FYS3400 - Vår 2020 (Kondenserte fasers fysikk)...On 22/4 10-12 Origin of the band gap; Nearly free...

FYS3400 - Vår 2020 (Kondenserte fasers fysikk)http://www.uio.no/studier/emner/matnat/fys/FYS3410/v17/index.html

Pensum: Introduction to Solid State Physics

by Charles Kittel (Chapters 1-9 and 17 - 20)

Andrej Kuznetsov

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

Tel: +47-22857762,

e-post: [email protected]

visiting address: MiNaLab, Gaustadaleen 23a

http://www.uio.no/studier/emner/matnat/fys/FYS3410/v16/index.html

2020 FYS3400 Lecture Plan (based on C.Kittel’s Introduction to SSP, Chapters 1-9, 17-20 + guest lectutes)

Module I – Periodity and Disorder (Chapters 1-3, 19, 20) calender week

To 16/1 12-13 Introduction.

On 22/1 10-12 Crystal bonding. Periodicity and lattices. Lattice planes and Miller indices. Reciprocal space. 4

To 23/1 12-13 Bragg diffraction and Laue condition

On 29/1 10-12 Ewald construction, interpretation of a diffraction experiment, Bragg planes and Brillouin zones 5

To 30/1 12-13 Surfaces and interfaces. Disorder. Defects crystals. Equilibrium concentration of vacancies

On 5/2 10-12 Mechanical properties of solids. Diffusion phenomena in solids; Summary of Module I 6

Module II – Phonons (Chapters 4, 5, and 18 pp.557-561)

To 6/2 12-13 Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D

On 12/2 10-12 Lattice heat capacity: Dulong-Petit and Einstein models 7

To 13/2 12-13 Effect of temperature - Planck distribution;

On 19/2 10-12 canceled 8

To 20/2 12-13 Periodic boundary conditions (Born – von Karman); phonons and its density of states (DOS); Debye models

On 26/2 10-12 Comparison of different lattice heat capacity models; Thermal conductivity. 9

To 27/2 12-13 Thermal expansion

On 4/3 10-12 Summary of Module II 10

Module III – Electrons I (Chapters 6, 7, 11 - pp 315-317, 18 - pp.528-530, 19, and Appendix D)

To 5/3 12-13 Free electron gas (FEG) versus free electron Fermi gas (FEFG);

On 11/3 10-12 DOS of FEFG in 3D; Effect of temperature – Fermi-Dirac distribution. 11

To 12/3 12-13 canceled

teaching free week 12

Module IV – Disordered systems (guest lecture slides)

On 25/3 10-12 Thermal properties of glasses: Model of two level systems (Joakim Bergli) 13

To 26/3 12-13 Experiments in porous media (Gaute Linga)

On 1/4 10-12 Electron transport in disordered solids: wave localization and hopping (Joakim Bergli) 14

To 2/4 12-13 Theory of porous media (Gaute Linga)

Easter 15

Module V – Electrons II (Chapters 8, 9 pp 223-231, and 17, 19)

On 15/4 10-12 After Easter repetition; Heat capacity of FEFG in 3D 16

To 16/4 12-13 DOS of FEFG in 2D - quantum well, DOS in 1D – quantum wire, and in 0D – quantum dot

On 22/4 10-12 Origin of the band gap; Nearly free electron model; Kronig-Penney model 17

To 23/4 12-13 Effective mass method

On 29/4 10-12 Effective mass method for calculating localized energy levels for defects in crystals 18

To 30/4 12-13 Intrinsic and extrinsic electrons and holes in semiconductors

On 06/5 10-12 Carrier statistics in semiconductors 19

To 07/5 12-13 p-n junction

On 13/5 10-12 Summary of Modules III and V

To 14/5 12-13 Repetition - course in a nutshell

Exam: oral examination

May 28th – 29th

Intrinsic and extrinsic electrons and holes in semiconductors; carrier statistics

• Band-to-band transitions

•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

4

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 electronsBand-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


Lecture: Electrons and holes in semiconductors; carrier statistics







Intrinsic and extrinsic semiconductors

iE

pi

ni

EgiE

p0

n0

Eg

iE

p0

n0

Eg


p0

n0

Eg

Ed

Ei

EIDnd


Hydrogen like impurities in semiconductors

d

Hydrogen-like donorHydrogen 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

Hydrogen atom - Bohr model

eV613)4(2 20

40

H .qm

E

eV0.05

2

0s

0

0

*neV613

) π(42

q*n2

0s

4

d

Km

m.

K

mE

eV613)4(2 20

40

H .qm

E

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

EIDnd


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


FYS3400 - Vår 2020 (Kondenserte fasers fysikk)...On 22/4 10-12 Origin of the band gap; Nearly free...

Documents

Transcript of FYS3400 - Vår 2020 (Kondenserte fasers fysikk)...On 22/4 10-12 Origin of the band gap; Nearly free...