8. Semiconductor Crystals

• Band Gap

• Equations of Motion

• Intrinsic Carrier Concentration

• Impurity Conductivity

• Thermoelectric Effects

• Semimetals

• Superlattices

Transistors, switches, diodes, photovoltaic cells, detectors, thermistors, …

IV: Si, GeIII-V: InSb, GaAsII-VI: ZnS, CdSIV-IV: SiC

Strong T dependence

Insulator: ρ > 1014 Ω cm

III IV V VI

B C N O

Al Si P S

Ga Ge As Se

In Sn Sb Te

Tl Pb Bi Po

SemiC: 109 > ρ > 10–2 Ω cm

Eg = 0.66 eV Eg = 1.11 eV

/g BE k T

in e

Intrinsic temperature range:σ indep of impurities

Band Gap

k k K

gE

k k k k

gE k k

210 eV

Excitons not shown

For γ & e of same energy,210F

e

vk

ck For ph & e of same k,

310ph

F

ph

e

v

v

ph emitted,low T.

InSb

Another way for determining Eg : σi (T) or ni (T) determined from RH .

III IV V VI

B C N O

Al Si P S

Ga Ge As Se

In Sn Sb Te

Tl Pb Bi Po

Equations of Motion

• Physical Derivation of k = F

• Holes

• Effective Mass

• Physical Interpretation of the Effective Mass

• Effective Masses in Semiconductors

• Silicon and Germanium

Wave packet: g kv1 k group velocity

Particle subjected to force F: kkg t v F

→

g k v

d

dt

kF

Lorentz force:1d

qdt c

k

kE B

→ Particles in contant B field move on surface of constant energy perpendicular to B.

Physical Derivation of dk/dt = F

Plane wave expansion: nin

n

C e k G rk r r k k

Electron momentum: el i P k k

* 3

,

1m ni i

m n nn m

C C d e e k G r k G rk k k G rV

2

n nn

C k k G

2

n nn

C k k G

21n

n

C k

2

el n nn

C kP k G k k

2

lat n nn

C kP G k k

tot el lat P P P k t Fd

dt

kF

Holes

h ek kfilled band

k 0

→

h h k

ε = 0 at top of valence band:

k k

→ e e e e k k

k kno spin-orbit interaction:

h e k

e e h h k k

Inversion symmetry →

hh h hkv k

e e ek k ev

h ev v

h em m see next section

1hh

de

dt c

kE v B

1ee

de

dt c

kE v B

e moves toward –kx ; so does h

C.B.

V.B.

e ee j v

E E

h hej v

e j v e v e v

Effective Mass1

g kv

1gd d

dt dt

k

v

1 d

dt

k k

k

2

1 k kF

2

1g ij

ji j

dvF

dt k k

2

2

1j

j i j

Fk k

1

* jj i j

Fm

* gd

dt

vm F

2

2

1

* i j i jm k k

= effective mass tensor ( of electrons )

Near zone boundary :

/2/2

21 g

k g KUU

2 2

2k

k

m

2

gk K

2

* 2 2/ /

1

m K

/221

1 g

m U

1

/22*1gm

m U

CBVB

CBVB

m* < 0 near top of VB

1

*gd

dt

vF

m

/22 g

U

band width

band gap

4

U << λg/2

Physical Interpretation of the Effective Mass

0 1

i k G xi k xC e C e

PW k + Bragg reflected k−G(p transferred to lattice)

vice versa

C0 / C−1 = 1 → standing wave

m* < 0

m* > 0

m* < 0

Effective Masses in Semiconductors

m* determined by cyclotron resonance (rf) at low carrier concentration.

*c

q B

m c

2 2

2vhh

khh

m

2 2

2vlh

klh

m

2 2

2vsoh

ksoh

m

1c

Condition for complete orbit without collison:

c Bk T

cyclotron frequency

Landau levels:

1

2n cE n

For m* = 0.1 and ωc = 24GHz,we have B = 852 gauss.

Prob 9.8 → m* Eg for direct-gap crystals

0.015 0.026 0.073, ,

0.23 0.43 1.42c

g

m

m E

For InSb, InAs, InP

0.065, 0.060, 0.051 Eg from Table 1

Silicon and Germanium

2 2 4 2 2 2 2 2 2 23/2 x y y z z xAk B k C k k k k k k k

VB at k=0 : p3/2 + p1/2

21/2 Ak k

CB of Ge with B in (110).CB edge at L.4 mass spheroids along [111]; 2 of which are equivalent in (110) plane.ml = 1.59 m, mt = 0.082 m.

2 2

2 2

1 cos sin

c t t lm m m m

θ = angle with

longitudinal axis

Si

GaAsDirect-gap

Spheroids along <100>.CB edges on Δ line near point X.ml = 0.92 m, mt = 0.19 m.

A=−6.89, B=−4.5, C=6.2, Δ =0.341

Isotropic mc = 0.067m.

Intrinsic Carrier Concentration

1

1ef

e

1 e 1

Bk T

Near CB edge:2

2ce

E k k km

Isotropic band:2 2

2ce

kE

m k

3/2

2 2

21

2e

e c

mD E

c

e eEn d D f

3/2

2 2

21

2 c

e

E

me d e

3/2

22

2cEem

e

*e cm m

11

h e

ef f

e

1 e

Near VB edge:2

2vh

E k k km

Isotropic band:2 2

2vh

kE

m k

3/2

2 2

21

2h

h v

mD E

vE

h hp d D f

3/2

22

2vEhm

e

1

1ef

e

1

1e

2

2

1

h i ji jk k

k 0

m

3/2

22

2cEem

n e

→ 3

3/2

2

14

2gE

e hn p m m e

np values at 300K:19 6 26 6 12 62.1 10 2.89 10 6.55 10

Si Ge GaAs

cm cm cm

(independent of doping)

0hm

1

* i j

m

Black body radiation:

d nA T B T n p

dt

A T

B T n pe h

At equilibrium:

A Tn p

B T = const at given T

Intrinsic carrier concentration: 3/2

3/4 /2

2

12

2gE

i i e hn p m m e

Carrier compensation: n+p is reduced by increasing either n or p through doping.

3/2

22

2c iEe

i

mn e

Pure sample:

→

3/2

22

2i vEh

i

mp e

3/2

2 c vi E Eh

e

me e

m

1 3ln

2 4h

i c v Be

mE E k T

m

Intrinsic Mobility

v EMobility μ of single type of carriers:

0

e hne pe

Electrical conductivity σ of semiconductor:

ee

e

e

m

hh

h

e

m

q

m

E

T /2gE

i in p e μh < μe due to interband scattering

Ionic crystals:h moves by hopping.

Self-trapped via Jahn-Teller effect

Eg small → m* small→ μ large, esp D-G

Impurity Conductivity

• Donor States

• Acceptor States

• Thermal Ionization of Donors and Acceptors

Stoichiometric deficiency → Deficit semiconductorsImpurities → Doped semiconductors

e.g., 10–5 B → σ = 103 σi for Si at 300K

Donor States

Donor = Impurity atom that tends to give up an electron

Bohr model:4

2 22e

d

e mE

2

13.6 emeV

m

Bohr radius:

2

2de

am e

0.53e

mA

m

Valid when ad >> atomic distance.& Ed << Eg .

Anisotropy need be considered for Si & Ge

III IV V VI

B C N O

Al Si P S

Ga Ge As Se

In Sn Sb Te

Tl Pb Bi Po

20 29.8

5 9.05

isotropic anisotropic

Bohr model

30

80

da A

Si

Ge

Impurity band formed at low impurity concentrations.Mott (metal-insulator) transition.

Conduction in impurity band is by hopping. Occurs at lower concentration in compensated materials.

300 26Bk K meV

Acceptor StatesAcceptor = Impurity atom that tends to capture an electron

III IV V VI

B C N O

Al Si P S

Ga Ge As Se

In Sn Sb Te

Tl Pb Bi Po

Complication:VB degeneracy.

Ultra pure Ge:imp conc < 10–11

active impurities ~ 21010 cm−3

intrinicregion

Electrically inactive impurities in Ge: H, O, Si, C.Can’t be reduced below 1012 – 1014 cm–3 .

13 31.7 10in cm at T = 300K with ρi 43 Ω cm

Thermal Ionization of Donors and Acceptors

/20

dEdn n N e

1dE

3/2

0 22

2h Bm k T

p

/20

aEap p N e

3/2

0 22

2e Bm k T

n

No acceptors present:

1aE

0

c iEin n e

D e D

0dEd

d

n Nn e

N

0

cEn n e

Reminder:

Extrinsic region:

2in p n e h

Thermoelectric Effects

Electrical conductivity: q J E=0T 2

q

q

n q

m

Thermal conductivity: Q T J0q J2 2

3q B

q

n k T

m

Seebeck effect: S T E=

Peltier effect: Q qJ J0T

0q J S = Seebeck coeff.(Thermal power)

b = carrier mobility

qB

B

k T

q k T

cq

v

E efor

E h

q qn q b

Π = Peltier coeff.

Heat current density JQ :

i ii

dU dQ Y dx dN → U Q N J J J Steady state

Q NU J J qq Bk T

q

J

NU J

→

2

ST Kelvin relation(derived from thermodynamics)

2 2

23Bk

Tq

B AQ I

Thermoelectric Effects: Boltzmann Eq Ref: Haug, IV.B.Kittel, App F

Boltzmann eq.: d ff

dt r pv F

C

f

t

1 p k

1E kv

0

C

f ff

t

Relaxation time

approximation

Linearization: 0

d ff

dt r pv F

C

f

t

00

ff E

E

k k0f

E

v

00

ff T

T

r r

2

1

E

E

Ee

T TT

e

r

0

1

1Ef

e

0f ET

E T T

r

Ee semicond

metals

0f

t

A-current density:

3

0322

A

df f A

kJ v 02 dE E f f A v

ET

T T

rv F

A-current density:

3

0322

A

df f A

kJ v

30

322

fdA

E

kv

02f E

dE E A TE T T

rv v F

20/ /2 j

j

fK dE E E v

E

For isotropic materials, JA is a linear combination of integrals

00

ff f

E

E

0

C

fE fT

T T E t

rv F 0f f

0f

E

1 00

q qq

d K K dTJ q K q

d z T d z

E

2

2 1 01 0

2q q qQ q

d K K K dTJ K K q

d z T d z

E

1 0

20 0

1q q qJ K K ddT

q K qK T d z q d z

E

21 0 1 0 2

0 0

qQ q

K K K K K dTJ J

qK K T d z

→

Electrical conductivity: qJ E0T 20q K

Thermal conductivity: q

dTJ

d z0qJ

20 2 1

0

K K K

K T

Seebeck effect:dT

Sd z

E= 1 0

0

qK KS

qK T

Peltier effect: Q qJ J0T

ST

0qJ Seebeck coeff.(Thermal power)

1 0

0

qK K

qK

Kelvin relation(derived from thermodynamics)

For spherical energy surfaces: ( 1)!

jqj B

n bK j k T

q b = carrier

mobility 2 qB

B

k T

q k T

cq

v

E efor

E h

02 ln qB

q

nk T

q n

Peltier coefficent of Si

Semimetals

2 Group V atoms in primitive cell→ insulator

Band overlap→ semimetal

SuperlatticesSuperlattice: lattice with long period created by stacking layers of atoms.

Ref: J.Singh,”Physics of Semiconductors & Their Heterostructures”

(GaAs)1 (InAs)1

(GaAs)2 (InAs)2

Bloch Oscillator

Bloch Oscillator:For a collisionless electron accelerated across a Brillouin zone, the motion is periodic.

2G

A

e T E A = superlattice constant along

2B

e A

T

E Bloch frequency

Simple TBM: 0 1 cosk kA

1 dv

d k

0 sin

AkA

z dt v dtdk v

d k 0

0

sink A

dk kAe

E

d ke

dt E

0 cos 1kAe

E0 cos 1

e At

e

EE

0 0z

Zener Tunneling (field-induced interband tunneling):

Tilting of band by

→ different bands at same ε

→ Zener tunneling (breakdown)

Heavily doped p-n junction

Strong reverse bias→ Zener breakdown

I-V curve