Solid state physics for Nano

Lecture 6: electronic band structure of solids

Prof. Dr. U. Pietsch

)cos(2),( kaEmkE mm

at

m

Band of allowed states Overlap between two neighboured atomic states results in 2 states : bonding and antibonding state. Bonding between 1023 atomic in solids results in 2 1023 solution, which are such as dense that they overlap and create „bands“ separated by „forbidden“ zones. Empirically described by

describes decrease of energy due to bonding, 4 measures the band width between k=0 and k=p/a. Many properties of solids can be described in terms of „electron gas“ (Sommerfeld) : electrons are moving as non-interacting particles with spin1/2

)cos(2),( kaEmkE mm

at

m

It describes a dispersion relation between energy E and momentum k

m

pv

mE

2

²²

2These „free electrons have kinetic energy

m

kE

2

²²

electron gas in solid

ikx

k e0

What happens, if k is approaching k=p/a ?

If k< p/a l>a no interaction of electron wave with lattice if k≈p/a l ≈ a electron wave „reflected“ at lattice planes

Ansatz: Electron wave propagates free as long as k< p/a

For k≈p/a =G/2 wave is Bragg reflected

ikx

k e

0

)sin(2)(

)cos(2)(

00

00

kxiee

kxee

ikxikx

ikxikx

1²|| ikxikxee)²(sin²||

)²(cos²||

kx

kx

²||

Largest probability between atoms

²||

Largest probability at atoms

Vkm

kE

Vkm

kE

²2

²)(

²2

²)(

In terms of energy:

gEVV Energy gap

Valid also k=2p/a ; k=3p/a….. Extended zone scheme

Parabel origin also at k=p/a ; k=2p/a….. Overlap of peridodic zone scheme Folding into -p/a < k < p/a 1st Brillouin zone

Reduced zone scheme

“effective mass” of electrons in solid

Due to specific badnd curvature

k

Ev

1

dt

dk

k

E

dt

dv

²

²1

eEF

dt

dk

Fk

E

dt

dv

²

²

²

1

amdt

dv

k

EF *

²

²

²

²

²

²

1

*

1

k

E

m

effective mass

Number of states per band in 1st BZ: )³2

(a

Vp

a

ka

pp

1 or 3 electrons in upper band

³³ NaL )³2

(L

Vp

N

La)³

2/()³

2(

pp

Because of spin 2N states per energy band

2 or 4 electrons in upper band: Be, Mg, Ca, ……Sn T>18°C metal Si, Ge – typical semiconductors, Eg 0.1..2 eV C, SiC … isolators Eg = 10eV

Na: 1s²2s²2p6,3s1= Ne 3s1 upper band

is filled with 1 electron only

Higherst populed energy – FERMI energy

EF

Density of states in 1D (DOS) particle in box model, standing waves

nn E

xm

²

²

2

²0)()0( Lnn

F

N

n

Nn

n

n

NEN

LmE

nLm

EEF

3

1

2/

1

2/

1

³2

²2

1

2

²

²²2

1

2

²22

dE

dn

ervallenergy

statesofnumberED 2

int

__)(

EE

mL

n

mL

dE

dnED

1

2

41

²

²82)(

Total energy is:

)²4(

²

2

²²

22

²

L

N

mL

n

mE F

F

N=2n due to spin

Fermi energy

Fermi energy in 3D Standing waves in 3D )sin()sin()sin()(

³8 zyxr

L

n

L

n

L

n

Lnzyx ppp

Alternative rki

Vk er 1)(

),,(),,(),,( LzyxzLyxzyLx kkk

........;;2

;...;4

;2

,0 zyx kkL

n

LLk

ppp

xiknLiniLnxiLLxniLxik xx eeeeee pppp 22/2)/)(2()(

)(2

²

2

²²)( 222

zyx kkkmm

kkE

m

kkE F

2

²²)(

Fermi sphere with radius kF

Boundary conditions

Dispersion relation for electrons

Cu Si

33

²3)³

2/(

3

42 F

F kV

L

kN

p

pp 3/2

3

)²3

(2

²

²3

V

N

mE

V

Nk

F

F

p

p

FF mvk

3/1)²3

(V

N

mvF

p

FF kTE

Fermi momentum 3/1)²3

(V

NmvF

p

Fermi velocity

Fermi temperature

Example for Li: N/V= 4.6 1022 cm-3 =1 el/Li atom

kF = 1.1 108 /cm = 1.1 /A vF = 1.3 108 cm/s = 1.3 10³ km/s= 0.04% c EF = 4.7 eV TF = 55.000K

https://www.zahlen-kern.de/editor/

Number of electrons in sphere with radius kF

Analogous to phonons: Density of states ||

)³2

(2)(Egrad

dFLED

kp

For perfect sphere:

²²/²

²4)³

2(2)(

mkV

mk

kLED

p

p

p

m

kEgradk

²||

²4 kdF p

EmV

ED

2/3

²

2

²)(

p

Density of states in 3D (DOS)

EF

Influence of Temperature on DOS Bonding energy for lowest energy levels are large, all levels occupied

F

F

EEfor

EEforEf

___0

__1)(At T=0 probability of occupation

At T>0 , thermal energy kT, only levels close to EF can be redistributed 1)exp(

1),(

kT

µETEf

FERMI – DIRAC distribution µ ≈ EF chemical potential

Impact of electrons to specific heat

Only electrons close to EF can gain energy of about kT this results in range EF- ½ kT non occupied states (2) and in range EF + ½ kT (1) and newly occopied states, all other electrons cannot contribute to specific heat

2

1 )),(1()()()2(

),()()()1(

0

TEfEDEEdE

TEfEDEEdEE

F

F

E

F

E

Fel

0

)()(T

fEDEEdE

T

Ec Fel

Integral differs from zero only close to EF

]²1)[exp(

)exp(

²

kT

EEkT

EE

F

F

F

kT

EE

T

f

For kT << EF : D(E) = D(EF) TkEDT

fEDEEdE

T

Ec FFel ²)(²)()(

31

0

p

FF EmV

ED

2/3

²

2

²)(

p

3/2)²3

(2

²

V

N

mEF

pusing

TT

TNk

E

kTNkc

FF

el pp ²²21

21

Separation between electron and phonon contribution to specific heat

C/T

T²

³ATTctotal

Impact of electrons to specific heat

m

Ev F

F

2

eNN

jv

pe

D)(

Electrons close to EF contribute to charge transport only. Their characteristic velocity is Fermi velocity,

expressing the maximum velocity an electron can reach before scattering at another electron, defect, phonon aso. Is is much larger vF >> drift velocity

Electrical charge transport

By collosion with defects, phonons etc. Fermi sphere is shifted constantly

dt

dk

dt

dpeEF

Means, Fermi sphere is constantly shifted

m

Fdk – is collision time

Ne, Np – concentration of electrons and holes, vD is the real velocity of signal transport . Force, F, acting on an electron within electric field E

In metals n is constant (and high) , µ and j increase with decreasing the temperature or decrease with increasing the temperature due to the increase of scattering at thermally excited phonons and defects. Opposite for semiconductors, the µ and j increases for increasing temperature because more and more electrons are able to occupy states within the conduction band and can contribute to charge transport.

Electric transport

ETeTnETTj )()()()(

µ - is the mobility (Beweglichkeit) [cm²/Vs], n – number of „free“ charge carrier.

Fmdvdk 4m

eEvdv D

m

Enenedvj

² using

ne is transported amount of charge, – scattering time Gained energy in field E is donated to lattice

²²²

2

)²2(21

Em

Enedvm

t

Q

Joules heat

Temperature dependence

m

ne

²

Cyclotron resonance spectroscopy

Hall effect

BevF rBermF ²rv m

eBc

c

eBm

*

m* from exp. c

vx

Bz

Fy

][ BveF

zxy BevF creates Hall field EH

zxH BeveE xx

zxH

nevj

BvE

zxHzxH BjRBjne

E 1 RH – Hall constant

Sign and concentration Of charge carriers

Specific band structure of semiconductors

Generell insolator structure: with Eg >kT, 0.2…3 eV

j )exp(0kT

Eg

Band structure in k-space:

[100] [111] G – origin of Brillouin zone

Direct band semiconductor

Dispersion of electrons Dispersion of hole states

*2

²²)(

e

VBCBm

kEkE

*2

²²)(

h

VBm

kkE

Indirect band semiconductor

Absorption vs direct gap = 3.4 eV generation of electron hole-pairs Recombination via indirect gap : 1.1 eV

phononglight EE

Few band structure parameters

DOS of Semiconductors

1)exp(

1),(

kT

µETEf E-µ >>kT; EF≈µ )exp(),(

kT

µETEf

Probability to occupy a state in CB:

*2

²²)(

e

VBCBm

kEkE

ge

e EEmV

ED

2/3*

²

2

²2)(

p

Number of electrons on CB:

0

*

0

)exp(²

2

²2

1),()(

kT

EEE

mdETEfEDn

g

ge

ep

)exp(²

22

2/3*

kT

EkTmn

ge

p

Probability to find holes )exp(),(1kT

µETEff eh

E

mVED h

h

2/3*

²

2

²2)(

p

Number of holes in VB: )exp(²

22

2/3*

kT

µkTmp h

p

For intrinsic semiconductor yields : n=p )exp(exp( 2/32/3

kt

µm

kT

Eµm h

g

e

kT

Eµ

m

m g

h

e

2

ln2

3*

*

*

*

ln4

3

2 h

eg

m

m

kT

Eµ

Because 2nd term is very small : µ=EF= Eg/2

Thermally excited charge carriers: )2

exp()(²

22 4/3**

2/3

kT

Emm

kTpn

g

hn

p

pe eµTpeµTnkT

Eg)()()

2exp(0

Carrier concentration in CB and VB

eVESicmpn

eVEGecmpn

gii

gii

14.1__10*4.1

67.0__10*5.2

310

313

µe(Si) = 1300 Vs/cm²; µh(Si)=500Vs/cm² µe(GaAs) = 8800 Vs/cm²; µh(GaAs)=400Vs/cm²

Temperature dependence of intrinsic charge carrier concentration

Doping of semiconductors

Energy states of dopands calculated by effective mass theory: (H-atom modell): replacing e²e²/e and m me,h. where e is the statistic dielectric constant of host material.

Effective number of charge carriers can be increased by DOPING:

replacement of very few Si atoms by P, As,…(donators) or by B, Al,…(acceptors).

²

1

²6.13

,

,nm

meVE

he

ADe

Donator energy

Radius of donator electron

²0529.0,

, nm

mnmr

he

AD

e

²²2

4

n

meEH

6meV (Ge), 20meV (Si)

8nm (Ge), 3nm(Si)

Donator doped semiconductors : n- conductivity Acceptor doped semicductor : p - conductivity

)exp(

)exp(

kT

Ep

kT

En

A

d

Temperature dependence of doped semiconductors

Low temperatures

high temperatures

)2

exp(kT

Epn

g

Excitation of dopands

Formation of e-h pairs