17. Surface and Interface Physics

Reconstruction and Relaxation Surface Crystallography Reflection High-Energy Electron Diffraction Surface Electronic Structure Work Function Thermionic Emission Surface States Tangential Surface Transport Magnetoresistance in a Two-Dimensional Channel Integra] Quantized Hall Effect (IQHE) IQHE in Real Systems Fractional Quantized Hall Effect (FQHE) P-N Junctions Rectification Solar Cells and Photovoltaic Detectors Schottky Barrier Heterostructures

n

-

N

Heterojunction Semiconductor Lasers Light-Emitting Diodes

Reconstruction and Relaxation

Dangling bonds in diamond Surface  3 outermost atomic layers.

Unreconstructed surface : Outermost layer same as in bulk except for a contraction in interlayer distance.

C.f., interatomic distances in molecules are smaller than those in solid.

Reconstructed surface (occurs mostly in nonmetals): Bonding on surface rearranged to eliminate dangling bonds.

Superstructure may (Si) or may not (GaAs) be formed. Surfaces of nominally high indices can be built up from low ibdex ones by steps.

Attachment energies are low at steps, hence facilitate chemical activities.

Surface Crystallography

5 distinct 2-D nets: oblique, square, hexagonal, rectangular, & centered rectangular. Substrate net // surface is used as reference.

E.g., for the (111) surface of a cubic substrate, the substrate net is hexagonal & the surface net is referred to these axes.

Let

c

1 ,

c

2 be translational vectors for the surface mesh &

a

1 ,

a

2 be those for the substrate.

c c

1 

P a a

   

P

11

P

21

P

12

P

22

a

  

a

If the angles of the 2 meshes are equal, Wood’s notation is often used:  

c a

1 1 

c

2

a

1  

R



α

is the angle of relative rotation between the meshes (

R α

is omitted if α = 0 ).

p

= primitive

c

= centered

Reciprocal lattice vectors

c

j

*

for the surface are defined by *

c c

i

  2  

i j

c

3  1

c*

Reciprocal net points represented in 3-D as rods  surface.

Diffracted beam given by intercepts of Ewald sphere with rods.

Low Energy Electron Diffraction ( LEED ):

E

~ 10-1k eV Pt (111) 51eV 63.5eV

gracing angle

Reflection High-Energy Electron Diffraction

RHEED Radius

k

of Ewald sphere for 100 keV

e

’s  → Ewald sphere ~ flat plane → Intercept with rods ~ line 10 3 A –1 >> 2π /

a

 1 A –1 .

Surface Electronic Structure: Work Function

Work function

W

 ε vac – μ Operationally, the vacuum level ε vac is the energy of an The chemical potential μ is often called the Fermi level.

e

more than 100A outside the metal.

W

depends on surface orientation because of the surface dipoles.

Einstein relation for photoemission:   

W

Thermionic Emission

Consider an

e

-gas in vacuum in equilibrium with a metal.

 

ext



k T B n

ln

n Q n Q



m k T B

2  2  3/ 2 By definition: 

ext W

→

n



n e



B

Flux of

e

leaving metal when all

e

’s are drawn off is equal to the incident flux:

J n

 1 4

n v



n k T B

2 

m

Charge flux:

J e



e J n

    2

e m k T B

2  2 3 2  

e



W k T

Richardson-Dushman equation

Surface States

Weak binding approximation: Let the outward normal to the surface be in the +

x

direction.

U

    0 

G U G e iG x

for

x

 0

x

 0  

vacuum

 

G



n



a



out



e



s x s

,

x

> 0    2 2

s

2

m



in



e



C k



C e



iG x



x

< 0

ψ ψ in

real →

J

= 0.

real only if

k

=

G

/ 2

.

→ 

in



e q x



C G

/ 2

e iG x

/ 2 

C



G

/ 2

e



iG x

/ 2 

C



G

/ 2 

C G

* / 2

s

,

q

are determined by the conditions that

ψ

&

ψ

 are continuous at the surface.

The bound state energy ε is obtained by solving the 2-component secular equation.

Tangential Surface Transport

Surface bound states affects the thermal distributions of

e μ

= same everywhere → band-bending.

&

h

near surface (

μ

shifted ).

Surface highly conducting Inversion layer on

n

-type semiC.

Accumulation layer on

n

-type semiC.

Thickness & carrier conc

n S

of surface layer controlled by 

E

→ MOSFET

C gate



V gate

conductance 

Width Length n e b S b

= mobility Prob 2

Magnetoresistance in a Two-Dimensional Channel

The static magnetoconductivity tensor in 3D was obtained in Prob 6.9 as

B



B j z

 1   0   

C

 2     1  

C

0   

C

1 0 0 0 1    

C

 2      

E E y E x z

   0 

ne

2 

m

For a surface 

z

-axis, we set  

C

1 →

n



n S



N L

2 

xx

 

yy

 0 

x y



B

Alternatively, in a cross field

E y

&

B z

, we have from Chap 6

v D



c E y B z



C



eB mc

along

x

.

In a frame moving with

v D

,

e

is stationary → there is extra field

E y

  

c z

 

E y

In the lab frame,

j x



D



B z E y

→ 

x y



B

I x



j L x y



n e c S B

Hall resistance

y



H



V y I x



n e c S V y B



B n e c S j x j y

 

xx

 

E x yx E x

 

xy E y

 

yy E y σ xx

=

σ yy

= 0 →

j y

= 0 →

E y

 

xy



yy E x

 * 

j x E x

  →

j x

   

xx

  2

xy



yy

 

E x

Integral Quantized Hall Effect (IQHE)

• •

V pp

~ 0 (σ* ~  ) at some

V g

.

Plateaus in

V H

near such

V g

.

•

V H

/

I SD

=

h

/

e

2

s

at plateaus ( IQHE )

h e

2  25.813

k

 Strong field 

C

→ Landau levels are either filled or empty If ε F falls on a Landau level:

s e B s h c



n S B

 = 18T.

T

= 1.5K.

I SD

= 1 μA.

 Pauli Exclusion principle → only inelastic scattering possible.

Low

T

→ required phonons not available.

→ σ* ~ 

H



h s e

2  2

s c

   

e

2

c

 1 137 Explanation too simplified

IQHE in Real Systems

Ideal crystal Real crystal Landau levels are broadened by impurities / defects in real crystals.

Also, some Landau levels are partially filled unless ε F Yet in the IQHE,

ρ H

→ Better model needed ( & provided by Laughlin ) = Landau level.

is accurately quantized in dirty samples & over a range of

V g

.

Laughlin’s thought experiment: 2-D plane rolled into cylinder.

B



B



B

I



I x



I



U



t

 

V I x x



I c

 

t

→

V y



V z



V H I



c

 

U



• • In a dirty system, there’re 2 types of carrier states: Extended states: continuous around loop.

Localized states: not continuous around loop.

Extended & localized states do not coexist at the same

E.

• • In the presence of Φ : Extended states: enclose Φ →

E

changes with period δΦ = Localized states: do not enclose Φ →

E h c

/

e

. not change; effect like gauge transformation.

When ε F falls in the localized states, all extended states below ε F before & after a flux change δΦ. But an integral number

N

of

e

’s will be transferred (usually 1

e

are filled both per Landau level).

N

is integral because the system is identical before & after the flux change.

The corresponding energy change is δ

U

=

N e V H

.

→

I



c

 

U

  2

N e V H h



H



V H I



h N e

2

Fractional Quantized Hall Effect (FQHE)

FQHE : Hall effect with ρ H quantized to fractional values.

unbiased junction

p-n

Junctions

p-n

junction = single crystal with different dopings.

Interface may be less than 10 –4 cm thick. Majority carriers will diffuse into the other side.

The excess charges left behind set up an

E

field directed from

n

to

p

to oppose further diffusion.

In equilibrium, μ for all carriers must be a constant everywhere.

For

h

: For

e

: 

h



k T B

ln

p



e



k T B

ln

n



e

 

e

 

const



const

The absence of net current flow is accomplished in the junction by the exact cancelling between the generating and recombination current.

J nr



V



J n g

 0

p-n

junction in Ge Similarly for holes.

Rectification

Reverse biased:

J nr J n g



J nr



J n g

→

n

  

J nr e



e V

/ 

J nr



e



e V

/  1  Forward biased:

J J nr n g



J nr



J n g

→

n

  

J nr e e V

/ 

J nr



e e V

/

I



I S



e e V

/  1  Well satisfied in Ge, but not so much in others.

 1 

Solar Cells and Photovoltaic Detectors

Light with 

ω > E g

on

p-n

junction →

e-h

pair

e-h

pair diffused into junction: separated by built-in

E

→ Forward voltage across junction ( Photovoltaic effect )

Schottky Barrier

Schottky barrier : metal-SemiC junction 4 

ne

→

d

2 

d x

2   4  

ne

→    2  

ne x

2 where

x

= 0 indicates the right-hand edge of barrier and the contact is at

x

= –

x b

. Let the potential at the contact be

φ

0 relative to the right-hand side:

x b

   2  0

ne

ε =16,

e φ

0 = 0.5 eV,

n

= 10 16 cm –3 →

x b

= 0.3 μ m.

Heterostructures

Heterostructures : Layers of 2 or more different semiC.

→ band structure design.

Lattice mistmatch negligible → heterojunction = single crystal with different site occupancies across junction E.g., Ge / GaAs (

a

 5.65A ) 3 types of band edge offsets: GaAs / Ge Good matches: AlAs / GaAs InAs / GaSb GaP / Si ZnSe / GaAs GaAs / (Al,Ga)As Both

e

&

h

on the right are GaSb / InAs

n

-

N

Heterojunction

Junction similar to Schottky barrier.

Quantum well created for

e

on

n

-side.

If

n

is lightly doped, impurity scattering will be negligible in well → mobility limited by lattice scattering, which falls off sharply for low

T

.

E.g., μ  10 7 cm 2 V –1 s –1 observed in GaAs / (Al,Ga)As.

If thickness of

N

is reduced below depletion layer thickness, all

e

conduction // interface will be on the

n

side.

( high mobility

e

on

n

-side separated from their donors on

N

-side ) → 2-D

e

-gas, high-speed FET, …

Semiconductor Lasers

Direct gap well 

r e h

Inversion condition (in well): 

c

 

v

 

g



n

 

p



eV

 

g

Structure itself is an EM cavity ( flat // ends: radiation emitted in plane of junction ) (Al,Ga)As / GaAs / (Al,Ga)As

p n

GaAs: 8383A (1.48eV, near IR) 50% power to light conversion efficiency.

90% differential efficiency for small changes.

For optical fibre transmissions Ga x In 1-x P y As minimize loss.

1–y are used to

Light-Emitting Diodes

In GaAs, inter-band photon are absorbed within 1 μm (strong absorption).

The direct gap ternary GaAs 1–x P x shortens λ with increasing

x

.

→ 1 st visible (red) LED.

Blue LED uses In x Ga 1–x N – Al y Ga 1–y N.