Introduction to Quantum Properties of Solid State...

Introduction to Quantum Properties

of Solid State Devices

Please pay attention that these are only lectures notes

prepared for using in the classroom by the instructor. For

more detail refer to any textbook related to physics of solid

state devices.

Principles of Quantum Mechanics

1) The principle of Energy Quanta:

Experiments which showed inconsistency

between experimental results and classical

theories:


Thermal Radiation

• Classical Physics: the frequency of the radiation

of a heated cavity should increase monotonically

as the thermal energy of the cavity increases.

• Observations: the frequency of the radiation of a

heated cavity increases only up to a certain

energy then decreases.


When a particle jumps from one

discrete state to another, it either

absorbs or emits energy quanta (most

frequently it is a photon) which

corresponds to a particular wavelength

associated with the difference of the two

states energy

Max Planck’s postulate (1900):

explains blackbody radiation

hE

Typical light intensity vs its freq. when

the object heated to a certain

temperature, e.g.6000 K


Hydrogen Atoms:• Classical Physics: The atomic spectra should be

continuous.

• Observations: The atomic spectra contain only some discrete lines.

Photoelectric effect:• Classical Physics: The absorption of optical energy by

the electrons in a metal is independent of light frequency.

• Observations: The absorption of optical energy by the electrons in a metal depends on optical frequency.


• Planck’s postulate in 1900 that radiation

from a heated sample is emitted in

discrete units of energy, called quanta.

E =hυ

h is the Planck’s constant, υ is frequency of the radiation


Photoelectric effect:• Classical Physics: The

absorption of optical energy by the electrons in a metal is independent of light frequency.

• Observations: The absorption of optical energy by the electrons in a metal depends on optical frequency.

• Einstein in 1905 took a step further to interpret the photoelectric results by suggesting that the electromagnetic radiation exists in the form of packets of energy. This particle-like packet of energy is called photon.


2) Wave-Particle Duality Principle

The nature of light in the following observations:

• Photoelectric Effect (particle-like)

• Compton Effect (particle-like)

• Diffraction pattern by electrons (wave-like)

• Since waves behave as particles, then particles

should be expected to show wave-like

properties.

• Louis de Broglie hypothesized that the

wavelength of a particle can be expressed as

=h/p

• The momentum of a photon is then: p = h/ =ħk


3) The Uncertainty Principle (Heisenberg)

• First statement:

• It is impossible to simultaneously describe with absolute accuracy the position and momentum of a particle.

• x p ħ

• ħ is very small so this principle only applies to very small particles

• Second statement:

• It is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy.

• E t ħ

A Particle in a Box

Classical Mechanics vs. Quantum Mechanics

• CM: The particle remains trapped inside the box and can acquire any energy available to it .

• QM: A particle trapped in such a box cannot have any energy, and will have only certain allowed energy.

• CM: If no other forces are in the system, then the particle should haveBrownian motion and has an equal probability of being anywhere andeverywhere in the box.

• QM: The particle has certain preferable points, where the probability offinding the particle is more than in other parts.

• CM: The exact position and momentum of the particle should be obtainablesimultaneously for the particle in question.

• QM: We cannot measure the momentum and the position of the particle inthe box simultaneously without some uncertainty.

A Particle in a Box

Classical Mechanics vs. Quantum Mechanics

• CM: At a particular moment, the exact energy of a particle should bemeasurable.

• QM: At any instant of time, the energy is not measurable above acertain extent.

• CM: The particle must be confined in the box and has no chance ofescaping the box as long as its energy is lower than that of the box.

• QM: The particle has a finite chance of escaping the system evenwith energy lower than the box.

• CM: If by any chance the particle attains energy larger than that ofthe box, the particle will certainly leave the box.

• QM: If by any chance the particle attains energy larger than that ofthe box, there remains certain probability that particle might still betrapped inside.

Principle of Quantum Mechanics

Basic Postulates

– Late 1920’s Ervin Schrodinger developed

wave mechanics based on Planck’s theory

and de Broglie’s idea of the wave nature of

matter.

– Warner Heisenberg formulated an alternative

approach in terms of matrix algebra called

Matrix Mechanics.


Basic Postulates

Dynamical Variables in Physics: position, momentum, energy

• In classical mechanics the state of a system with a number of particles at any time is defined by designating the particle and momentum coordinates of all particles.

• In quantum mechanics the state of a system is defined by a state function Ψ that contains all the information we can obtain about the system.

• To approach quantum mechanics we consider several postulates that are assumed to be true


Basic Postulates

• Postulate I: (in one-dimensional system)

– There exists a state function Ψ(x;t) which contains all the measurable information about each particle of a physical system. (x;t) is also called wave function.

• Postulate II:

– Every dynamical variable has a corresponding operator. This operator is operated on the state function to obtain measurable information about the system.


Basic Postulates

In a one-dimensional system

Dynamical Variables:

• Position x

• Momentum p

• Total Energy E

• Potential Energy V(x)


Basic Postulates

Dynamical Variable Quantum Mechanic Operator

Position x

Momentum p

Total Energy E

Potential Energy V(x)

XX ˆ

XiPx

ˆ

tiE

ˆ

)()(ˆ xVxV


Basic Postulates

• Postulate III:– The state function Ψ(x;t) and its space derivative

/x must be continuous, finite and single-valued for

all values of x.

• Postulate IV

The state function must be normalized i.e

– Ψ* is the complex conjugate of Ψ. Obviously Ψ Ψ* is

a positive and real number and is equal to /Ψ/².

1

dx


Basic Postulates

• Assume a single particle-like electron;

then Ψ Ψ* is interpreted as the statistical

probability that the particle is found in a

distance element dx at any instant of time.

Thus Ψ Ψ* represents the probability

density.


Basic Postulates

Postulate V.• The average value Q of any variable corresponding to the state

function is given by expectation value:

• Q is the expected value of any observation.

• Once the state function corresponding to any particle is found, it is

possible to calculate the average position, energy and momentum of

the particle within the limit of the uncertainty principle.

dxQQ


Schrodinger Equations

Assume:

where

Recall the operators were defined in postulate II;

),,(2

2

zyxVm

pE

2222

zyx pppp

222

22

mm

p ),,(),,( zyxVzyxV

ti

E

Schrodinger Equations:

The potential energy term forms the necessary link between quantum

mechanics and the real world. It contains all environmental factors

that could influence the state of a particle.

Note than p(r)= * = * i.e. probability density is time -independent and define an stationary state

trrVm

trt

i ,2

, 22

Time dependant Schrödinger’s Equation

single-particle 3D time-dependent

Schrödinger equation

Let’s assume :

Or

And

Time-independent

Schrödinger equation

Consequences

1

2the expectation value for any

time-independent operator is

also time-independent

The general solution to equation will be a linear

combination of these particular solutions

Schrödinger’s Equation in 3D

zUyUxUrU zyx

if

separate the 3D equation

into three sets of decoupled

1D equation and solve

these three 1D equations

independently

zZyYxXr

xXxUdx

d

mxXE xx

2

22

2

yYyUdy

d

myYE yy

2

22

2

zZzUdz

d

mzZE zz

2

22

2

plane wave

If there is no force acting on the particle, then the potential function V(r) will be constant

and we must have E>V(r).

assume V(r)=0

0

222

2

x

mE

x

x

mEixmEix BeAex 22 tEiet

time dependant part

EtmEx

iEtmEx

i

BeAetx22

,

tkxiAetx ,in +x direction

k=2π/λ λ=h/√(2mE)or

λ=h/p

a free particle with a well

defined energy will also

have a well defined

wavelength and

momentum.

ψ(x,t)ψ*(x,t)=AA* is a constant independent of position. The probability density function

a free particle with a well defined momentum can be found

anywhere with equal probability.

Particle in a box

1D Infinite Quantum Well

Discrete X axis

Functio

n f

(x)

U=0

U=8 U=8

X=1

X=0

X=n

X=n+1

0

222

2

x

mE

x

x

V(x) =0 inside the box

kxAkxAx sincos 21

2

2

mEk is the wave number.

A particular form of

solution

First boundary condition,

second boundary condition

x=0, ψ(0)=0 then A1=0 kxAx sin2

x=a, ψ(a)=0 0sin2 kaAa

k=nπ/a

from the normalization condition 1sin0

22

2 dxkxA

a

A2=√(2/a)

Final solution

a

xn

ax

sin

2

2

222

2

22

2 2

2

ma

nEE

a

nmEn

energy for

this system is

quantized

Particle in a box…

p = h/ =ħk

Particle in a box

n=1

n=2

n=3

n=4

Eψ

n

|ψn|2

x=

0

x=

0

x=

a

x=

a(a) (b) (c)

Particle in an infinite potential well:

a) Four lowest discrete energy levels;

b) Corresponding wave functions

c) Corresponding probability functions

Reflection: Step potential

0

2122

1

2

x

mE

x

x

T.I.S.E

011

111

xeBeAxxjkxjk

21

2

mEk

vi and vr are the magnitude of the incident and reflected wave velocity.

0

22022

2

2

xEV

m

x

x

region II the Schrödinger

equation (E<Vo)

022

222

xeBeAxxkxk

2

02

2

EVmk

B2=0 is zero due to fact that function must be

finite everywhere and potential is a step function xk

eAx 2

22

boundary x=0 ψ1(0)= ψ2(0) A1+B1=A2. 0

2

0

1

xx xx

jk1A1-jk1B1=-k2A2

22

1

2

2

21

2

1

2

221

2

1

2

2

*

11

*

11 22

kk

kjkkkkjkkk

AA

BB

Continue

kinetic energy K.E.=mv2/2

mvvmmv

mmEk

2

222

21

221

22

0.14

22

1

2

2

2

2

2

1

22

1

2

2

*

11

*

11

kk

kkkk

AAv

BBvR

i

r

Barrier potential and tunneling

quantum particles can tunnel

trough the barrier and get out on

the other side

E<U0 for left-

side incident

particle

k0 = p√(2mE)/ħ,

Tunneling

For wide barriers,

The tunneling rate is a very sensitive function of L

222

4

2)4( n

qmE

o

o

n

e

0/

2/3

0

100

11 arq

a

Energy States & Energy Bands

Pauli’s Exclusion Principle

1

s

2

s

2

p

1

s

2

s

2

p

1

s

2

s

2

p

1s

2s

2

p

1

s

The potential wells due to the interactions

between 2 atoms (in one molecule). Some

electrons are shared between the atoms. Due

to the interactions between electrons-

electrons, nucleons-nucleons, and electrons-

nucleons, the energy levels split, creating 1s,

2s, 2p,… doublets.

The potential experienced by an electron due to the coulomb

interactions around an atom. 1s, 2s, 2p,… are the energy

levels that the electron can occupy.

Larger molecules, larger splitting.

In Solid with n≈ 10e23 atoms, the sublevels are extremely close to each other.

They coalesce and form an energy band. 1s, 2s, 2p… energy bands.

Classification of Solid Structures

Amorphous: Atoms (molecules)

bond to form a very short-range

(few atoms) periodic structure.

Polycrystalline: made of pieces of

crystalline structures (called grain)

each oriented at different direction

(intermediate-range-ordered)

Crystals: Atoms (molecules) bond to

form a long-range periodic structure.

The constant bonds (coordination) ,

bond distance and angles between

bonds are the characteristics of a

crystal structure

Represents an atom or a

molecule

An IDEAL CRYSTAL is constructed by the infinite repetition of identical

structural units in space.

Crystals

Crystalline Structures

(b)

Lattice Structures

• A LATTICE represents a set of points in space that form a periodic structure. Each point sees exactly the same environment. The lattice is by itself a mathematical abstraction.

• A building block of atoms called the BASIS is then attached to each lattice point yielding the crystal structure.

• LATTICE + BASIS = CRYSTAL STRUCTURE

• The identical structure units that have small volume are called UNIT CELL.

O

O'

A 2-D lattice showing translation of a unit cell by R = 4a + 2b

Crystal lattices

Cubic P Cubic I Cubic F

Tetragonal

P

Tetragonal I

Orthorhombi

c P

Orthorhombi

c COrthorhombi

c I

Orthorhombi

c F

Monoclinic P Monoclinic C Triclinic

Trigonal R Trigonal and

Hexagonal R

a

c

b

Lattice Structures

Lattice Constant

Side diagonal

Body diagonal

a

2a

3a

Simple CubeSC

Body Centred CubeBCC

Face Centred CubeFCC

Example of FCC Structure

(a) The crystal structure of copper

is face centered cubic (FCC).

The atoms are positioned

at well defined sites arranged

periodically and there is a

long range order in the

crystal.

(b) An FCC unit cell with closed

packed spheres.

(c) Reduced sphere representation

of the unit cell. Examples: Ag,

Al, Au, Ca, Cu, Ni.

Number of atoms per unit cell

8 corners = 8 × (1/8)=1

6 faces = 6 × (1/2) = 3

Z

X

y

Z

X

y

Z

X

y

Z

X

y

(110)(100)

(111) (200)

Some popular lattice planes

Crystals

Parameters that characterize a unit cell:

• Lattice structure (e.g. cubic, tetragonal, etc.).

• Basis.

• Number of atoms in a unit cell.

• Crystal planes e.g. {100}, {110}, {111}, Miller Indices.

• Number of atoms in each plane.

• Chemical binding (e.g. metallic, covalent, …).

• Number of nearest (nn) and next nearest (nnn) atoms to each atom.

ab

c

x y

z

(100)

Example of (100)

Example of (110)

ab

c

x y

z

Example of (111)

ab

c

x y

z

Diamond Structure

• Top view of an extended (100) plane of the diamond lattice

structure.

100

101

000

001011

010

110

111

z

y

x

The unit cell of diamond (zinc blend) lattice structure. The

position of each lattice point is shown with respect to the 000

lattice point.

43

41

43

2

10

2

1

41

41

41

43

43

41

41

43

43

021

21

a

b

c

xy

z

a/2

a

a

Common Planes

• {100} Plane

• {110} Plane

• {111} Plane

a

a

a – Lattice Constant

For Silicon

a = 5.34 Ao

Two Interpenetrating Face-Centered Cubic Lattices

Silicon – Diamond Structure

Some Properties of Some Important Semiconductors

Compound Eg Gap(eV) Transition λ(nm) BandgapDiamond 5.4 230 indirect

ZnS 3.75 331 direct

ZnO 3.3 376 indirect

TiO2 3 413 indirect

CdS 2.5 496 direct

CdSe 1.8 689 direct

CdTe 1.55 800 direct

GaAs 1.5 827 direct

InP 1.4 886 direct

Si 1.2 1033 indirect

AgCl 0.32 3875 indirect

PbS 0.3 4133 direct

AgI 0.28 4429 direct

PbTe 0.25 4960 indirect

Periodical Potential

r

r

Non-Periodical Potential

r

r

)(rV

2)(r

)(rV

2)(r

jkrerur )()(

The state function for electrons in a single crystal has a form of:

rkj

kk erUr .)()(

where

)()(

RrUrU kk

is the periodic potential in single crystals of semiconductors. R = aX + bY +cZ, a, b, and c are integers.

)()()()( ...).(

reeerUeRrURr k

RkjRkjrkj

k

Rrkj

kk

Therefore:

22

)()(

rRr kk

Which means that the probability of finding electrons is periodic.

Potential Function

The Concepts of:

1) Effective Mass

*2

2

2

11

mdk

Ed

2) Negative Mass

3) Positive charge

4) Holes

Density of states

Pauli exclusion principle, only one electron can occupy a given quantum state.

The number of carriers that can contribute to the

conduction process is a function of the number of

available energy or quantum states

density of electronic quantum energy states are called density of

states (DOS).

Bulk density of states

zyxV

zyxV

,,

0,,

elswhere

azayax 0,0,0for

2

22222222

2

2

annnkkkk

mEzyxzyx

Schrödinger’s equitation solution

the distance between two states in k space

aan

ankk xxxx

11

the differential density of quantum states in k space

3

2

2

3

2

81

42 a

dkk

a

dkkdkkgT

converting the states into energy space

mEk

mEk

222

2

dEE

mdk

2

1

dEEmh

adE

E

mmEadEEgT

2

3

24

2

123

3

22

3

Eh

mEg

3

23

24

Density of

states in K

space

Density of states

in energy space

EEh

mEg

EEh

mEg

v

p

v

cn

c

3

2/3*

3

2/3*

)2(4)(

)2(4)(

Density of State.

Electrons Distribution.

Density of states in thin films: 2D structures

zyppx EEEEEE

2

222222

2

2

annkkk

mEzyzy

if

2D system to give a 2D plane of k

values

2

241

22 a

kdk

a

kdkdkkgT

dEh

madE

E

mmEadEEgT 2

2

2

2 4

2

12

converting the k

space into energy

space

2

4

h

mEg

Density of state in Nanowire: 1D structure

dispersion relation

2

2222

2

2

ankk

mEzz

adk

a

dkdkkgT

2

12

length associated with

every quantum state Lk=π/a

dEE

m

h

adE

E

madEEgT

2

2

1

E

m

hEg

21

Converting the k space

to energy space

Density of states

Energy States vs Size of Structures.

kT

EEEf

F

F

exp1

1)(

Fermi-Dirac Distribution:

E EF

0

1/2

1.0

T = 0

T = T1

T = T2 > T1

The Fermi probability function versus energy for

differential temperatures.

f F(E

)

kT

EEEf

F

F

exp1

1)(

Density and Current operator

*2 yprobabilit 13* rd

many body system Fermi-Dirac

distribution

mmm EfEn 2

1

2

m

mmm EfdEEnn

current conservation, more popularly

known as the continuity equation trJtrt ,,

A particle of charge e

2 e

tte

te

te

t

**

*2

from Schrödinger’s equation trrVm

trt

i ,2

, 22

Multiplying both sides by ψ*(r,t).e/iħ

trrVtretrtrm

eitrt

tre i ,,,,2

,, *2**

For the second term we take the complex conjugate and multiply both sides by

ψ(r,t)

Jm

eirVrVi

etr

meitrtr

meitr

t

***

*22*

2,

2,,

2,

Finally **

2

meiJ

kT

EEnn FiF

i

)(exp0

kT

EEnp FiF

i

)(exp0

Carrier Transport:

Current in Semiconductors: (Drift, Diffusion, Carriers Rcommendation,

Carriers Generation)

Drift Current:

EpnqJ pndrf )(

dx

dpqDJ

dx

dnqDJ

ppdif

nndif

Diffusion Current

)()(dx

dpD

dx

dnDqEpnqJ pnpntotal

Under Equilibrium Condition J(total) = 0.

q

kTDD

p

p

n

n

Einstein Relation

kT/q at room temperature = 0.0259 V

pono

pn

tntnRR

)()(

Generation - Recombination

t

nng

x

En

x

nE

x

nD

t

ppg

x

Ep

x

pE

x

pD

n

nnn

p

ppp

)(

)(

2

2

2

2

Continuity Equations

Quantum Mechanics in nanostructures: some issues

1 Validity of effective mass approximation

2 Doping and effects in nanostructures

3 Surface Effects

4 Recombination

5 Hot Carrier Effects

6 Exciton

7 Density of States and Coulomb Blockade

Validity of effective mass approximation

Crystal can be thought to be effectively made of only periodic atoms

applies Bloch’s theorem and uses Krönig-Penny model to treat the macroscopic potential due

to applied bias or macroscopic space charge, and gets the band diagrams of the material.

lowest parts of the conduction band, is approximated parabolic.

.

an electron in an external electric field E

acceleration

reduced Planck's constant

wave number

From the

external

electric field

alone,

charge.

This concept of effective mass is valid for bulk materials where we can

assume the dispersion relation to be periodic, but in nanostructures the

periodicity is no longer there as dimensions get smaller. Hence the band

diagram also fails for nanostructure and we can no longer use the effective

mass approximation. Experiments have shown that below 5 nm, we can no

longer use the effective mass approximation.

http://en.wikipedia.org/wiki/Planck%27s_constant

http://en.wikipedia.org/wiki/Wave_number

Doping and effects in nanostructures

In nanostructure, the semiconductor material could be small, specially in quantum dots,

and a substitutional impurity should no longer be regarded as doping and a full quantum

mechanical treatment is needed. But it has been experimentally found that still the

concept of doping does remain somewhat valid in the range of QDs and p doped or n

doped QD are readily available.

Ec

Ev

Ef

Ea

+3

Al

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

Eg Valence hole

+5

As

+4

Si+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

+4

Si

Eg

Ec

Ev

Ef

Ed

Conduction

electron

Surface Effects

In bulk material, surface pose only as the loss of continuity in otherwise perfectly

periodic crystal, hence surface is considered a defect.

The unsatisfied bonding makes dangling bond at the surface

In nanostructures, the ratio of surface to volume is very high and a considerable

number of atoms may constitute the surface for a nanowire or a quantum dot. In these

cases, surface plays a very important part in its physical, chemical and electronic

property.

Some bulk indirect material becomes direct due to the surface

recombination of electron holes, while some material conduct highly

through the surface.

Some materials (eg Cs/p-type GaAs) can even be activated to negative electron

affinity, and such materials form a potent source of electrons, which can also be

spin-polarized as a result of the band structure.

Hot Carrier Effects

The term 'hot carriers' refers to either holes or electrons (also referred

to as 'hot electrons') that have gained very high kinetic energy after

being accelerated by a strong electric field in areas of high field

intensities within a semiconductor (especially MOS) device.

The term 'hot carrier effects', refers to device degradation or instability caused

by hot carrier injection.

1) the drain avalanche hot carrier injection;

2) the channel hot electron injection;

3) the substrate hot electron injection;

4) the secondary generated hot electron injection.

four commonly encountered hot carrier injection mechanisms:

The drain avalanche hot carrier (DAHC) injection is said to produce

the worst device degradation under normal operating temperature

range

under non-saturated conditions (VD>VG) high voltage applied at the drain

Other Implications

• Effective resistance

(ballistic & effects of splitting energy level)

Effective resistance may rise dramatically as current approaches saturation level

• Conductance is quantized and has a maximum value for a channel with one level

• Familiar voltage divider and current divider rule may not be valid on submicron scales

h

qG

22

Lh

qG

22 2

Conclusions

Quantum Confinement

• Transport properties are function of

confinement length in quantum structures

because of the change in the Density of States

• Relative strength of each scattering is different

from bulk materials

• Electrons tend to stay away from the interface

as wave function vanishes near the interface

Conclusions

High-Field Driven Transport

• Electric field puts an order into otherwise completely random motion

• Higher mobility may not necessary lead to higher saturation velocity

C onclusions

Failure of Ohm’s Law

• Effective resistance may rise

dramatically as current approaches

saturation level

• Familiar voltage divider and current

divider rule may not be valid on

submicron scales

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