Chem 253, UC, Berkeley Fermi Surfacenanowires.berkeley.edu/teaching/253b/2016/253B-2016-02.pdf ·...

1

Chem 253, UC, Berkeley

Fermi Surface

EF

KF

K

m

kkE x

2)(

22

ar

ra

4.02

22

Chem 253, UC, Berkeley With periodic boundary conditions:

1 zzyyxx LikLikLik eee

z

zz

y

yy

x

xx

L

nk

L

nk

L

nk

2

2

2

2D k space:Area per k point:

yx LL

22

3D k space:Area per k point: VLLL zyx

38222

A region of k space of volume will contain: allowedk values.

33 8

)8

(V

V

2


For divalent elements: free –electron model

ar

ra

56.0

22


3

Chem 253, UC, BerkeleyFor nearly free electron:

1. Interaction of electron with periodic potential opens gap at zone boundary

2. Almost always Fermi surface will intersect zone Boundaries perpendicularly.

3. The total volume enclosed by the Fermi surface depends only on total electron concentration, not on interaction


Alkali Metal Na, Cs: spherical Fermi surface

Alk. Earth metal: Be, Mg:: nearly spherical Fermi surface

ar

ra

4.02

22

ar

ra

56.0

22

2D case

4



5



6



7



8


Brillouin Zone of Diamond and Zincblende Structure (FCC Lattice)

Sign Convention Zone Edge or

surface : Latin alphabets

Interior of Zone: Greek alphabets

Center of Zone or origin:

Chem 253, UC, BerkeleyBand Structure of 3D Free Electron in FCC in reduced zone scheme

E(k)=(2/2m) (kx2+ ky

2 + kz2)

Notation:

<=>[100] direction

X<=>BZ edge along [100] direction

<=>[111] direction

L<=>BZ edge along [111] direction

9


Comparison between Free Electron and Real Electron Band Structure of Si


*m

e

E

vd

10


h


11


Motion of carrier in field:

Group velocity: transmission velocityof a wave packet

dk

dE

dk

dvg

1

Parabolic

*

22

*

22

2

2

hv

ec

m

kEE

m

kEE

Wave packet made of wavefunctions near a particular wavevector k

Acceleration:

dt

dk

dk

Ed

dkdt

Ed

dt

da g

2

22 11


Motion of carrier in field:

Group velocity: transmission velocityof a wave packet

dk

dE

dk

dvg

1

Parabolic

*

22

*

22

2

2

hv

ec

m

kEE

m

kEE

Wave packet made of wavefunctions near a particular wavevector k

Acceleration:dt

dk

dk

Ed

dkdt

Ed

dt

da g

2

22 11

dt

dk

mdt

dv

kvmp

*

*

2

2

2

1

*

1

dk

Ed

m

Effective mass

12



2

2

2

1

*

1

dk

Ed

m

Positive m*: the band has upward curvature 02

2

dk

Ed

If the energy in a band depend only weakly on k, then m* very large

1/* mm When very small. 2

2

dk

Ed

Heavy carrier

13


)(22

)( 222222

zyx kkkmm

kkE

2

2

2

1

*

1

dk

Ed

m


2

2

2

1

*

1

dk

Ed

m E

k

14


Heavy hole

Light hole


Group Material Electron me Hole mh

IVSi (300K) 1.08 0.56

Ge 0.55 0.37

III-VGaAs 0.067 0.45

InSb 0.013 0.6

II-VIZnO 0.29 1.21

ZnSe 0.17 1.44

15


Excitons The annihilation of a photon in exciting an

electron from the valence band to the conduction band in a semiconductor can be written as an equation: e+h.

Since there is a Coulomb attraction between the electron and hole, the photon energy required is lowered than the band gap by this attraction

To correctly calculate the absorption coefficient we have to introduce a two-particle stateconsisting of an electron attracted to a hole known as an exciton


– Excitons represent the elementary excitation of a semiconductor. In the ground state the semiconductor has only filled or empty bands. The simplest excitation is to excite one electron from a filled band to an empty band and so creating an electron and a hole

– Exciton is neutral over all but carries an electric dipole moment and therefore can be excited by either a photon or an electron

16


222

422 2

2 n

k

hn

em

r

eE e

nn

rm

nhv

e2

2

22

r

e

r

vme

02

22

22

4an

em

hnr

en

r1=Bohr radius =a0=0.529 Ao

r

e

r

e

r

e

r

evmE e

2

)(2

1

)(2

1

2

22

22

k= 13.606 eV


Excitons0

222

22

4an

em

hnr

en

**

222

4

2

1112

)(

he

n

mm

n

eEE

r

erU

Binding energy:

en m

evn

eEE

2222

4

6.132

Exciton Bohr Radius:

e

heex

m

mme

hr 529.0)

11(

4 **22

2

22

422 2

2 hn

em

r

eE e

nn

H

)(22

)( 222222

zyx kkkmm

kkE

Reduced mass

17


mevER nx

26.13

m

r 529.0


SemiconductorEnergy gap

(eV)at 273 K

Effective mass m*/m Dielectric

constantElectrons Holes

Ge 0.67 0.2 0.3 16

Si 1.14 0.33 0.5 12

InSb 0.16 0.013 0.6 18

InAs 0.33 0.02 0.4 14.5

InP 1.29 0.07 0.4 14

GaSb 0.67 0.047 0.5 15

GaAs 1.39 0.072 0.5 13

18


ExcitonSemiconductor Eg Rx or Eex

meV

rex

nm

Si 1.11 0.33; 0.50 14.7 4.9

Ge 0.67 0.2; 0.3 4.15 17.7

GaAs 1.42 0.0616

(0.066, 0.5)

4.2 11.3

CdSe 1.74 (0.13, 0.45) 15 5.2

Bi 0 0.001 small >50

ZnO 3.4 (0.27, ?) 59 3

GaN 3.4 (0.19, 0.60) 25 11

)/;/(

/........**

hhee mmmm

m

mevEn 2

6.13

m

r 529.0


h

hh

Exciton bindingenergy

Related to DOSotherwise dissociates

19


h


20


h


21



h

hh

22


Implication in solar cell


Thin film PV

Si PV

23



24



25



h

26


Fermi’s Golden Rule:

Transition Rate

Atkins, Molecular Quantum Mechanics, Oxford

•Time-dependent perturbation theory: treat excitations which depend on time•Optical transition: view the solid with unperturbed Hamiltonian H0 as being perturbed by the time-dependent EM field H’(t) generated by the incident photon flux.

)('0 tHHH

)('2 2

mlml EElHm

is the photon energy+: emission-: absorption


27



Assume the state m and l are the valence and conduction band states, then

vcHcHvlHm ''' Define: Joint density of states


Fermi’s Golden Rule: S

dkn

Let's introduce an energy surface S in k-space such that Ec − Ev =

dk =dSdkn

))()((8

2)(

3

kEkEdk vcvc

S is the surface of all possible direct optical transitions with h = Ec - Ev

28



*

22

*

22

2

2

hv

ec

m

kEE

m

kEE

At critical point where

Large JDOS contribution.



29


h

hh


ifM 22)(


() Density of states

M: transition matrix elements

Spontaneous emission rate

dvVM fiif Operator for the physical interactionthat couples the initial and final states

30


Selection Rule: Electric Dipole (E1) Transition

Light interaction with dipole moment (p=ex): xeEpEH

In general, the wavelength of the type of electromagnetic radiation which induces, or is emitted during, transitions between different atomic energy levels is much larger than the typical size of an atom.

1ikxeElectric dipole approximation

)(0),( tkxieEtxE

Light as harmonic EM plane wave

02

xkxx

dVxeEH n

V

mPmn )0(Transition dipole moment

rdzM

rdyM

rdxM

32112

32112

32112

For x, y, z polarized light


Selection Rule: Electric Dipole (E1) Transition

Dipole Moment

Matrix element (dipole moment) is non-zero allowed electric dipole transition

Parity of wavefunction: sign change under inversion about the origineven parity: f(-x)=f(x)odd parity: f(-x)=-f(x)

Initial/final wavefunctions must have different paritiesfor allowed electric dipole transition!

22)( M

rdzM

rdyM

rdxM

32112

32112

32112

31


Electronic Transitions in H atoms

Hydrogen atom: lowest state 1S, optical transition between 1S & 2S?Both states are symmetric, angular momentum l=0

No electronic transition between 1S and 2S!


Electronic Transitions in H atoms

Hydrogen atom: lowest state 1S, optical transition between 1S & 2P?2P is asymmetric, angular momentum l=1

Electronic transition between 1S and 2P is allowed!

32


Selection Rules

Symmetric function (gerade):Asymmetric function (ungerade):

The operator of the electric field: -(x)=-x

)()(

)()(

xx

xx

Transition between two gerade functions:

Transition between gerade and ungerade functions:

0)0(

0)0(

21

21

gdVuugdVH

udVgugdVH

P

PForbidden

Allowed

Selection rule for electronic transition: 1l


Electronic Transitions: Particle in a box

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