch/iii_v/qwr.htm#S1.7
Formed from reorganisation of a
sequence of AlGaAs and strained
InGaAs epitaxial films grown on
GaAs (311)B substrates by MOCV
The size of the quantum dots are as
small as 20 nm
Coupled QWRs -Evidence fortunneling and electronic coupling
shown - Wire is GaAs, barrirer is
AlGaAs
Etched Quantum Dots By E Beam Lithograph
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Etched Quantum Dots By E-Beam Lithograph
E-beam lithography used forAu-liftoff etch mask
Mask size =15-22 nm
SiCl4/SiF4 RIE etch Dot Size= 15-25 nm
Dot Density = 3x1010cm-2
GaAsAlGaAs
AlGaAsGaAs
QW
Etched dots have poor optical quality Dot density is low
Device applications require regrowth Courtesy: P.Bhattacharya,Courtesy: P.Bhattacharya,University of MichiganUniversity of Michigan
esearc ers e o c eve a uan um o aser
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esearc ers e o c eve a uan um- o aser(Physics Today, May 1996)
K. Kamath, P. Bhattacharya, T. Sosnowksi, J. Phillips, and T. Norris, Electron. Lett., 30, 1374, 1996.
Room temperature quantum dot laser
Courtesy:Courtesy:
P.Bhattacharya,P.Bhattacharya,
University ofUniversity of
MichiganMichigan
Tunnel Injection QD Lasers Grown by MBE
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The laser heterostructures are grown by solid
source molecular beam epitaxy
The quantum dots are grown at 530C, the quantum
well is grown at 490C, and the rest of the structureat 630C
The high straindue to the In0.25Ga0.75As QW limits th
number of dot layers to less than 4.
The energy separation between the quantum well
injector layer ground state and quantum dot ground
state is tuned by adjusting the In and Ga charge in
Tunnel Injection QD Lasers Grown by MBE
Single mode ridge waveguide lasers
W=3mL=200-1300m
p-AlGaAs
n-AlGaAs
Quantum dots
Active region
650GaAs
1.5m p- Al0.55Ga0.45Ascladding layer
5m n- Al0.55Ga0.45Asadding layer
750 GaAs
95In0.25Ga0.75AsInjector well
In0.4Ga0.6Asquantum dots
18 GaAsbarriers
20 Al0.55Ga0.45Asbarrier
hLO
0
0.5
1
1.5
2
2.5
850 900 950 1000 1050
Quantum Dot(~980nm)
Injector(~950nm)
T=12K
urtesy:urtesy:
hattacharya,hattacharya,
iversity ofiversity ofchiganchigan
History of Heterostructure Lasers
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History of Heterostructure Lasers
10
100
1000
10000
100000
1000000
1960 1970 1980 1990 2000 2010
Year
ThresholdCurren
tDensity(A/cm
2)
GaAs pnQW Miller et. al.
QD Kamath et. al.Mirin et. al.Shoji et. al.
QD Ledenstov et. al.
QD Liu et. al.
QW Dupuis et. al.
QW Tsang
QW Alferov et. al.Chand et. al.
DHS
Alferovet. al.
DHS QWAlferov et. al.Hayashi et. al.
T=300K
DHS - Diode HeterostructureQW - Quantum WellQD - Quantum Dot
Courtesy:Courtesy:
P.Bhattacharya,P.Bhattacharya,
University ofUniversity of
MichiganMichigan