quantum wells wires dots- lecture 8-2005

8/3/2019 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

quantum wells wires dots- lecture 8-2005

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