FYS3410 Condensed matter physics

Lecture 15: Carrier statistics in semiconductors

Randi Haakenaasen

randi.haakenaasen@ffi.no, 63 80 73 09

UniK/UiO

Forsvarets forskningsinstitutt

11.03.2015

Outline

• Repetition: energy band structure and filling of bands

• The Fermi level

• Carrier concentrations at equilibrium

• Electrical conductivity

• Fermi level at equilibrium and in electric field

• Semiconductor growth and characterization: HgCdTe at FFI

Remember this…

• Pauli exclusion principle: no two electrons in interacting system can be in same state

• When atoms are brought together to form a solid, the wavefunctions start to overlap,

and the electron levels split and form bands

• For a crystal with N atoms, each overlapping electron level will split into N levels

which can accomodate 2N electrons

• The periodicity of the crystal potential introduces band gaps in the free electron

bands at the Brillouin zone boundaries

and this…

• Material with odd number of valence electrons per prim. cell -> metal

• Even number of valence electrons per prim. cell – semiconductor or insulator if filled

shell, metal or semi-metal if overlapping bands

• In a crystal we can decribe the motion of electrons as if they were free electrons but

with an effective mass m* which includes the effect of the crystal on the electron

𝑚 ∗= ℏ2𝑑2𝐸(𝑘)

𝑑𝑘2

−1

Holes in VB are treated as charge carriers with positive charge and positive mass

The hole energy increases downwards

SC or metal or metal

insulator semimetal

Goal

• Electronic properties of semiconductors -> Solid state device behaviour

– s electrical conductivity

• For e- to accelerate in applied E-field, they must be able to move into new energy

states

– > Can only get a current if there are empty states available

– > current in SC carried by e- in conduction band and holes in valence band

• Need to calculate carrier concentrations

• Flexibility in controlling carr. Conc.: temperature, band gap, doping, optical

excitation..

• Majority carrier conc. often determined by doping

– > we must find equations for minority carr. conc.

• Start with carr. conc. for undoped SC -> intrinsic SC -> largely controlled by Eg/2kT

Inctrinsics and carr. conc. largely controlled by Eg/kT

At finite T, carriers are thermally excited from valence band VB to

conduction band CB and create electron-hole pairs EHP. The

intrinsic carrier concentrations are therefore equal:

𝑛 = 𝑝 = 𝑛𝑖

In equilibrium EHP recombine at same rate as they are generated

At any temp T 𝑟𝑖 𝑇 = 𝛼𝑟𝑛0𝑝0 = 𝛼𝑟𝑛𝑖2 = 𝑔𝑖

.

Intrinsic semiconductors

An intrinsic semiconductor has no impurities or defects -> no carriers at 0 K

Thermal equilibrium: no net flows of matter, charge or energy, no phase changes, and no

unbalanced potentials (or driving forces) within the system. (a sample at constant temperature,

in the dark, with no fields applied). A system in thermodyn. equil. experiences no changes

when isolated from its surroundings

distribution of e- over allowed energy levels at thermal equilibrium

𝑓 𝐸 =1

1+𝑒 𝐸−𝜇 𝑘𝐵𝑇 =1

1+𝑒(𝐸−𝐸𝐹) 𝑘𝐵𝑇

= probability that available state at E is occupied by an e- at T

In SC physics 𝐸𝐹 is called the Fermi level although only strictly

correct at T=0. The chemical potential m is a function of T

At 𝑇 = 0, 𝑓 𝐸 = 1 𝐸 < 𝐸𝐹 and 𝑓 𝐸 = 0 𝐸 > 𝐸𝐹

All states are filled up to 𝐸𝐹

At T>0 some e- thermally excited to higher energies

𝑓(𝐸𝐹 + ∆𝐸) = 1 − 𝑓(𝐸𝐹 − ∆𝐸) symmetric about EF at all T

In metals, 𝐸𝐹 2-10 eV, 𝑉𝐹 1-2 x108 cm/s and 𝑘𝐹 1-2 x108 cm-1,

while kBT 0.026eV at RT

𝐸𝐹 natural reference point in calculation of SC e- and h conc.

Fermi-Dirac distribution function f(E)

The Fermi level EF

• At EF the prob of being occupied is ½:

• From symmetry about EF and intrinsic 𝑛𝑖 = 𝑝𝑖

– > EF close to middle of Eg

– (in middle if DOS in VB and CB are equal)

• At room temp f(EC) and (1-f(EV)) are quite small:

– In Si at 300K

while DOS

– As are large, small changes in f(E)

can result in significant changes in n and p

• At finite T, both CB and VB are partially filled and

contibute to electrical conduction

310105.1 cmi

pi

n

2

1

11

1

1

1)(

/)(

TkEEFBFFe

Ef

31910)(),( cmEgEg VC

)(),(V

EgC

Eg

Prob. of occupancy of available

states – but no states in band gap

In VB, electrons can hop and fill in

the empty states

– This can be treated as a

pos.charged hole moving

in the opposite direction

• In n-type material:

– n in CB > (>>) p in VB

– > f(E) must be above intrinsic position so that

– Thus gives a measure of n

• In p-type material:

– p in VB > (>>) n in CB

– FD must be below middle of gap so below

is larger than above

– Thus gives a measure of p

• For given T, if you know EF and DOS of CB and VB

– > n and p

EF in extrinsic semiconductors

)(1)( VC EfEf

FCEE

)(1 Ef VE

)( CEf CE

VF EE

Carrier concentrations at equilibrium - qualitative

Concentration of e- in conduction band

𝑛0 =1

𝑉 𝐷(𝐸)𝑓(𝐸, 𝑇)𝑑𝐸∞

𝐸𝐶

𝐷 𝐸 = 𝑉

2𝜋22𝑚

ℏ2

32

𝐸12 increases with E

f(E) decreases faster with energy

-> Very few e- far above EC

Prob of finding a hole also decreases

rapidly below EV

We get carrier densities as indicated in

figure

Can replace integral over CB by

effective DOS at EC: 𝑛0 = 𝑁𝐶𝑓(𝐸𝐶)

𝑝0 =1

𝑉 𝑔(𝐸) 1 − 𝑓(𝐸, 𝑇) 𝑑𝐸𝐸𝑉

−∞

For holes, the energy increases downwards since

E scale refers to electron energy

Electron and hole concentrations at equilibrium

Assumptions:

1. 𝐸𝐶 − 𝐸𝐹 ≫ 𝑘𝐵𝑇 then 𝑓(𝐸, 𝑇) =1

1+𝑒(𝐸−𝐸𝐹) 𝑘𝑇 ≅ 𝑒−(𝐸−𝐸𝐹) 𝑘𝐵𝑇 in CB kBT = 0.026 eV

This means that most n in CB close to CBM and most p in VB close to VBM

2. For holes in VB 1 − 𝑓(𝐸, 𝑇) = 1 − 1

1+𝑒(𝐸−𝐸𝐹) 𝑘𝑇 ≅ 𝑒−(𝐸𝐹−𝐸) 𝑘𝐵𝑇

3. Near CBM and VBM bands are nearly parabolic, so we approximate bands by parabolas with

(constant) effective masses 𝑚𝑒∗ and 𝑚ℎ

∗ given by the curvature of the band

𝑛0 =1

𝑉

𝑉

2𝜋2∞

0

2𝑚𝑒∗

ℏ2

3 2

𝐸1 2 𝑒−(𝐸−𝐸𝐹) 𝑘𝑇 𝑑𝐸

=2𝑚𝑒

∗ 3 2

2𝜋2ℏ3𝑒𝐸𝐹 𝑘𝑇 𝐸1 2

∞

0𝑒−𝐸 𝑘𝑇 𝑑𝐸 𝑥1 2

∞

0𝑒−𝑎𝑥𝑑𝑥 =

𝜋

2𝑎 𝑎

= 22𝜋𝑚𝑒

∗𝑘𝑇

ℎ2

3 2

𝑒𝐸𝐹 𝑘𝑇 refer to bottom of CB as Ec instead of 0

= 22𝜋𝑚𝑒

∗𝑘𝑇

ℎ2

3 2

𝑒−(𝐸𝐶−𝐸𝐹) 𝑘𝑇

= 𝑁𝐶𝑒−(𝐸𝐶−𝐸𝐹) 𝑘𝑇 where 𝑁𝐶 = 2

2𝜋𝑚𝑒∗𝑘𝑇

ℎ2

3 2

Valid for material in thermal equilibrium, both intrinsic and doped:

𝑛0 = 𝑁𝐶𝑒−(𝐸𝐶−𝐸𝐹) 𝑘𝑇 where 𝑁𝐶 = 2

2𝜋𝑚𝑒∗𝑘𝑇

ℎ2

3 2

𝑝0 = 𝑁𝑉𝑒−(𝐸𝐹−𝐸𝑉) 𝑘𝑇 where 𝑁𝑉 = 2

2𝜋𝑚ℎ∗𝑘𝑇

ℎ2

3 2

Integration over distributed electron states in CB represented by an effective DOS NC at EC

The concentration of holes in the VB similarly be represented by an effective hole DOS NV at

EV

In equilibrium, the product of 𝑛0 and 𝑝0 is constant at a given temperature, independent of EF

(and thereby independent of doping) – law of mass action:

𝑛0𝑝0 = (𝑁𝐶𝑒−(𝐸𝐶−𝐸𝐹) 𝑘𝑇 )(𝑁𝑉𝑒

−(𝐸𝐹−𝐸𝑉) 𝑘𝑇 )= 𝑁𝐶𝑁𝑉𝑒−(𝐸𝐶−𝐸𝑉) 𝑘𝑇 = 𝑁𝐶𝑁𝑉𝑒

−𝐸𝑔 𝑘𝑇

𝑛0𝑝0 = 𝑁𝐶𝑁𝑉𝑒−𝐸𝑔 𝑘𝑇

For intrinsic material we get: 𝑛𝑖 = 𝑝𝑖 = 𝑁𝐶𝑁𝑉𝑒−𝐸𝑔 2𝑘𝑇

While for doped materials 𝑛0𝑝0 = 𝑛𝑖2

𝑛0 = 𝑛𝑖 𝑒−(𝐸𝐹−𝐸𝑖) 𝑘𝑇 𝑝0 = 𝑛𝑖 𝑒

−(𝐸𝑖−𝐸𝐹) 𝑘𝑇

• For intrinsic material we also get

• If effective masses for electrons and holes are

equal, EF is in the middle of the gap

– If not, then it changes with temperature

• For example: light electrons, heavy holes

- many more holes than electrons generated as temperature increases, unless EF moves up with temperature

• Heavy holes: bands formed from wavefunctions

with little overlap (inner or core e-) such as 4f e- in

rare earth metals

– Slow tunneling from one ion to the next is

reflected in heavy mass

𝐸𝐹 =𝐸𝑔

2+3

4𝑘𝐵𝑇 ln

𝑚ℎ∗

𝑚𝑒∗

Simplified view of band edge structure of a direct band gap SC

Temperature dependence of ni

• 𝑛𝑖 𝑇 = 2 2𝜋𝑘𝐵𝑇

ℎ2

3 2 𝑚𝑒∗𝑚ℎ∗ 3 4 𝑒

−𝐸𝑔 2𝑘𝑇

• Temperature dependence in exponential, but also in T3/2

from DOS and in EF (or Eg)

• Usually ni is known for given material

• Plot (neglect T dependence of ni and Eg)

• ni strongly temp dep, much smaller than those of metals

• El. conductivity increases with T, incontrast to metals where s=ne2t/m decreases with T due to electron - phonon scattering

Eg

• GaAs 1.43 eV

• Si 1.11eV

• Ge 0.67 eV

• HgCdTe 0 – 1.6 eV

Example

1.1/2=0.55

0.550-0.407=0.143

Ec-EF/kT=0.143/0.026=5.5 ok to use approximations

Extrinsic region

Raise T from T=0:

Ionization region -> extrinsic region -> intrinsic region

Compensation and space charge neutrality

• A SC can have both donors and acceptors

• Nd > Na n-type, Fermi level above middle, Ea is

filled with electrons

• With EF > Ei, cannot have hole conc. in valence

band same as acceptor concentration

• Compensation; e- in CB fill holes in VB, giving

n=Nd-Na

• From space charge neutrality: if material is electrostatically neutral, the sum of

positive charges must balance the sum of negative charges:

• 𝑝0 + 𝑁𝑑+ = 𝑛0 +𝑁𝑎

−

• 𝑛0 = 𝑝0 + 𝑁𝑑+ − 𝑁𝑎

−

• If doped n-type 𝑛0 ≫ 𝑝0 and impurities ionized

• 𝑛0 ≅ 𝑁𝑑 − 𝑁𝑎, 𝑝0 = 𝑛𝑖2 𝑁𝑑 − 𝑁𝑎

• Intrinsic SC and doping atoms are neutral, this is maintained at equilibrium.

Conductivity and mobility

Have n and p -> calculate current in ℇ − and B- field

– But need also mobility of carriers (how fast the carriers flow through the

crystal)

The charge carriers scatter from phonons, impurities and defects -> mean free time t

The current density for electrons in an ℇ -field in the x-direction is (𝑣𝑥 is drift velocity)

𝐽𝑥 = −𝑞𝑛𝑣𝑥 = −𝑞𝑛 −𝑞𝜏

𝑚𝑒∗ℇ𝑥 = 𝜎ℇ𝑥 (C/electron electrons/cm3 cm/s = A/cm2)

𝜎 =𝑛𝑞2𝜏

𝑚𝑒∗ = 𝑞𝑛𝜇𝑒 conductivity in (Ω−1𝑐𝑚−1)

𝜇𝑒 =𝑞𝜏

𝑚𝑒∗ 𝜇ℎ =

𝑞𝜏

𝑚ℎ∗ mobility in (cm/Vs)

For both e- and hole contributions: 𝐽𝑥 = 𝑞 𝑛𝜇𝑒 + 𝑝𝜇ℎ ℇ𝑥 = σℇ𝑥 𝜎 = 1 𝜌

𝑣𝑥=−𝜇𝑒ℇ𝑥

Mobility

Small effective mass gives high mobility -> fast detectors

• Hall measurements LPE155ann

– ND-NA,77 = 1.3E14 cm-3

– m77= 140 000 cm2/Vs

𝜇𝑒

• GaAs 8000

• Si 1350

• Ge 3600

• Hg0.8Cd0.2Te 140000 at 77K

Invariance of Fermi level at equilibrium

Consider two materials in contact (two SCs, n- and p-type, metal/SC, nonuniform doping)

No current, no net charge transport, no net transfer of energy

For each energy E, a transfer of e- from 1 to 2 must be balanced by opposite transfer of

e- from 2 to 1

Electron energy levels in ℇ -field

• Electrons drift in direction opposite to field

• -> Potential energy increases in direction of field

• The electrostatic potential 𝑉 𝑥 = 𝐸 𝑥 −𝑞

• The electric field ℇ 𝑥 = −𝑑𝑉 𝑥 𝑑𝑥

• Relate ℇ 𝑥 to the electron potential energy in band

diagram, choosing 𝐸𝑖 as reference

• ℇ 𝑥 = −𝑑𝑉 𝑥 𝑑𝑥 = −𝑑

𝑑𝑥

𝐸𝑖

−𝑞=1

𝑞

𝑑𝐸𝑖

𝑑𝑥

• To remember: e- drift ‘downhill’ in an ℇ -field, while

holes float up like bubbles

Measurements of Eg, 𝑚𝑒∗ and 𝑚ℎ

∗

• Need Eg, 𝑚𝑒∗ and 𝑚ℎ

∗ to calculate carrier concentrations

• Get Eg from absorption experiments

• Get effective masses from cyclotron resonance experiments

• In a magnetic field B, electrons move in spirals around the field with

wC=Bq/me*

• Strong absorption of radio frequency radiation when wr=wC

FTIR cut-on and detector cut-off l

• Fourier Transform Infrared spectroscopy or optical absorption experiments

– Send spectrum of IR radiation onto sample, measure transmission spectrum

– Photons with energy h = hc /l > Eg can create an EHP and are then absorbed

– Photons with energy h = hc /l > Eg can not create an EHP and are transmitted

– Band gap where transmission increases steeply

– Oscillations on transmitted spectrum are interference fringes due to film thickness

• Detector cut-off l: below cut-off spectral response QE positive verdier

– Industri standard: der QE er 50% av maksimum

3 4 50

1

2

3

4

5

6

7

8

Ap

pare

nt

Q.E

.

Wavelength [mm]

Some semiconductor properties

Summary

• Semiconductors: 0 < Eg < 2-3 eV, thermally excited carriers at resonable T

• Fermi level chemical potential: f(EF)=1/2, FD symmetric about EF at all T

– Position of EF -> n and p

• Intrinsic semiconductors in thermal equil: 𝑛 = 𝑝 = 𝑛𝑖

– Fermi level close to middle of gap

– 𝑛𝑖 = 𝑝𝑖 = 22𝜋𝑘

𝐵𝑇

ℎ2

3 2

𝑚𝑒∗𝑚ℎ

∗ 3 4 𝑒−𝐸𝑔 2𝑘𝑇

• For doped material in thermal equil. 𝑛0𝑝0 = 𝑛𝑖2 law of mass action

𝑛0 = 𝑁𝐶𝑒−(𝐸𝐶−𝐸𝐹) 𝑘𝑇 = 𝑛𝑖 𝑒

−(𝐸𝐹−𝐸𝑖) 𝑘𝑇

𝑝0 = 𝑁𝑉𝑒−(𝐸𝐹−𝐸𝑉) 𝑘𝑇 = 𝑛𝑖 𝑒

−(𝐸𝑖−𝐸𝐹) 𝑘𝑇

• Current density in SC in E-field: 𝐽𝑥 = 𝑞 𝑛𝜇𝑒 + 𝑝𝜇ℎ 𝐸𝑥 = σ𝐸𝑥