Handout 26 2D Nanostructures: Semiconductor Quantum...

1

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Handout 26

2D Nanostructures: Semiconductor Quantum Wells

In this lecture you will learn:

• Effective mass equation for heterojunctions• Electron reflection and transmission at interfaces• Semiconductor quantum wells• Density of states in semiconductor quantum wells

Leo Esaki (1925-)Nobel Prize

Nick Holonyak Jr. (1928-) Charles H. Henry (1937-)


Transmission and Reflection at Heterojunctions

The solution is:

1221

1221

1221 11

12

xxxx

xxxx

xxxx kmkmkmkm

rkmkm

t

Where:

Special case: If the RHS in the above equation is negative, then kx2 becomes imaginary and the wavefunction decays exponentially for x>0 (in semiconductor 2). In this case:

and the electron is completely reflected from the hetero-interface

1r

1cE2cE

cE

x0

r t

zyeffx

x

x

x

zz

z

yy

yc

x

x

x

x

kkVmk

mk

mmk

mm

kE

mk

mk

,22

112

11222

1

21

2

2

22

2

12

22

12

22

1

21

2

2

22

2

2


Semiconductor Quantum Wells

1cE2cE

cE2cE

1vE

2vE2vE

AlGaAs AlGaAsGaAsA thin (~1-10 nm) narrow bandgap material sandwiched between two wide bandgap materials

GaAs

GaAsInGaAs quantum well (1-10 nm)

Semiconductor quantum wells can be composed of pretty much any semiconductor from the groups II, III, IV, V, and VI of the periodic table

TEM micrograph

GaAs

GaAs

InG

aA

s


Semiconductor Quantum Well: Conduction Band Solution

1cE

2cE

cE2cE

x

0Assumptions and solutions:

e

cc mk

EkE2

22

11

e

cc mk

EkE2

22

22

rErEm

rEriE

ce

c

111

22

111

2

ˆ

rErEm

rEriE

ce

c

222

22

222

2

ˆ

zkykix

zkykix

zy

zy

exk

exkAr

sin

cos1

2

2

2

2 Lxee

eeBr zkykiLx

zkykiLx

zy

zy

2

2

2

2 Lxee

eeBr zkykiLx

zkykiLx

zy

zy

L

Symmetric

Anti-symmetric

3


22

2||

22

2

2||

22

1

2

22

xce

ec

e

xc

kEm

m

kE

m

kkEE

1cE2cE

cE2cE

x

0

L

Energy conservation condition:

The two unknowns A and B can be found by imposing the continuity of the wavefunction condition and the probability current continuity condition to get the following conditions for the wavevector kx:

x

xce

x

x

x

xce

x

x

k

kEm

kLk

k

kEm

kLk

22

22

2

2cot

2

2tan

Wavevector kx cannot be arbitrary!Its value must satisfy these transcendental equations

222|| zy kkk

Semiconductor Quantum Well: Conduction Band Solution


1cE

2cE

cE2cE

x

0

L

x

xce

x

x

x

xce

x

x

k

kEm

kLk

k

kEm

kLk

22

22

2

2cot

2

2tan

2Lkx2

23 2

250

Different red curves for Increasing Ec values

Graphical solution:

• Values of kx are quantized• Only a finite number of solutions are possible – depending on the value of Ec

In the limit Ec ∞ the values of kxare:

Lpkx ( p = 1,2,3……..

Semiconductor Quantum Well: Conduction Band Solution r

4


Electrons in Quantum Wells: A 2D Fermi Gas

1cE

2cE

cE2cE

x

0

L

epc

ee

xc

m

kEE

m

k

mk

EE

2

222||

2

1

2||

222

1

Since values of kx are quantized, the energy dispersion can be written as:

p = 1,2,3……..

• We say that the motion in the x-direction is quantized (the energy associated with that motion can only take a discrete set of values)• The freedom of motion is now available only in the y and z directions (i.e. in directions that are in the plane of the quantum well)• Electrons in the quantum well are essentially a two dimensional Fermi gas!

1E

2E

In the limit Ec ∞ the values of Ep are:22

2

Lp

mE

ep

p = 1,2,3……..

222|| zy kkk


1cE

2cE

cE2cE

x

0

L1E

2E

Energy Subbands in Quantum Wells

e

pcc m

kEEkpE

2,

2||

2

1||

p =1,2,3……..

The energy dispersion for electrons in the quantum wells can be plotted as shown

It consists of energy subbands (i.e. subbands of the conduction band)

Electrons in each subband constitute a 2D Fermi gas||k

1cE11 EEc

21 EEc

E

kz

ky

31 EEc

E

222|| zy kkk

5


1cE

2cE

cE2cE

x

0

L1E

2E

Density of States in Quantum Wells

Suppose, given a Fermi level position Ef , we need to find the electron density:We can add the electron present in each subband as follows:

pfc EkpEf

kdn ||2

||2

,2

2

fEIf we want to write the above as:

fQWE

EEfEgdEnc

1

Then the question is what is the density of states gQW(E ) ?||k

1cE

11 EEc

21 EEc 31 EEc

||k

1cE

11 EEc

21 EEc 31 EEc


Density of States in Quantum Wells

fE

||k

1cE

11 EEc

21 EEc 31 EEc

||k

1cE

11 EEc

21 EEc 31 EEc

pfc EkpEf

kdn ||2

||2

,2

2

Start from:

e

pcc m

kEEkpE

2,

2||

2

1||

And convert the k-space integral to energy space:

fp

pce

E

pf

e

EE

EEfEEEm

dE

EEfm

dEn

c

pc

12

2

1

1

This implies:

ppc

eQW EEE

mEg 12

EgQW

1cE 11 EEc 21 EEc 31 EEc

2em

22

em

23

em

6


Density of States: From Bulk (3D) to QW (2D)

The modification of the density of states by quantum confinement in nanostructures can be used to:

i) Control and design custom energy levels for laser and optoelectronic applicationsii) Control and design carrier scattering rates, recombination rates, mobilities, for electronic applications iii) Achieve ultra low-power electronic and optoelectronic devices

Eg D3k

1cE

E E E

Eg D2

2em

22

em

23

em

||k

1cE

11 EEc

21 EEc

31 EEc

EE

Eg D2

2em

22

em

23

em

||k

1cE

11 EEc

21 EEc

31 EEc

E

EgQW


Semiconductor Quantum Well: Valence Band Solution1vE

2vEvE2vE

x

0Assumptions and solutions:

h

vv mk

EkE2

22

11

v

vv mk

EkE2

22

22

rErEm

rErEm

rEriE

vh

vh

v

111

22

111

22

111

2

2

ˆ

rErEm

rErEm

rEriE

vh

vh

v

222

22

222

22

222

2

2

ˆ

zkykix

zkykix

zy

zy

exk

exkAr

sin

cos1

2

2

2

2 Lxee

eeBr zkykiLx

zkykiLx

zy

zy

2

2

2

2 Lxee

eeBr zkykiLx

zkykiLx

zy

zy

L

Symmetric

Anti-symmetric

7


22

2||

22

2

2||

22

1

2

22

xvh

ev

xv

kEm

m

kE

mh

kkEE

Energy conservation condition:

The two unknowns A and B can be found by imposing the continuity of the wavefunction condition and the probability current conservation condition to get the following conditions for the wavevector kx:

x

xvh

x

x

x

xvh

x

x

k

kEm

kLk

k

kEm

kLk

22

22

2

2cot

2

2tan

Wavevector kx cannot be arbitrary!

1vE

2vEvE2vE

x

0

L

Semiconductor Quantum Well: Valence Band Solution


1vE

2vE

vE

2vE

x

0

L

x

xvh

x

x

x

xvh

x

x

k

kEm

kLk

k

kEm

kLk

22

22

2

2cot

2

2tan

2Lkx2

23 2

250

Different red curves for Increasing Ev values

Graphical solution:

• Values of kx are quantized• Only a finite number of solutions are possible – depending on the value of Ev

In the limit Ev ∞ the values of kxare:

Lpkx ( p = 1,2,3……..

Semiconductor Quantum Well: Valence Band Solution

8


1vE

2vE

vE

2vEx

0

L

hpv

hh

xv

m

kEE

m

k

mk

EE

2

222||

2

1

2||

222

1

Since values of kx are quantized, the energy dispersion can be written as:

p = 1,2,3……..

• We say that the motion in the x-direction is quantized (the energy associated with that motion can only take a discrete set of values)• The freedom of motion is now available only in the y and z directions (i.e. in directions that are in the plane of the quantum well)• Electrons (or holes) in the quantum well are essentially a two dimensional Fermi gas!

1E

2E

In the limit Ev ∞ the values of Ep are:22

2

Lp

mE

hp

p = 1,2,3……..

Semiconductor Quantum Wells: A 2D Fermi Gas

Light-hole/heavy-hole degeneracy breaks!


Density of States in Quantum Wells: Valence Band

fE

pfv EkpEf

kdp ||2

||2

,12

2

Start from:

h

pvv m

kEEkpE

2,

2||

2

1||

And convert the k-space integral to energy space:

fp

pvh

E

pf

hEE

EEfEEEm

dE

EEfm

dEp

v

pv

1

1

12

2

1

1

This implies:

ppv

hQW EEE

mEg 12

EgQW

1vE11 EEv 21 EEv 31 EEv

2hm

22

hm

23hm

||k

1vE

11 EEv

21 EEv

31 EEv

E

9


photon

stimulated andspontaneousemission

non-radiativerecombination

holes

electrons

A quantum well laser (band diagram)

N-doped P-doped

metal

A ridge waveguide laser structure

Example (Photonics): Semiconductor Quantum Well Lasers

All lasers used in fiber optical communication systems are semiconductor quantum well lasers

Some advantages of quantum wells for laser applications:• Low laser threshold currents due to reduced density of states• High speed laser current modulation due to large differential gain• Ability to control emission wavelength via quantum size effect


InP

850 nm

980 nm

1300 nm1550 nm

Compound Semiconductors and their Alloys: Groups IV, III-V, II-VI

CdS

10


Example (Electronics): Silicon MOSFET

cE

1EEc fE

NMOS Band diagram

Inversion layer(2D Electron gas)

2EEc

100 nm

A 50 nm gate MOS transistor

Si

SiO2

x

y

z x

x


cEEEc fE

NMOS Band diagram

tc EE

Si

SiO2

f(x)

cEEEc fE

NMOS Band diagram

tc EE

Si

SiO2

f(x)


zikzik

ctt

zyexfr

rErrUEzmymxm

2

22

2

22

2

22

222

t

z

t

yc

t

z

t

yc

mk

m

kEEE

xfmk

m

kEExfrU

xm

22

222

2222

2222

2

22

For minima 1 and 2:

For minima 3 and 4:

1 2

3

4

5

6

zikzik zyexgr

EEt

x

x

y

z

t

zytc

t

zyc

t

mk

m

kEEE

xgmk

m

kEExgrU

xm

22

222

2222

2222

2

22

11


cEEEc fE

NMOS Band diagram

tc EE

Si

SiO2

f(x)

cEEEc fE

NMOS Band diagram

tc EE

Si

SiO2

f(x)


For minima 5 and 6:

1 2

3

4

5

6

zikzik zyexgr

x

y

z

x

mk

m

kEEE

xgmk

m

kEExgrU

xm

z

t

ytc

z

t

yc

t

22

222

2222

2222

2

22

Advantage of Quantum Confinement and Quantization:

• As a result of quantum confinement the degeneracy among the states in the 6 valleys or pockets is lifted

• Most of the electrons (at least at low temperatures) occupy the two valleys (1 & 2) with the lower quantized energy (i.e. Eℓ )

• Electrons in the lower energy valleys have a lighter mass (i.e. mt) in the directions parallel to the interface (y-z plane) and, therefore, a higher mobility

EEt


Example (Electronics): HEMTs (High Electron Mobility Transistors)

GaAs (Substrate)

Source

AlGaAs

DrainGate

GaAs

AlGaAs

Metal

InGaAsQW

GaAs

AlGaAs

Metal

InGaAsQW

Band diagram in equilibrium

The HEMT operates like a MOS transistor:The application of a positive or negative bias on the gate can increase or decrease the electron density in the quantum well channel thereby changing the current density

Modulation doping

InGaAs (Quantum Well)

Handout 26 2D Nanostructures: Semiconductor Quantum...

Documents

Transcript of Handout 26 2D Nanostructures: Semiconductor Quantum...