FYS3410 - Vår 2014 (Kondenserte fasers fysikk)...• Kronig-Penney model ... oscillatory solution...

FYS3410 - Vår 2014 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html

Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11)

Andrej Kuznetsov

delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO

Tel: +47-22857762,

e-post: [email protected]

visiting address: MiNaLab, Gaustadaleen 23c

http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html

Lecture schedule (based on P.Hofmann’s Solid State Physics,chapters 1-7 and 11)

Module I – Periodic Structures and Defects 20/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h

21/1 Laue condition, Ewald construction, interpretation of a diffraction experimen 2h

22/1 Bragg planes and Brillouin zones (BZ) 2h

23/1 Elastic strain and structural defects 2h

23/1 Atomic diffusion and summary of Module I 2h

Module II - Phonons 03/2 Vibrations, phonons, density of states, and Planck distribution 4h

04/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models 2h

05/2 Comparison of different models 2h

06/2 Thermal conductivity 2h

07/2 Thermal expansion and summary of Module II 2h

Module III – Electrons 24/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 4h

25/2 Effect of temperature – Fermi- Dirac distribution 2h

26/2 FEFG in 2D and 1D, and DOS in nanostructures 2h

27/2 Origin of the band gap and nearly free electron model 2h

28/2 Number of orbitals in a band and general form of the electronic states 2h

Module IV – Semiconductors 10/3 Energy bands and effective mass method 4h

11/3 Density of states in 3D semiconductors and nanostructures 2h

12/3 Intrinsic semiconductors 2h

13/3 Impurity states in semiconductors and carrier statistics 2h

14/3 p-n junctions and optoelectronic devices 2h

Lecture 18: Energy bands in solids

• Bloch theorem

• Kronig-Penney model

• Empty lattice approximation

• Number of states in a band and filing of the bands

• Interpretation of the effective mass

• Effective mass method for hydrogen-like impurities


• Origin of the band gap and Bloch theorem






Why do we get a gap?

E

k p/a -p/a

At the interface (BZ), we have two counter-propagating waves eikx, with k = p/a, that Bragg reflect and form standing waves y

Its periodically extended partner

Let us start with a free electron in a periodic crystal, but ignore the atomic potentials for now

Why do we get a gap?

E

k p/a -p/a

y- y+

Its periodically extended partner

y+ ~ cos(px/a) peaks at atomic sites

y- ~ sin(px/a) peaks in between

Let’s now turn on the atomic potential

The y+ solution sees the atomic potential and increases its energy

The y- solution does not see this potential (as it lies between atoms)

Thus their energies separate and a gap appears at the BZ

k p/a -p/a

y+

y-

|U0|

This happens only at the BZ where we have standing waves

Bloch theorem

If V(r) is periodic with the periodicity of the lattice, then

the solutions of the one-electron Schrödinger eq.

Where uk(r) is periodic with the periodicity of the direct lattice.

uk(r) = uk(r+T); T is the translation vector of lattice.

“ The eigenfunctions of the wave equation for a periodic potential

are the product of a plane wave exp(ik·r) times a function uk(r)

with the periodicity of the crystal lattice.”








• Kronig and Penney assumed that an electron

experiences an infinite one-dimensional array of finite

potential wells.

• Each potential well models attraction to an atom in the

lattice, so the size of the wells must correspond roughly

to the lattice spacing.

Kronig-Penney Model

• An effective way to understand the energy gap in

semiconductors is to model the interaction between

the electrons and the lattice of atoms.

• R. de L. Kronig and W. G. Penney developed a

useful one-dimensional model of the electron lattice

interaction in 1931.

Kronig-Penney Model

• Since the electrons are not free their energies are less

than the height V0 of each of the potentials, but the

electron is essentially free in the gap 0 < x < a, where

it has a wave function of the form

where the wave number k is given by the usual

relation:

Kronig-Penney Model

• In the region between a < x < a + b the electron can

tunnel through and the wave function loses its

oscillatory solution and becomes exponential:

Kronig-Penney Model

• The left-hand side is limited to values between +1 and

−1 for all values of K.

• Plotting this it is observed there exist restricted (shaded)

forbidden zones for solutions.

Kronig-Penney Model

• Matching solutions at the boundary, Kronig and

Penney find

Here K is another wave number. Let’s label the

equation above as KPE – Kronig-Penney equation

Kronig-Penney Model

(a) Plot of the left side of Equation

(KPE) versus ka for κ2ba / 2 = 3π / 2.

Allowed energy values must

correspond to the values of k for

for which the plotted

function lies between -1 and +1.

Forbidden values are shaded in light

blue.

(b) The corresponding plot of energy

versus ka for κ2ba / 2 = 3π / 2, showing

the forbidden energy zones (gaps).

Kronig-Penney Model








Suppose that we have empty lattice where the periodic

V(x)=0. Then the e-s in the lattice are free, so that

Empty lattice approximation

in 3-dim.


In simple cubic (SC) empty lattice


Everything can be described

within the 1st B.Z.









k →

E

→

d

p

E

→

d

p2

[100] [110]

k →

Effective gap

o90Sin dk

p

p

Sin d

nk

o45Sin dk

p

Number of states in a band

Monovalent metals

Divalent metals

Monovalent metals: Ag, Cu, Au → 1 e in the outermost orbital

outermost energy band is only half filled

Divalent metals: Mg, Be → overlapping conduction and valence bands

they conduct even if the valence band is full

Trivalent metals: Al → similar to monovalent metals!!!

outermost energy band is only half filled !!!

Filing of the bands

SEMICONDUCTORS and ISOLATORS

Band gap

Elements of the 4th column (C, Si, Ge, Sn, Pb) → valence band full but no

overlap of valence and conduction bands

Diamond → PE as strong function of the position in the crystal

Band gap is 5.4 eV

Down the 4th column the outermost orbital is farther away from the nucleus

and less bound the electron is less strong a function of the position

in the crystal reducing band gap down the column

Filing of the bands

Multiple Quantum Wells (MQWs) for advanced LEDs

MQWs

Single quantum well - repetitions of

ZnO/ZnCdO/ZnO periods








• The electron is subject to internal forces from the

lattice (ions and core electrons) AND external forces

such as electric fields

• In a crystal lattice, the net force may be opposite the

external force, however:

+ + + + +

Ep(x)

-

Fext =-qE

Fint =-dEp/dx

Internal and external forces affecting an electron in crystal

• electron acceleration is not equal to Fext/me, but rather…

• a = (Fext + Fint)/me == Fext/m*

• The dispersion relation E(K) compensates for the internal

forces due to the crystal and allows us to use classical

concepts for the electron as long as its mass is taken as m*

+ + + + +

Ep(x)

-

Fext =-qE

Fint =-dEp/dx

Internal and external forces affecting an electron in crystal

Hole - an electron near the top of an energy band

• The hole can be understood as an electron with negative effective mass

• An electron near the top of an energy band will have a negative effective mass

• A negatively charged particle with a negative mass will be accelerated like a positive particle with a positive mass (a hole!)

p/a

E(K)

K

F = m*a = QE

Without the crystal lattice, the hole would not exist!

The hole is a pure consequence of the periodic potential

operating in the crystal!!!

E(K) and E(x)

p/a

E(K)

K

conduction band

valence band

EC

EV +

-

x

E(x)

Eg

Generation and Recombination of electron-hole pairs

conduction band

valence band

EC

EV +

-

x

E(x)

+

-

MQWs

Single quantum well - repetitions of

ZnO/ZnCdO/ZnO periods

Generation and Recombination of electron-hole pairs

a

b

x

y

p/a

E(Kx)

Kx

p/b

E(Ky)

Ky

Different lattice spacings lead to different curvatures for E(K)

and effective masses that depend on the direction of motion.

Real 3D lattices, e.g. FCC, BCC, diamond, etc.

light m*

(larger d2E/dK2)

heavy m*

(smaller d2E/dK2)


21

, 2

1c ij

i j

Em

k k


Ge

Si

GaAs

• energy (E) and momentum (ħK) must be conserved

• energy is released when a quasi-free electron recombines with a hole in the valence band:

ΔE = Eg

– does this energy produce light (photon) or heat (phonon)?

• indirect bandgap: ΔK is large

– but for a direct bandgap: ΔK=0

• photons have very low momentum

– but lattice vibrations (heat, phonons) have large momentum

• Conclusion: recombination (e-+h+) creates

– light in direct bandgap materials (GaAs, GaN, etc)

– heat in indirect bandgap materials (Si, Ge)

Direct and inderect band gap in semiconductors

Motion of free electrons

0 d d

mdt dt

p r

F p

0

E E E E Eˆ ˆ ˆ ˆˆ ˆ x

x x y z

x x y z

pv v x v y v z x y z

m p p p p

v =

p

px

E(p) Consider free electron

0

Dispersion relation

Newton’s law of motion

2 2 21

0 2 2 2

E E E=

x y z

mp p p

2 2 2 2

0 0 0E / 2 =( + + ) / 2 x y zp m p p p m m p v

Velocity

Mass

Effective mass

23

0 02, 1

31

, 0

, 1

1( ) ( )( ) ...

2

1... ( )

2

g i i j j

i j i j

g c ij i j i i i

i j

EE E k k k k

k k

E m p p p k k

k

E

ky

kx

k0

C.B. minimum

kox

koy

23

0 0

, 1

1( ) ( )( ) ...

2g i i j j

i j i j

EE E k k k k

k k

k

Taylor series near CB minimum

Effective mass tensor 2

1

, 2

1c ij

i j

Em

k k

Density of states

kx

kz

ky

Parallelepiped of Volume V=LxLyLz

Allowed states satisfy boundary conditions: 2i i ik L np

Each state occupies volume in k space

3 3 18 /( ) 8k x y zV L L L V p p

Consider a conduction valley along x or [100] direction

2 22 2 2 2

( )2 2 2

yx zg

L T T

kk kE k E

m m m

2 22 2 2 2

12 ( ) 2 ( ) 2 ( )

yx z

L g T g T g

kk k

m E E m E E m E E

2 2 2

2 ( ) 2 ( ) 2 ( ), = , = ;

L g T g T g

x y z

m E E m E E m E Ea a a

Half axes

The volume in k-space containing energies less than E

3/ 2

1/ 2 3/ 2

2

4 4 2( ) ( )

3 3k x y z L T gV E a a a m m E Ep p

Density of states

kx

kz

ky

3/ 2

1/ 2 3/ 2

2

4 2( ) ( )

3k L T gV E m m E Ep

3/ 2

1/ 2 3/ 2

2 2

2( ) 2 ( ) / ( )

3k k L T c

VN E V E V m m E E

p

Number of states with energies less than E

Multiply by number of valleys -G , divide by volume take derivative

3/ 23/ 2 *1 1/ 2 1/ 2 1/ 2

2 2 2 2

2( ) 2 1( ) ( ) ( )

2 2

nc L T g g

mN E Gg E G V m m E E E E

E p p

Number of states per unit volume per unit energy –density of states

Effective D.O.S. mass 1/3

* 2/3 2

n L Tm G m m

In GaAs *

n cm m

Density of states in the valence band

k

E

3/ 23/ 23/ 2 **1/ 2 1/ 2 1/ 2

2 2 2 2 2 2

22 21 1 1( ) ( ) ( )

2 2 2

plh hhv lh hh

mm mg E g E g E E E E

p p p

Count energy down

Effective D.O.S. mass is 2/3

* 3/ 2 3/ 2 ~p hh lh hhm m m m

Lecture 18: Energy bands in periodic crystalls






• Effective mass method for hydrogen-like impurities in semiconductors

Hydrogen like impurities in semiconductors

d

Hydrogen-like donor Hydrogen atom

P donor in Si can be modeled as hydrogen-like atom

Hydrogen atom - Bohr model

222

0

42

2

0

2

0

22

0

2

2

0

2

2

0

2

22

0

22

0

2

0

2

0

2

2

2

2

0

1

2)4(

2)4( :energy Total

242

1 :energy Kinetic

44 :energy Potential

4

1

4

4)(44

...3,2,1 ,for 4

1

n

emZEK

r

ZeVKE

r

ZemvK

r

Zedr

r

ZeV

n

Ze

mr

nv

mZe

nr

mr

n

mr

nmrrmvZe

nnmvrLr

vm

r

Ze

r

pp

p

pp

p

p

ppp

p


eV613)4(2 2

0

40

H .qm

E p

eV0.05

2

0s

0

0

*neV613

) π(42

q*n

20s

4

d

Km

m.

K

mE

eV613)4(2 2

0

40

H .qm

E p

Instead of m0, we have to use mn*.

Instead of o, we have to use Ks o.

Ks is the relative dielectric constant

of Si (Ks, Si = 11.8).

Hydrogen-like donor



















11/3 Energy band structure (continuation) 2h




Lecture 19: Energy band structure (continuation)

• Dynamics of electrons in a band

• Band-to-band transitions

Dynamics of electrons in a band

The external electric field causes a change in the k vectors of all electrons:

Eedt

kdF

If the electrons are in a partially filled band, this

will break the symmetry of electron states in the

1st BZ and produce a net current. But if they are in

a filled band, even though all electrons change k

vectors, the symmetry remains, so J = 0.

E

kx

ap

ap

v

kx

When an electron reaches the 1st BZ edge (at k =

p/a) it immediately reappears at the opposite edge

(k = -p/a) and continues to increase its k value.

As an electron’s k value increases, its velocity

increases, then decreases to zero and then becomes

negative when it re-emerges at k = -p/a!!

E e

dt

k d

Dynamics of electrons in a band

52

Band-to-band transitions

valence

band

conduction

band


gap size

(eV)

InSb 0.18

InAs 0.36

Ge 0.67

Si 1.11

GaAs 1.43

SiC 2.3

diamond 5.5

MgF2 11

valence

band

conduction

band


valence

band

conduction

band


electrons in the conduction band (CB)

missing electrons (holes) in the valence band (VB)



VB maximum

as E=0

conduction band

valence band

free electrons Band-to-band transitions

electrons in the conduction band (CB)

missing electrons (holes) in the valence band (VB)


for the conduction band

for the valence band

Both are Boltzmann distributions!

This is called the non-degenerate case.



VBM

CBM

μ


In quantum wells


FYS3410 - Vår 2014 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html

Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11)

Andrej Kuznetsov

delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO

Tel: +47-22857762,

e-post: [email protected]

visiting address: MiNaLab, Gaustadaleen 23c




















11/3 Energy bands (continuation) 2h




Lecture 21: Impurity states in semiconductors and carrier statistics

• Intrinsic and extrinsic semiconductors

• hydrogen-like impurities

• n- and p-type semiconductors

• equilibrium charge carrier concentration

• carriers in non-eqilibrium conditions: diffusion, generation and recombination







Intrinsic and extrinsic semiconductors

iE

pi

ni

Eg

iE

p0

n0

Eg

iE

p0

n0

Eg


p0

n0

Eg

Ed

Ei

EID nd









d

Hydrogen-like donor Hydrogen atom

P donor in Si can be modeled as hydrogen-like atom


222

0

42

2

0

2

0

22

0

2

2

0

2

2

0

2

22

0

22

0

2

0

2

0

2

2

2

2

0

1

2)4(

2)4( :energy Total

242

1 :energy Kinetic

44 :energy Potential

4

1

4

4)(44

...3,2,1 ,for 4

1

n

emZEK

r

ZeVKE

r

ZemvK

r

Zedr

r

ZeV

n

Ze

mr

nv

mZe

nr

mr

n

mr

nmrrmvZe

nnmvrLr

vm

r

Ze

r

pp

p

pp

p

p

ppp

p


eV613)4(2 2

0

40

H .qm

E p

eV0.05

2

0s

0

0

*neV613

) π(42

q*n

20s

4

d

Km

m.

K

mE

eV613)4(2 2

0

40

H .qm

E p

Instead of m0, we have to use mn*.

Instead of o, we have to use Ks o.

Ks is the relative dielectric constant

of Si (Ks, Si = 11.8).

Hydrogen-like donor


p0=0

n0=0

Eg

Ed

Ei

EID

p0

n0

Eg

Ed

Ei

EID nd








a) Energy level diagrams showing the excitation of an electron from the valence band to the conduction band.

The resultant free electron can freely move under the application of electric field.

b) Equal electron & hole concentrations in an intrinsic semiconductor created by the thermal excitation of

electrons across the band gap

N- and p-type semiconductors

Intrinsic semiconductor

a) Donor level in an n-type semiconductor.

b) The ionization of donor impurities creates an increased electron concentration distribution.


n-type Semiconductor

a) Acceptor level in an p-type semiconductor.

b) The ionization of acceptor impurities creates an increased hole concentration distribution


p-type Semiconductor


• Intrinsic material: A perfect material with no impurities.

• Extrinsic material: donor or acceptor type semiconductors.

• Majority carriers: electrons in n-type or holes in p-type.

• Minority carriers: holes in n-type or electrons in p-type.

)2

exp(Tk

Enpn

B

g

i

ly.respective ionsconcentrat intrinsic & hole electron, theare && inpn

e.Temperatur is energy, gap theis TEg

2

inpn


donor: impurity atom that increases n

acceptor: impurity atom that increases p

N-type material: contains more electrons than holes

P-type material: contains more holes than electrons

majority carrier: the most abundant carrier

minority carrier: the least abundant carrier

intrinsic semiconductor: n = p = ni

extrinsic semiconductor: doped semiconductor








Consider conditions for charge neutrality.

The net charge in a small portion of a uniformly doped semiconductor

should be zero. Otherwise, there will be a net flow of charge from one

point to another resulting in current flow (that is against out assumption

of thermal equilibrium).

Charge/cm3 = q p – q n + q ND+ q NA

= 0 or

p – n + ND+ NA

= 0

where ND+ = # of ionized donors/cm3 and NA

= # of ionized acceptors

per cm3. Assuming total ionization of dopants, we can write:

Equilibrium charge carrier concentration in semiconductors

Assume a non-degenerately doped semiconductor and assume total ionization of

dopants. Then,

n p = ni2 ; electron concentration hole concentration = ni

2

p n + ND NA = 0; net charge in a given volume is zero.

Solve for n and p in terms of ND and NA

We get:

(ni2 / n) n + ND NA = 0

n2 n (ND NA) ni2 = 0

Solve this quadratic equation for the free electron concentration, n.

From n p = ni2 equation, calculate free hole concentration, p.


• Intrinsic semiconductor:

ND= 0 and NA = 0 p = n = ni

• Doped semiconductors where | ND NA | >> ni

n = ND NA ; p = ni2 / n if ND > NA

p = NA ND ; n = ni2 / p if NA > ND

• Compensated semiconductor

n = p = ni when ni >> | ND NA |

When | ND NA | is comparable to ni,, we need to use the charge neutrality

equation to determine n and p.


33

17

20

0

2

0 1025.210

1025.2

cmn

np i

kTiEFE

enn i

)(

0

eVn

nkTEE

i

iF 407.0105.1

10ln0259.0ln

10

17

0

Si is doped with 1017 As Atom/cm3. What is the equilibrium

hole concentra-tion p0 at 300°K? Where is EF relative to Ei

Example


0 0a dn N p N








Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion.

Charge carriers in non-eqilibrium conditions

dx

dnqDJ ndiffn,

dx

dpqDJ pdiffp,

D is the diffusion constant, or diffusivity.


dx

dnqDqnJJJ nndiffndriftnn ε ,,

dx

dpqDqpJJJ ppdiffpdriftpp ε ,,

pn JJJ


• The position of EF relative to the band edges is determined

by the carrier concentrations, which is determined by the

net dopant concentration.

• In equilibrium EF is constant; therefore, the band-edge

energies vary with position in a non-uniformly doped

semiconductor:

Ev(x)

Ec(x)

EF


The ratio of carrier densities at two points depends exponentially on

the potential difference between these points:

1

2i2i112

1

2

i

1

i

2i2i1

i

2Fi2

i

1Fi1

i

1i1F

ln1

lnlnln Therefore

ln Similarly,

ln ln

n

n

q

kTEE

qVV

n

nkT

n

n

n

nkTEE

n

nkTEE

n

nkTEE

n

nkTEE


n-type semiconductor

Decreasing donor concentration

Ec(x)

Ef

Ev(x)

dx

dEe

kT

N

dx

dn ckTEEc Fc /)(

dx

dE

kT

n c

kTEE

cFceNn

/)(

Consider a piece of a non-uniformly doped semiconductor:

Ev(x)

Ec(x)

EF

εqkT

n


If the dopant concentration profile varies gradually with position, then the majority-carrier concentration distribution does not differ much from the dopant concentration distribution.

– n-type material:

– p-type material:

in n-type material

)()()( AD xNxNxn

)()()( DA xNxNxp

)()()()( AD xnxNxpxN

dx

dN

Nq

kT

dx

dn

nq

kT D

D

11


Band-to-Band R-G Center Impact Ionization


Direct R-G Center Auger


Little change in momentum

is required for

recombination

momentum is conserved

by photon emission

Large change in momentum

is required for recombination

momentum is conserved by

phonon + photon emission

Energy (E) vs. momentum (ħk) Diagrams

Direct: Indirect:


0nnn

0ppp

Charge neutrality condition:

pn

equilibrium values


• Often the disturbance from equilibrium is small, such that

the majority-carrier concentration is not affected

significantly:

– For an n-type material:

– For a p-type material:

However, the minority carrier concentration can be

significantly affected.

so |||| 00 nnnpn

so |||| 00 ppppn


n

n

t

n

p

p

t

p

for electrons in p-type material

for holes in n-type material

Consider a semiconductor with no current flow in which

thermal equilibrium is disturbed by the sudden creation

of excess holes and electrons.


Uniformly doped p-type and n-

type semiconductors before the junction is formed. Internal electric-field occurs in

a depletion region of a p-n

junction in thermal equilibrium


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