Metals, Semiconductors and Insulators.

2nd half of the course by D.Krizhanovskii ( E16, d.krizhanovskii@shef.ac.uk)

Topics to cover1. Optical transitions2. Doping3. Electron statistics4. Hall effect5. Cyclotron resonance6. Scattering and mobility7. Plasma oscillations8. Amorphous materials9. Quantum wells

20nm GaAs

InGaAs

Nanostructures.Single quasiparticleQuantum bit=> novel data storage/memory elements, quantum computing, quantum cryptography…..

Graphene- one monolayer of Graphite. Much faster

Carbon-based nanoelectronics.

Nanophotonics, plasmonics.

Integrated optical nanocircuits

etc, etc and etc……

Topic 10: Optical absorption

1. Optical Absorption-onset at band gap

2. Absorption process for direct gap semiconductors.Vertical process.

3. Absorption coefficient proportional to density of states.For a bulk material ~ E1/2.

4. Excitons. Bound electron-hole state.

5. Role of phonons for indirect gap semiconductors. Temperature dependence of absorption spectra.

6. Optical absorption in doped semiconductors (the shift of the absorption edge- the Moss-Burstein effect)

Onset of optical absorption is at ; is the photon energy

Energy band gaps in various materials

GaN > 3 eV

GaInP 2-2.3 eV

GaAs 1.4 eV

Si 1 eV

Ge 0.7 eV

InSb 0.2 eV

gE>ω ω

Optical absorption in semiconductors

Conduction band(C-band)

Valence band(V-band)

Schematic diagramm of electron energy spectrum

gE

Optical absorption in direct gap semiconductors

One photon with energy creates electron-hole pairConservation of energy and momentum:

Electronic transition due to photon absorptionis vertical on energetic schematic diagramm

*22*22 2/2/ hheeg mkmkE ++=ω

heph kkk += m*e,h-effective electron (e) and hole (h) masses;

ke,h –electron and hole wavevectorskph-photon wavevector

ωh

ke at the edge of BZ ~ 1010~a

πm-1 (a is lattice spacing)

610~2

λπ=phk m-1

|||| he kk ≈

Direct transition

Schematic of vertical electronic transition

Example of energy spectrum ofGaAs crystal, which is direct gap material (minima of C and V bands coincide)

Energy dependence of absorption coefficient in direct band gap material

Absorption coefficient is defined as the relative rateof decrease of light intensity along its propagation path

)( ωα h)(ωI

α is proportional to the density of states associated with optical transition at given frequency

)()(2

2

)2(

8)()( 2/1

32

2/3

**

**

3

2

ωωππ

πωω

dEmm

mm

dkkdN g

he

he

−

+

==

)( ωhN

2/1)(~)( gE−ωωα

)(

)()(

ωωωα

dxI

dIh =

The effect of free carriers on optical absorption spectra

The Moss-Burstein effectConsider semiconductor material, where C band is partially filledwith electrons (n-doped material)

Only optical transitions from V-band to empty states in C-band allowed

*22*22 2/2/ hheFgonset mkmkE ++=ωOnset of optical absortption at

Fk - Fermi wavevector

)/1(2/ ***22heeFgonset mmmkE ++= ω

Indirect optical transitions

Schematic of indirect optical transitions

Indirect gap semiconductors

Conduction band minimum not at k=0

Phonon with momentum kphonon andenergy is involved

Momentum and energy conservationkel+kh=kphonon ;

1. Second order process andthus very weak ~104 times weaker

1. Phonon energy smallcompared to Eg

Ω

Ω±= GEω

Indirect gap semiconductors

Ω+= hEh Gω

Low temperature.Phonons are not present and thus absorptionoccurs with emission of phonon

The absorption threshold is at

High temperaturePhonons are present and a phonon can be absorbed alongwith a photon. Threshold energy Ω−= hEh Gω

Excitons

Formation of stable bound e-h bound state by analogy with Hydrogenatom. This is exciton in semiconductors.

The energy levels of bound states (Hydrogenic model) lie below Eg

So far we assumed thatelectrons interact only with crystal lattice.

Consider attractive interaction between electron and hole.

; n=1,2….

; me,h*-electron and hole effective masses

Absorption spectrum revealing excitonic levels

Summary:

1. Optical transition in direct and indirect gap materials.Role ofphonons at high and low T in case of indirect transitions

2. Energy dependence of absorption coefficient. Concept of joint density of states.

3. Moss-Burstein effect in doped matearials

4. Excitonic effects. Energy spectrum of bound e-h state.

Ref. Charles Kittel, Introduction to Solid State Physics, 4th Edition p 364, 614

Previous Lecture on Topic 10, Optical transitions

Direct transitionIndirect transition

Topic11: Doping of Semiconductors. Donors and Acceptors

1. Control of conductivity. Central of microelectronics

2. Analogy with hydrogen atom

3. Derivation of expressions for binding energy and Bohrradius

4. For GaAs EB~5.8 meV, aB~99 A

5 Consistent with approximation of macroscopical dielectricconstant and effective mass

He

n Rm

mE

=

2

* 1

ε Be

n am

mr

=

*

ε

Group IV semiconductors Si, Ge, (C)

Donors- group V P, As, SbAcceptors- group III B, Ga, Al

Group III-V semiconductors GaAs, InSb

Donors –group VI (on As site) S, Se, Tegroup IV (on Ga site) Si

Acceptors- group IV (on As site) C, Sigroup II ( on Ga site) Zn, Be, Cd

Doping of semiconductors by donors and acceptors

Binding energy of electron(hole) on donor (acceptor)

Use analogy with hydrogen atom

Hydrogen Rydberg

Electron Bohr radius in Hydrogen

eVem

R en 6.13

)4(2 20

2

4

==πε

He

een R

m

m

nn

emE ×

−=−= 2

*

220

22

4* 11

)4(2 επεε

me – free electron mass; e-electron charge;

Aem

nr

en 53.04 02

22

== πε(for n=1)

For semiconductors one needs to take into account electron motion in crystal, rather than in vacuum. Need to consider effective electron mass

and dielectric constant

*em

ε

(for n=1)

02*

22

4πεεem

nr

en

=

Donor bound states Donor Bohr radius

For GaAs ee mm 067.0* =

12=ε

meVRm

memE H

e

ee 8.51

)4(2 2

*

20

2

4*

1 −=×

−=−=

επεε

Donor binding energy (n=1)

Donor Bohr radius

Aem

nr

en 984 02*

22

== πεε

The bound state wavefunction of electron on donor extends over many atomic diameters=> approximation of effective mass and dielectric constant is valid

Band gaps at room temperature

Ge 0.66 meVSi 1.1 eVGaAs 1.4 eV

Donor binding energiesGe 10 meVSi 45 meVGaAs 5.7 meV

Acceptor binding energiesGe 11 meVSi 46 meVGaAs 27 meV

Binding energies

1. Very much less than band gap

2. Of order kBT at 300 K

High degree of ionisation at room T => Effective doping allows control of conductivity

Make sure you understand (know)

1. Concepts of donor and acceptor2. Example of donors (acceptors )3. Hydrogen model for the energy levels of electron (hole)on donor (acceptor)4. Why this model works well5. Binding energies in comparison to thermal energies are small

J.R.Hook and H.E.Hall, Second edition P136-138

Topic 12: Semiconductor statistics, Intrinsic and Extrinsic Semiconductors

1. Intrinsic semiconductor. Fermi level close tocentre of gap.

2. Extrinsic conduction, result of doping.

3.Extrinsic, saturation and exhaustion regimes.Variation of carrier concentration with temperature.

4. Compensation. Compensation ratio. Number of ionised impurities at low and high temperature

Fermi distribution

The probability of occupation of a state with energy E is given by Fermi distribution

If there is no doping where is the position ofFermi level EF in band gap?

At final T since populations in bands small, EF must remain close tothe middle of the band gap

EF-Fermi level. By definition

TVF N

FE ,)(

∂∂= ; F- Free energy

Consider the case of T=0

Calculation of electron(hole) density as a function of T

)exp(2

22/3

2

*

Tk

EE

h

Tkmn

B

gFBee

−

= π

Electron density

2/3

2

*22

=

h

TkmN Be

c

π -effective density of states

(Eq. 1)

Intrinsic regime.

Band to band excitation of electrons and holes.Electron ne and hole np concentrations are equal

Here Nc, v –effective densities of states in conduction and valence bands respectively.

Example:At 300 K nintrinsic=2x1019 m-3 (Ge)

= 1x1016 m-3 (Si)

)2

exp(Tk

ENNnn

B

gvcpe

−==

Extrinsic conduction as a resultof doping

Consider material doped only with donors

At very low temperatures DB ETk <<ED- donor binding energy

most donors are not ionised

Fermi Level is close to 2/DgF EEE −≈

)2

exp()( 2/1

Tk

ENNn

B

DDce

−=Precise concentration of electrons

Compensation

For ND>NA,

At high T (kT>>ED) all impurities are ionised:ne = ND-NA

Both donors and acceptors are present

ND,A-concentration of donors (D) or acceptors (A)

At low T (kT<<ED):

ND>NA => n-type semiconductorND<NA=> p-type semiconductor

DgF EEE −≈)exp(

Tk

ENn

B

Dce

−≈

Temperature dependence of carrier numberin doped material

Temperature dependence of Fermi level and carrier number in doped material

Summary

1. Concept of Fermi level.

2. Position of Fermi level in doped and undoped material.

3. Temperature behaviour of Fermi level

4. Derivation of expression for electronconcentration using Fermi distribution

5. Compensation in material doped with donors and acceptors. Electron concentration at low and high T. Number of ionised impurities at low and high T.

J.R.Hook and H.E.Hall, Second edition p139-147

Topic 13: Hall effect

1. Motion of charge carrier in crossed electric and magnetic fields (d.c. ω=0, ω≠0 corresponds to cyclotronresonance )

2. Solve equation of motion

3. Hall coefficient RH=-1/(ne), expressions for mobility, conductivity

4. Hall field exactly balances the Lorentz force

5. Method to determine sign of charge carriers in semiconductor and metals

6. RH negative for e.g. simple alkali metals. Can be positive for divalent metals due to overlapping bands

)()(* BVEeV

dt

Vdm

ee ×+−=+

τ

Geometry of Hall effect

The Lorentz force acting on electrons

The Lorentz force is compensated by electric force

BVe ×−

Consider geometry of Hall bar

Assume that electrons are the main carriers

HEe−BandV are electron velocity and external magnetic field

HE is generated Hall electric field

Motion of charge carriers in constantelectric and magnetic fields

General equation of motion

)()(* BVEeV

dt

Vdm

ee ×+−=+

τ*em Electron effective mass

eτ Electron scattering time

);;( zyx VVVV =

Electron velocity

Calculation of Hall field

0=yV Since current cannot flow out of the bar

Electric field yE (Hall field) has value, since it exactly Counter balances the Loretz force due to B-field

Hall coefficient

Define Hall coefficientBj

ER

x

yH =

Here electric current in X direction (ne-electron concentration )

xe

eexex E

m

eneVnj ⋅=−= *

2τ

( )enmBEen

mEeBR

eexee

exeH

1

/

/*2

*

−=−=ττ

Electron concentration can be extracted from Hall effect

Electron mobility *e

e

m

eτμ =

Conductivity μσ ene=

Measuring both σ and ne we can obtain μ

If both electron and holes are present in the system, it is morecomplicated

Write equations of motion for both electrons and holes

If the mobility of minority carriers is high these carries may determine the sign of the Hall coefficient

( )( )eh

ehH npe

npR

μμμμ

+−=

22

Final answer:Here n, p are electron and hole concentrations

RH negative for e.g. simple alkali metals. Can be

positive for divalent metals due to overlapping bands

Summary

1. Hall experiment is used to determine carrier concentrationand mobility

2. Sign of Hall coefficient is determined by the sign of the main carriers

3. If both electrons and holes are present the mobility of minority carriers may define the sign of Hall

coefficient

J.R.Hook and H.E.Hall, Second edition p152-154

Topic 14 Cyclotron Resonance

1. Method to determine effective mass in semiconductors

2. Electron moves in circular orbits in constant

magnetic field B. The precession angular

frequency *ce

eB

mω =

3. a.c electric field ~ exp( ); ( ~ )cE i tω ω ω perpendicular B results in strong acceleration of electrons. => effect of cyclotron resonance

4. Condition for observation of cyclotron resonance 1cω τ >

τ − electron scattering time.

5. To observe cyclotron resonance one uses low temperature and pure samples to maximise τ , and higher magnetic fields to maximise ωc

6. Quantum mechanical description of cyclotron

resonance. Electron spectrum consists of Landau levels. Transition between Landau levels due to

photon absorption of energy cω

Electron motion in magnetic field B.

Simple treatment

*

* 2

e

e

dm e B

dt

m r e rB

υ υ

ω ω

= − ×

=

υ

-electron velocity r- radius of circular orbit

*em - electron effective mass

Electron moves in circular orbits about B with frequency of angular motion given by

*ce

eB

mω =

Electron motion in magnetic field B||Z and electric field

E

=(Ex , Ey)=(E cos(ωt), E sin(ωt)) in the xy plane.

*( )x x

x ye e

d eE B

dt m

υ υ υτ

+ = − +

*( )y y

y xe e

d eE B

dt m

υ υυ

τ+ = − −

,x yυ υ - Projections of electron velocity on X and Y τ − electron scattering time.

Introduce complex velocity

x yu iυ υ= +

*( exp( ) )

e e

du u eE i t iuB

dt mω

τ+ = − −

Solution u=u0 exp(iωt).

0 * ( / )e c e

ieEu

m iω ω τ=

− −

Cyclotron frequency *ce

eB

mω =

Maximum amplitude of electron velocity*

0 0u u at

~ cω ω

Radiation of frequency ~ cω ω will be absorbed.

Method to determine effective mass m* of either electron or hole.

Quantitative example

*/c eeB mω =

100

2c GHz

ωπ

=,

m*=0.07 me (me-free electron mass)

*

0.247e cmB T

e

ω= =

To observe cyclotron resonance one needs that a particle completes several orbits between collisions

1cω τ >

τ ∼1.5 ∗10−12 sec

The higher magnetic field, the higher the experimental frequency and thus the shorter scattering times that can be tolerated To observe cyclotron resonance one maximises scattering time by using low temperatures and investigating pure samples

It is possible to use cyclotron resonance to determine the sign whether charge carriers are electrons or holes. In the above sign convention for electron

*ce

eB

mω =

is positive=>

the electric field used is σ+ circularly polarised

E

=(Ex , Ey)=(E cos(ωt), E sin(ωt)) **********************************************

For holes *ce

eB

mω = −

is negative, and thus the

radiation is σ− circularly polarised

E

=(Ex , Ey)=(E cos(ωt), -E sin(ωt))

Quantum mechanical treatment of cyclotron resonance. In external magnetic field B||Z electron motion is quantised in xy plane. Quantised Landau levels are formed.

2

*( 1/ 2)

2z

c

kE n

mω= + +

kz – momentum in Z direction

Cyclotron energy *ce

eB

mω =

Cyclotron resonance corresponds to transition from one Landau level to the next one accompanied by

absorption of photon of energy *ce

eB

mω =

Quantised cyclotron orbits in k-space

Cyclotron resonance in Si

Few resonances are observed in Si because dispersion is anysotropic

Summary

1. Cyclotron resonance can be used to determine effective mass and sign of carriers 2. To observe cyclotron resonance scattering time should be long enough to allow electron to complete few circular orbits 3. In quantum mechanical treatment cyclotron resonance corresponds to transition between Landau levels Reference: Hook, Hall, Second Edition, pages 155-159

Topic 15: Mobility and Scattering Processes in Semiconductors. 1. Two dominant scattering mechanisms of electrons

affecting conductivity: scattering with ionised impurities (or missing atoms –vacancies and other structural defects) and scattering with phonons.

2. Mobility *e

e

m

τμ =, τ−scattering time.

Mattheisen’s rule: 1 1 1

( ) ( )ph impT Tτ τ τ= +

,

τph , τimp –scattering times with phonons and impurities, respectively

1 1 1imp phμ μ μ− − −= +

3 / 2 3 / 2~ ( ) ~ ; ~ ( ) ~imp imp ph phT T T Tμ τ μ τ −

for the mobilities determined by ionised impurity and phonon scattering respectively

3. Comparison with metals. In semiconductors impurity scattering is temperature dependent since electron velocity is temperature dependent.

In metals impurity scattering is temperature independent.

Scattering by ionised impurities in semiconductors: Defect (impurity) scattering is temperature dependent, since mean velocity of electrons is temperature dependent. In case of nondegenerate gas electrons obey Boltzman distribution. Average kinetic energy of electrons ( v -electron velocity)

* 2 3

2 2e Bm v k T

=

Scattering by charged impurities significant

when thermal energy ( ~ Bk T ) is of the order of

Coulomb potential (

1~

r )

:

1~Bk T

r

*Scattering cross-sectional area of impurity

22

1~ ~A r

T

*Mean free path of electron 21

~ ~l TA ;

*Electron scattering time

23 / 2

1/ 2( ) ~ ~imp

l TT T

v Tτ =

*Mobility due to impurity scattering

2/3~)(~ TTimpimp τμ

Analogy with Rutherford scattering of alpha particles: mean free path associated with such scattering ~ to the square of electron energy

2~l v => scattering time 3( ) ~ph

lT v

vτ =

Scattering with phonons in semiconductors: Mobility due to phonon scattering

3 / 2~ ( ) ~ph ph T Tμ τ −

Predominant scattering with longitudinal acoustic phonons (LA), which produce series of compressions and dilations and modulate energies of conduction and valence bands locally. The LA phonons which scatter electron must

have wavelengths phλ at least as large as the

electron wavelength eλ :

2 2~

ph e

π πλ λ

Scattering of electron with energy Bk T by phonon involves a major change of its momentum but not its energy, since the energy

of phonon with momentum

2 2~

ph e

π πλ λ is small

compare to Bk T

Free mean path between collisions 1

~lT

=>3/ 2( ) ~ ( ) ~ph ph

lT T T

vμ τ −=

Comparison to metals: In metals at low temperature conductivity and thus mobility is independent of T

At low T scattering with defects/impurities is dominant. The scattering rate is temperature independent since the electron velocity given by Fermi energy is temperature independent.

Summary

1. Two dominant scattering mechanisms of electrons: scattering with ionised impurities (or missing atoms –vacancies and other structural defects) and scattering with phonons.

2. Temperature dependence of the mobilities determined by ionised impurity and phonon scattering respectively in the limit of low and high T

3. Comparison to metals. In metals at low temperature conductivity and thus mobility is independent of T

Reference: J.S. Blackmore, Solid State Physics, Second Edition, pages 330-339

Topic 16: Plasma reflectivity, Plasma Oscillations

1. Collective response of electron gas to electromagnetic field 2. Dielectric constant in a material depends of the frequency of external

radiation ω and wavevector k

3. Oscillation of electron density at plasma angular frequency

0

22

εω

m

nep = occurs in the ultraviolet for typical metals. n-electron

density; m-electron effective mass;ε0-permittivity of free space 4. Dielectric function 22 /1)( ωωωε p−= ; Dispersion of plasmon

frequency as a function of k-vector derived

from222 ),( ckk =ωεω . c-speed of light in free space.

5. For pωω < , ε is real and negative, and hence k is imaginary. Wave is damped in metal, which has a very high reflectivity

6. For pωω > ε is real and positive , and k is real. wave propagates.

7. Colour of metals e.g. lithium, potassium, silver, copper gold 8. Quantised oscillations of plasma: plasmons. Observed in electron

energy loss experiments

Transverse electromagnetic field in a plasma Collective response of electron gas to an electromagnetic field Dielectric function in a metal ),( kωε depends on the wavevector and frequency of external radiation. Plasma frequency determines reflectivity of metal up to high photon energy (~3 eV)

Derivation of ε(ω) in presence of high density of carriers (electrons)

Motion of free electron in electromagnetic field E~exp(iωt)

Frequency dependence of dielectric function

Electromagnetic field in metal.

However colour of some metals arises from interband absorption. For Cu, light in with green and blue wavelengths is absorbed and red light is reflected. The same for Au. When semiconductors are heavily doped the plasma effect become significant

Longitudinal Optical modes in a Plasma. Plasmons

Don’t mix up with transverse modes!!

Plasmons are quanta of longitudinal plasma oscillations

Energy of plasmon phω

To observe, one measures energy loss spectra for electron reflected from metallic films

Summary.

1. Transverse optical modes in a plasma. 2. Reflection and propagation of transverse mode

3. Role of plasma frequency

4. Longitudinal modes in a plasma. Plasmons.

5. Experiments to reveal plasmons

Reference: Charles Kittel, Introduction to Solid State Physics, 4th Edition, p 269-278

Topic 17 Amorphous Semiconductors

1. No long range order, but local bonding still exists

2. Band edges strongly broadened

3. Electrons scattered by disorder-localised states near band edges

4. Defect states in gap due to dangling bonds

5. Dangling bond states can be removed by

hydrogen. Allows material to be doped. Permits applications

6. Growth is fast nonequilibrium process-atoms don’t

have time to reach equilibrium positions

7. Contrast with growth of crystals which is a “slow” equilibrium process

Amorphous semiconductors Local bonding present as in crystalline materials No long range order. Removes translational symmetry. Prepared as thin films by evaporation or sputtering Atom do not have time to achive crystalline order before positions become fixed on surface

Atoms are knocked out of target by high energy ions. Are then deposited on substrate

As a result of lack of translational symmetry momentum selection rule (∆∆∆∆ k=0) is relaxed Absorption edges broadened but bands and band gaps still exists- determined in part by local bonding Electrons strongly scattered by disorder-> localised states near band edges because disorder is very strong In addition there are many unsatisfied dangling bonds � large density of states in the middle of gap

Density of defect states is very high, up to 1025m-3

Compensates donor or acceptor doping and therefore doping is not effective However if amorphous Si is grown from SiH4 , dangling bonds are satisfied by H (“neutralises” defect states) and doping is effective Application – large area of displays, solar cells Very cheap to produce TFT – thin film transistors in laptops + solar cells

Topic 18 Quantum wells

1. Potential wells created by growth of thin layers of crystalline semiconductors

2. Quantum confinement of electron and hole states

3. New bad gap result

4. Many applications as light emitters

Quantum wells (crystalline) Thin layer of crystalline semiconductor of thickness < de Broglie wavelength of electrons (200 A, 50-100 A typical) leads to quantatisation of motion in Z direction. The layer consists of 20-40 atomic layers only In 3D electron kinetic energy

)(2

222

2

zyx kkkm

hE ++=

In case of electron motion in a thin layer z-motion is quantised=>

)()(2

22

2

zEkkm

hE nyx ++=

Motion in XY plane of the thin layer is unaffected

Alterating layer of smaller (GaAs) and larger (Al x Ga1-xAs) band gap materials. Creates potential well for electrons and holes.

Typical Quantum well

Potential well formed for electrons and holes

Eg –energy gap of bulk GaAs material. In case of quantum well above onset of optical absorption occurs at Eg

’, because the band gap is increased due to quantised motion in Z direction

Applications Laser, Light emitting diodes (LED), detectors Design of heterostructures based on quantum wells determines wavelength of emitted light: 1.5 µm for fibre optics communications 0.6 µm for DVDs 0.4 µm for Blue-Ray etc, etc……