fys3400 - vår 2020 (kondenserte fasers fysikk)

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

Lecture: P-N junction

• Repetition: intrinsic and extrinsic semiconductors

• Charge carrier transport mechanisms – diffusion and drift

• Band bending as a function of carrier concentration

• P-N junction in equilibrium

• Gauss and Poisson equations for the depletion region

•P-N junction with applied external bias

Intrinsic and extrinsic semiconductors

Summary of Charge CarriersIntrinsic and extrinsic semiconductors

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

Diffusion of charge carriers

dx

dnqDJ ndiffn,

dx

dpqDJ pdiffp,

D is the diffusion constant, or diffusivity.

Diffusion of charge carriers

• The process in which charge particles move because of an

electric field is called drift.

• Charge particles will move at a velocity that is proportional to

the electric field.

Ev

Ev

ne

ph

Drift of charge carriers

dx

dnqDqnJJJ nndiffndriftnn ε ,,

dx

dpqDqpJJJ ppdiffpdriftpp ε ,,

pn JJJ

Diffusion + drift of charge carriers

• 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

Band bending as a function of carrier concentration

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


dx

dEe

kT

N

dx

dn ckTEEc Fc /)(

dx

dE

kT

n c

kTEE

cFceNn

/)(

Ev(x)

Ec(x)

EF

εqkT

n


<= P-type, low EF

- = fixed ionized acceptors

+ = mobile holes, p

<= N-type, high EF

+ = fixed ionized donors

- = mobile electrons, n

What happens when these bandstructures come into contact?

• Fermi energy must be constant at equilibrium, so bandsmust bend near the interface

• Far from the interface, bandstructures are conserved

P-N junctions in equilibrium

Time < 0

P-type piece N-type piece

Time < 0, i.e. before the contact is established


At time = 0, slam the two pieces together

Time =0, the contact is “just” established


Hole gradient

Jp, diffusion = -qDp dp/dx = current right, holes right

Electron gradient

Jn,diffusion = -qDn dn/dx = current right, electrons rightleft


Question: How long this diffusion will og

Hole gradient

Jp, diffusion = -qDp dp/dx = current right, holes right

Electron gradient

Jn,diffusion = -qDn dn/dx = current right, electrons rightleft


Question:

How long the diffusion

will og on!?

• When the junction is first formed, mobile carriers diffuseacross the junction (due to the concentration gradients)

– Holes diffuse from the p side to the n side, leaving behind negatively charged immobile acceptor ions

– Electrons diffuse from the n side to the p side, leaving behind positively charged immobile donor ions

A region depleted of mobile carriers is formed at the junction.

• The space charge due to immobile ions in the depletion region establishes an electric field that opposes carrier diffusion.

+++++

––

–––

p n

acceptor ions donor ions


E

E

Depletion Region


How big is the built-in voltage?

RightiFLeftFibi EEEEqV )()(

P side N side

i

aLeftFi

kTEE

ia

a

n

NkTEE

enN

Np

Fi

ln)(

)(

i

dRightiF

kTEE

id

d

n

NkTEE

enN

Nn

iF

ln)(

)(


2ln

lnln

i

dabi

i

d

i

abi

n

NN

q

kTV

n

N

q

kT

n

N

q

kTV

Na acceptor level on the p side

Nd donor level on the n side

How big is the built-in voltage?


• One side of the junction is heavily doped, so that the Fermi level is close to the band edge.

• e.g. p+-n junction (heavy B implant into lightly doped Si substrate)

i

dGbi

i

DRightiF

GViLeftFi

n

N

q

kT

q

EV

n

NkTEE

EEEEE

ln2

ln)(

2/)(


"P+ - N" => N a >> Nd => xp << xn

"P - N+" => N a << Nd => xp >> xn


p-type n-type

DN AN

-

-

-

-

-

-

-

-

-

-

-

-

-

+++++++++++++

0E

biq

,p diffJ

,p driftJ

,n diffJ

,n driftJ

−

−

+

+

−

Thermal

generation

+

Thermal

generation


neutral p region

+++++

––

–––

p n

acceptor ions donor ions

depletion region neutral n region

charge density (C/cm3)

distance

Charge is stored in the depletion region.

Gauss and Poisson equations for the depletion region

2

2

( ) ( ) ( )d x dE x x

dx dx

00

1( ) ( ) ( )

x

xE x E x x dx

dE

dx

1 encl

S V

QE dA dV

Gauss’s law describes the relationship between the charge density and the

electric field.

Poisson’s equation describes the relationship between the electric field

distribution and the electric potential

00( ) ( ) ( )

x

xx x E x dx

Gauss and Poisson equations in one dimension


0 ( ) ( ) ( 0)apo po

s

qNE x x x x x

000

0

0

0 , 0 and 0

0 np

nd

paxxxxx

xxqN

xxqNx

xno x

x

-xpo

ρo(x)

-qNa

qNd

xno x

x

-xpo

E0(x)

s

nod

s

poa xqNxqNE

)0(0

00 0

0

( )( ) ( ) ( ) 0

( ) ( )

(0 )

noxd

no nox

s s

dno

s

no

x qNE x dx E x x x

qNE x x x

x x

Gauss’s Law

p n

p n


pn

x

E0(x)

s

nod

s

poa xqNxqNE

)0(0

xno-xpo

22

22 pos

a

nos

dx

qNx

qN

0(x)

xxno-xpo

Poisson’s Equation


• The quasi-neutral p and n regions have low resistivity, whereas the depletion region has high resistivity. Thus, when an external voltage VD is applied across the diode, almost all of this voltage is dropped across the depletion region.

• If VD > 0 (forward bias), the potential barrier to carrier diffusion is reduced by the applied voltage.

• If VD < 0 (reverse bias), the potential barrier to carrier diffusion is increased by the applied voltage.

p n

+++++

––

–––

VD

P-N junction with applied external bias

pn

x

E0(x)

s

nod

s

poa xqNxqNE

)0(0

xno-xpo

22

22 pos

a

nos

dx

qNx

qN

0(x)

xxno-xpo

bi

Built-in potential bi=

-xp xn

-xp xn

bi-qVD

Higher barrier leads to less current!


pn

x

E0(x)

s

nod

s

poa xqNxqNE

)0(0

xno-xpo

22

22 pos

a

nos

dx

qNx

qN

0(x)

xxno-xpo

bi

Built-in potential bi=

-xp xn

bi-qVD

Lower barrier lead to more current!

-xp xn


• As VD increases, the potential barrier to carrier

diffusion across the junction decreases*, and

current increases exponentially.

ID (Amperes)

VD (Volts)

* Hence, the width of the depletion region decreases.

p n

+++++

––

–––

VD > 0The carriers that diffuse across the

junction become minority carriers in

the quasi-neutral regions; they then

recombine with majority carriers,

“dying out” with distance.

D( 1)qV kT

D SI I e


• As |VD| increases, the potential barrier to carrier

diffusion across the junction increases*; thus, no

carriers diffuse across the junction.

ID (Amperes)

VD (Volts)

* Hence, the width of the depletion region increases.

p n

+++++

––

–––

VD < 0A very small amount of reverse

current (ID < 0) does flow, due to

minority carriers diffusing from the

quasi-neutral regions into the depletion

region and drifting across the junction.


• Diode IV relation is an exponential function

• This exponential is due to the Boltzmann distribution of carriers

versus energy

• For reverse bias the current saturations to the drift current due to

minority carriers

1dqV

kTd SI I e

dqV

kT

d

s

I

I

1

( )d d SI V I

Diode I-V Curve


When a large reverse bias voltage is applied, breakdown occurs and

current flows through the diode increases dramatically.


When a large reverse bias voltage

is applied, breakdown occurs and

current flows through the diode

increases dramatically.


• Zener breakdown or tunneling mechanism, occurs in a highly doped p-njunction, while the conduction and valance bands on opposite sides of thejunction become so close during the reverse-bias that the electrons on thep-side can tunnel from directly VB into the CB on the n-side.

• Avalanche breakdown mechanism occurs when electrons and holes

moving through the depletion region and acquire sufficient energy from

the electric field to break a bond i.e. create electron-hole pairs by colliding

with atomic electrons within the depletion region. These newly created

electrons and holes move in opposite directions due to the electric field

and thereby add to the existing reverse bias current.

fys3400 - vår 2020 (kondenserte fasers fysikk)

Documents