Fundamentals of Semiconductor Physics 万歆 Zhejiang Institute of Modern Physics...

Fundamentals of Semiconductor Physics

万歆Zhejiang Institute of Modern Physics

[email protected]

http://zimp.zju.edu.cn/~xinwan/

Fall 2006

In memory of Prof. Xie Xide (1921-2000)

Chapter 1. Fundamentals

1.1 Bonds and bands

1.2 Impurities and defects

1.3 Statistical distribution of charge carriers

1.4 Charge transport

Total 12 hours.

Resistivity of Conductors, Semiconductors & Insulators

Semiconductor-forming Elements

B C N O

Al Si P S

Zn Ga Ge As Se

Cd In Sn Sb Te


1.1 Bonds and bands– Crystal structures– Bond picture– Band picture

• “Nearly free” electron model• Tight-binding model (LCMO)• k·p perturbation




An Apparently Easy Problem

Solid

Nuclei Electrons

interaction

In principle, by solving Schrödinger's equation

Lattice & Unit Cell

Crystal = periodic array of atoms

Crystal structure = lattice + basis

The choice of lattice as well as its axes a,b,c is not unique. But it is usually convenient to choose with the consideration of symmetry

Unit cell: Parallelpiped spanned by a,b,c

Cubic Lattices

Q: How many atoms are there in each of the unit cells?

Miller Indices

Elemental Semiconductors

Si, Ge: 4 valence electrons

14Si: 1s22s22p63s23p2 32Ge: [Ar]3d104s24p2

Covalent Bond

Complex Lattices

(a) Diamond: Si, Ge

(b) Zinc blende: GaAs, ZnS

Interpenetrating fcc lattices

Wurtzite Lattice

ZnS

Band: An Alternative Picture

From hydrogen atom to molecule

H H

H2

bonding

antibonding

Band Formation

N degenerate levels evolve into an energy band

Silicon14Si: 1s22s22p63s23p2

Resistivity

Conduction band, valence band & band gap

constant, T 0 , T 0

Formal Treatment

Approximation 1

Separation of electrons into

valence electrons and core electrons

Ion cores = core electrons + nuclei

e.g. Si: [Ne] 3s23p2

Approximation 2

Born-Oppenheimer or adiabatic approximation

To electrons, ions are essentially stationary.

Ions only see a time-averaged adiabatic electronic potential.

electron-phonon interaction

ionic motion

Lattice vibration (phonons)

electronic motion

Separation of Motion

Approximation 3

Mean-field approximation: Every electron experiences the same average potential V(r)

V(r): by first principle (ab initio), or by semi-empirical approach

Can we calculate everything?

• Yes.

– First principle band calculations

– Slater, …

• No.

– Disordered & strongly correlated systems

– Mott, Anderson, …

Reciprocal Lattice

• Definition: The set of all wave vectors K that yield plane waves with

the periodicity of a given lattice is known as its reciprocal lattice.

1

iK r R iK r

iK R

R ma nb pc

e e

e

��

��

�� Try to verify the following bi form the reciprocal lattice of the lattice spanned by ai.

In 1D, we have simply K = 2n/ a, where a is the lattice constant.

FCC BCC

FCC Lattice & its Brillouin Zone

Lattice (fcc): real space

Reciprocal lattice (bcc)

Diamond structure

Construct the Brillouin Zone

A Digression on Group Theory

Ref.: Yu & Cardona, Section 2.3

Rotational Symmetry

Ref.: Yu & Cardona, Section 2.3

Translational Symmetry

• Define operator TR

• Discrete translational symmetry

• Bloch’s theorem:

( ) ( ) ( )

, 0R

R

T V x V x R V x

H T

( ) ( )RT f x f x R

( ) ( )

( ) ( )

ik rnk nk

R nk nk

r e u r

T u r u r

Example: 1D Empty Lattice

( ) ( ), ( )i k nG x ikx inGxnk nk nkk nG e e u x u x e

• V 0:

• We assume an imaginary periodicity of a. Define the reciprocal

lattice constant G = 2 / a. We can therefore restrict k within the

range of [-G/2, G/2].

2 2

( ) , ( )2

ikxkE k k e

m

2 2( )( )

2nk

k nGE x

m

Free Electrons in 1D

V 0: 22

( ) , ( )2

i k nG xnk nk

k nGE k k e

m

Comments

• The wave vector k is not momentum p/, since Hamiltonian does no

t have complete translational invariance. Rather, k is known as crys

tal momentum (quantum number characteristic of the translational s

ymmetry of a periodic potential).

• The wave vector k can be confined to the first Brillouin zone.

• More in A/M Chapter 8.

“Nearly Free” Electrons

( ) ( )

( ) iKxk

K

V x V x a

V x U e

2 2

22

2

2

K

K

kU

m

K kU

m

Nearly Free & Pseudopotential

Si

Overlap of Molecular Orbitals

Tight-binding or LCMO

hopping. .i j

ij

H t c c h c

2 cos( ) cos( )x yE t k a k a

E(k)

k

2D: z = 4 nearest neighbors

Si – Ge – -Sn

Comparison

Nearly free e-’s + pseudopotential

• Electrons nearly free

• Wave functions approximated

by plane waves

• Electrons in conduction band a

re delocalized, so can be appr

oximated well by nearly free el

ectrons

Tight-binding or LCMO approach

• e-’s tightly bound to nuclei

• Linear combination of atomic w

ave functions

• Valence electrons are concentr

ated mainly in the bonds and s

o they retain more of their ato

mic character.

Band DiagramConduction band

Valence band

E

k

Conduction band

Valence band

Allowed states

Allowed states

Forbidden band gap

Ec

Ei

Ev

2 2 2

2 2

1 1

2 * *

k d EE

m m dk

Eg

Direct Bandgap GaAs

Band Structure: Si & Ge

Electrons and Holes

vbvb filled band empty state empty state

i i i ii i i i

J ev ev ev ev

vbvb

ii

J ev

cbcb

ii

J ev

0

The k·p Method

Conduction Band (Nondegenerate)

Comment on k·p Method

• Band structure over the entire BZ can be extrapolated by the zone c

enter energy gaps and optical matrix elements.

• One can btain analytical expression for band dispersion and effectiv

e mass around high-symmetry points.

• Nondegenerate perturbation is applicable to the conduction band mi

nimum in direct-bandgap semiconductors (zinc-blende, wurtzite); de

generate perturbation to top valence band (diamond, zinc-blende, w

urtzite). Ref.: Yu & Cardona, Section 2.6

• The trend of m* can be explained.


1.1 Bonds and bands


– Classification of defects

– Point defects

– Shallow (hydrogenic) impurities



Classification of Defects

• Point defects

• Line defects

• Surface states

Point Defects

VAIA

CA

More Classifications

• Intrinsic vs extrinsic

– Intrinsic: native, such as vacancies or antisite defects

– Extrinsic: foreign, Si:P

• Shallow vs deep – “effective mass approximation”

• Donors, double donors, isovalent center

– Examples: Si:P, Si:Se, Si:C

Shallow Impurity States

r

eHH cryst

2

impurity potential

B C N O

Al Si P S

Zn Ga Ge As Se

Cd In Sn Sb Te

Si:P

Effective mass approximation

• Break translational symmetry

• No Bloch’s theorem!?

Screened Coulomb potential

Hydrogenic Wave Function

1s

E

2s2p

...

0( ) ( ) ( )kr r C r

Bloch wave

Hydrogenic envelope

22

*

1( ) ( )

2

eC r EC r

m r

Hydrogenic bound states 1s, 2s, 2p, ...

+ continuum (conduction band)

“Rydberg”

“Bohr radius”

~2 22

4*

em

~2*

2

em

10 meV (Ge)

30 meV (Si)

50 Å (Ge)

20 Å (Si)

Band Diagram

Donor (Si:P, Ge:As)

C. B.

V. B.

Acceptor (Si:B)

E

k

Conduction band

Valence band

Allowed states

Allowed states

Forbidden band gap

EcEd

Ei

EaEv

Effective Mass Approximation

1. Introduce Wannier Functions (indexed by lattice vector i

n real space): Fourier transforms of Bloch functions

For very localized electrons, Wannier functions are roug

hly atomic orbitals.

Effective Mass Approximation

2. Express H in the basis of Wannier functions

Assume

3. Parabolic, isotropic, nondegenerate

Envelope Wave Function

Approximation valid for large effective Bohr radius a*(small k).

Comment on EMA

• The net effect of the crystal potential on the donor electron

inside the crystal is to change the electron mass from the

value in free space to the effective mass m* and also to

contribute a dielectric constant of the host crystal.

• Only conduction band states over a small region of reciprocal

space around the band minimum contribute to the defect

wave function if the effective Bohr radius a* is much larger

than the lattice constant a0.

Heavy Doping

• Light doping: impurity atoms d

o not interact with each other

impurity level

• Heavy doping: perturb the ban

d structure of the host crystal

reduction of bandgap

E

(E)

Ec

Ed

Ev

Eg

Metal-Insulator Transition

• Average impurity-impurity distance = Bohr radius

• Mott criterionB

3 164dN a


1.1 Bonds and bands



– Thermal equilibrium

– Mass-action law

– Fermi level


Thermal Equilibrium

• Thermal equilibrium is a dynamic situation in which every

process is balanced by its inverse process.

• Thermal equilibrium means that time can run toward the

past as well as into the future.

E1

E2

Mass-Action Law

• Electron-hole pairs: generation rate = recombination rate

• Generation: G = f1(T) f1: determined by crystal physics and T

• Recombination: R = npf2(T)

– Electrons and holes must interact to recombine

• At equilibrium, G = R

• Intrinsic case (all carriers result from excitation across the forbidde

n gap): n = p = ni

2 1( ) ( )npf T f T 213

2

( )( )

( ) i

f Tnp f T n

f T

Fermi Level

• Fermi-Dirac distribution

• Boltzmann distribution

• Density of electrons

1

( )1 exp /

D

f B

F EE E k T

( ) exp

f

BB

E EF E

k T

f BE E k T

(not too heavily doped)

cb

( ) ( )D cn F E N E dE v

vb

1 ( ) ( )Dp F E N E dE

Intrinsic Semiconductors

A Parabolic Band

2 2 2

3 23

3/ 22

1/ 2

1

2

0

/

2ex

2 * *2

( ) ( ) 42

4 2 *( )

( ) ( ) ex pp c C c

B

c

c

B

C FC

B B

k kE E dE dk

m m

N E dE N k d k k dk

mN E E E

h

E En F E

E E E E E Ed

k T kN E dE N

Tk kT T

1

3/ 2

2

2exp , 2C F C B

C CB

E E m k Tn N N

k T h

Comment

• Multivalley, such as Si:

– additional factor of number of valleys

• Anisotropic band

– Effective mass: Geometrical average of mass components

• Valence band: NV

– Sum of heavy hole, light hole (neglecting split-off band)

– replace m* by hole effective mass

Density of states effective mass

Intrinsic Carrier Concentration

2

exp

exp

exp exp

C FC

B

F VV

B

gC Vi C V C V

B B

E En N

k T

E Ep N

k T

EE En np N N N N

k T k T

3/ 2

2

2exp , 2F V V B

V VB

E E m k Tp N N

k T h

Doped Semiconductors

• Assuming full ionization,

charge neutrality

• With intentional doping,

typically for n-type

2

1/ 222

2 2

d a

id a

d a d ai

N p N n

nn p n N N

n

N N N Nn n

2

d a i

d a

i

d a

N N n

n N N

np

N N

• Majority carriers

• Minority carriers

• Compensation

Extrinsic Semiconductors

exp

ln2 2 2

exp

exp

C ii C

B

C V V C VBi

C

f ii

B

i fi

B

E En N

k T

E E N E Ek TE

N

E En n

k T

E Ep n

k T

Conduction band

Valence band

EcEd

Ei

EaEv

EF

Carrier Concentration vs T

Fermi Energy in Si

Eg ↓

Nonmonotonic behavior

Occupation of Impurity level

/

/ /

/

1,

1

11

2

2

1,

1

d F B

d F B d F B

d F B

E E k T

d dE E k T E E k T

d

E E k Td

en N

e e

n

N e

0 Ed Ed 2Ed +

d a a dn n N p p N


1.1 Bonds and bands




Charge Transport

• Ohm’s law

• Equipartition of energy

• Room temperature (300K)

V IR

* 21 3

2 2th BE m v k T

2

2 3

7

~ 30meV, 0.5MeV, * / 0.07 (GaAs)

/ 3 / * ~ 10

~ 10 cm/s

B RT e e

th B

th

k T m c m m

v c k T m c

v

The complete notes for Chapter 1 are expected to be available after

Sept. 20, 2006.

Fundamentals of Semiconductor Physics 万歆 Zhejiang Institute of Modern Physics...

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Transcript of Fundamentals of Semiconductor Physics 万歆 Zhejiang Institute of Modern Physics...

Fundamentals of Semiconductor Physics 万 歆 Zhejiang Institute of Modern Physics...

Documents

Transcript of Fundamentals of Semiconductor Physics 万 歆 Zhejiang Institute of Modern Physics...

Fundamentals of Semiconductor Physics 万歆 Zhejiang Institute of Modern Physics...

Transcript of Fundamentals of Semiconductor Physics 万歆 Zhejiang Institute of Modern Physics...