overview - georgia institute of technologyweb.chemistry.gatech.edu/class/8873/semiconductor.pdf ·...

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Overview:Inorganic semiconductors and semiconductordevices+ links to organic semiconductors

Why inorganic semiconductors ?

They are “electronic materials“ (at this time probably the mostimportant ones)

Very well established models• often they can be applied to conjugated materials

• sometimes they cannot

• and sometimes nobody knows so far

• materials of interest and their crystal-structures

• band-structures of inorganic (and organic) materials(gap energies, reciprocal space, effective mass, density of states …)

• optical properties of semiconductors

• semiconductor statistics and doping

• semiconductor-heterostructures

• p-n junctions

• light emitting devices and lasers

• photodetectors and solar cells

• field effect transistors

2

MaterialsInorganic semiconductors (usually crystalline)

from: S.M. Sze, “Semiconductor Devices, Physics and Technology“

n

n

Organic semiconductors (crystalline or amorphous; much bigger variety !)

poly(acetylene)

poly(para-phenylene)

n

SS

n

poly(para-phenylenevinylene)

poly(thiophene)

polymers

Oligomers and molecules (molecular crystals)

sexiphenyl

pentacene

aluminum tris(8-hydroxyquinoline) Alq3

Cu-phthalocyanine

3

Typical crystal-structures of inorganic semiconductors

S. M. Sze, “Physics of semiconductor devices“

crystal structure: basis vectors

all lattice sites can be defined by:

At each lettice point one finds the basis

Diamond: fcc with 2 C in basis

c,b,a��

cnbmalR��

++=

Zinc blende and wurzite lattice generated from

tetragonal building blocks

from Ch. Weißmantel and C. Hamann “Grundlagen der Festkörperphysik”

Zinc blende wurzite

4


Wurzite lattice (CdS, ZnS …..)

GaAs: face-centered cubic (fcc) lattice with GaAs in the basis�� two interpenetrating fcc lattices

CdS: two interpenetratinghexagonal close-pacced (hcp) lattices

Crystal-structures of molecular crystals and crystallineconjugated polymersVery different packing forces: to a significant extent dispersion forces (van der Waals) + electrostatic contributions

especially molecular crystals: often unit cells with a much lower symmetry

b

a

χ

c

bcrystal structure of p-terphenyl• monoclinic• herringbone• layered

5

Inorganic semiconductors – strong interaction in all 3 dimensions�� we NEED to describe them by a 3D bandstructure

Band-structures of semiconductors

Would be a 4D plot ???

Yu, Cardona, „Fundamentals of Semiconductors“

Reciprocal spacedefine reciprocal basis vectors:

In k-space (momentum-space), they play the same role, the basis vectorsplay in real space; (they define the “repeat-unit“ in k-space)

• perpendicular to the faces of the unit cell• length inversely proportional to that of the corresponding vector in real space�� reciprocal lattice, where every point can be expressed by:

( ) ( ) ( )cba

bac,

cba

acb,

cba

cba ***

��

��

��

��

��

��

××=

××=

××= πππ 222

*** clbkahG��

++=for Bloch-type wavefunctions one finds for the associated energies:

( )[ ] ( )[ ]rErEGkk

��

+= ψψ

6

Unit cell in reciprocal space: 1st Brillouin zone

crystal structure: positions periodic in real space

band-structure: k-vectors “periodic“ in reciprocal space



Band-structures of “most important“ inorganic semiconductors

7

What can we easily see in the band-structure diagram:

• bandgap energy

• width of the bands

• number of bands

• effective mass (transport)

• direct vs. indirect gap (optical properties)

Different k-directions for bottom of the conduction band …(cannot be derived from symmetry arguments)

Shapes of constant energy surfaces in Ge, Si, and GaAs for the unoccupied bands


“simplified“ band-picture• k-dependence not explicitly considered• often very useful when discussing device

• shaded area denotes energy range, in which Bloch-states aseigenfunctions of the Schrödinger equation exist• x-axis: position !


8

Effective mass

Let‘s assume a free electron, which is described by:

Ψ=Ψ∆−=Ψ Em

H2

2�

�� eigenfunctions (plane waves):

energy eigenvalues:(energy-wavevector relations)

( ) rkier�� =Ψ

( )m

kkE

2

22�

��=

parabolic “band“

bands in semiconductors also (approximately) parabolic …..

(at least close to the band-edges)

let‘s define a “crystal electron“ with an effective mass m * so that the band structurecan be written as:

“crystal electron“ (QUASIPARTICLE) behaves like a free electron with mass m* !(indee the case, e.g., when applying a field)

How can we get m* from the actual band structure E(k) ?

( )*m

kkE

2

22�=

Hamaguchi, “Basic semiconductor physics“

9

Or for the more general 3D case (where m* is a second rank tensor)(in this way it also contains, e.g., anisotropic transport properties)

( )0

2

2

2

11

kk* dk

kEdm =

=�

( )0

2

2

11

kkjiij* kk

kEm

=∂∂

∂=�

�

The effect of the “surrounding“ (i.e., periodic lattice) is condensedinto the effective parameter m* !

approximation OK, for all future considerations in this course …

Density of statesSo far we know for which combination of k and E there exists an eigenstate …

Often it is sufficient to know, how many states exist between a certainenergy E and E+dE per unit volume V

(the derivation can be found in any standard solid-state physics textbook …)

3D extended system (i.e., bulk semiconductors):

for E > Ec in the case of conduction band statesand E < Ev for valence band states

• increases with increasing m*• square-root dependence on E

( ) V,C

*

EEm

Eg −��

��

�=

23

22

22

1�π

g(E)

10

Dpendence of the density of states on the dimensionalityof the electron-system

3D:

2D:

1D:

E > Ec for conduction band statesand E < Ev for valence band states

( ) V,CEEEg −∝

( ) .constEg =

( )V,CEE

Eg−

∝ 1

3D��

� �

1D

2D��

� �

��

� �

Semiconductor StatisticsNext question: Which of the states are occupied ?

for now: at the thermal equilibrium …

two cases: (i) intrinsic semiconductors (defect concentration << p.p.m.)

(ii) doped semiconductors


11

Fermi-Dirac distribution function(probability for a Fermion state – here an electron or hole state – to be occupied)

EF … Fermi-Energy (only“adjustable“ parameter)

( )��

��

� −+

=

TkEE

exp

EF

B

F1

1

Note: probability F istemperature-dependent !!!

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

0.0

0.2

0.4

0.6

0.8

1.0

��

�

�� ! �#" �$#%�& �

Approximations:(E-EF) > 3 kBT

(E-EF) < -3 kBT

i.e., Fermi-level deeplyinside the gap

( ) ��

��

� −−≈TkEE

expEFB

F

( ) ��

��

� −−−≈

TkEE

expEFB

F1

Number of electrons in the conduction band (per unit volume):

( ) ( )�=top

C

E

EC dEEFEgn

inserting the expressions for g(E) and F(E) this becomes:

��

��

� −+

��

��

�=

top

C

E

E

B

F

//

B*

dE

TkEE

exp

ETkmn

1

22

1 2123

22�π

Looks complicated – and in the general case does not have an analyticsolution ��

BUT: we can find a solution for the approximate case mentioned before(„EF is far enough away from the band-edges !“)

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In this case: number of electrons in the conduction band

Exponential decrease of the number of electrons in the conductionband with the „distance“ of the Fermi-level from the band-edge !

equivalently for the number of holes in the valenece band:

== number of electron states that are NOT occupied

23

222

/

Be*

CB

FCC

TkmNwith

TkEE

expNn ��

��

�=��

�

��

� −−≈

�π

23

222

/

Bh*

VB

VFV

TkmNwith

TkEE

expNp ��

��

�=��

�

��

� −−≈

�πNC and NV are called the effective densities of states in the conduction and valence bands

��

��

� −−=��

�

��

� −−��

�

��

� −−=

TkEE

expNNTkEE

expTkEE

expNNnpB

VCVC

B

VF

B

FCVC

or:

The product of p and n depends only on the material (EG, me*, mh*) and on temperature. It is independent of doping !

If n is increase by n-type doping, p has do decrease accordingly and vice versa !

��

��

�−=

TkE

expNNnpB

GVC

13

Intrinsic semiconductors:

inpn ≡=ni … intrinsic carrier density

Continous thermal excitations of electrons from the VB into the CBIn thermodynamic equilibrium: balanced by carrier recombination …

Where is the Fermi level in an intrinsic semiconductor ?

from n = p:

At 0K, EF is in the middle of the gap of an intrinsic semiconductor !

��

��

�−==

TkE

expNNnpnB

GVCi 2

��

��

�+

+=

e*

h*

BVCF m

mln

TkEEE

43

2

Simplified BS

Density of states (DOS)

Fermi-Dirac distribution


Carrier density

14

Doped Semiconductors

donor-level: neutral if filled by an electron and positive if emptyacceptor-level: neutral if filled by an electron and positive if empty

There are deep and shallow donors and acceptors.



15

Qualitatively:

donors increase n and consequentlydecrease p

acceptorsincrease p and consequentlydecrease n


Quantitative description:

add dopands (typically 1012 to 1018 cm-3)'' '' EF has to adjust to preserve charge neutrality

(reasonable) assumption for shallow dopands: all ionized !

donors with density ND: n=ND

or

acceptors with density NA: p=NA

Doping with donors: EF approaches the conduction band edge EC

Doping with acceptors: EF approaches the valence band edge EV

��

��

�−=

D

CBCF N

NlnTkEE

��

��

�−=

A

VBVF N

NlnTkEE

16

At low temperatures the complete ionization approximation does not apply!

Charge neutrality:

At very high temperatures intrinsic effects start to dominate !

+− +=+ DA NpNn


Optical properties of semiconductors

Energy- and momentumconservation !

Rosencher and Vinter, “Optoelectronics“

17

Comparison of Si and GaAsE

nerg

y(e

V)


direct excitations indirect excitations

Indirect excitations

simultaneous absorption of a photon and generation or annihilation of a phonon

(( (( very improbable process (3-particle process) !(( (( small associated absorption coefficient !(( (( vanishingly small emission probability !

Silicon can be (and is) used in solar cells but not in LEDsand lasers (only in combination with color centers)

18

Absorption coefficient of Ge as a function of temperature

G.G. Macfarlane et al, Phys. Rev. 108, 1377 (1957)

Direct (vertical) absorption

For the absorption process to happen there needs to be a filled state in the conduction band and an empty state in the valence band !

absorption coefficient proportional to the corresponding density of states !

( ) Geff Eg −��

��

�= ω

µπ

ω ��

�

23

22

2

21 with

h*

e*

eff mm111 +=

µ�� joint density of states

GE−∝ ωα �

19

Excitonic effects

Experimental spectrum for GaAs:

Absorption coefficient of GaAs as a function of temperature

M. D. Struge, Phys. Rev. 127, 768 (1962)

Not “exactly“ E1/2 dependence !

Reason for the abrupt onset of the absorption and peak: excitonic effects

So far: electron is excited from state in VB to a state in the CB

BUT (even if we stay within a single-particle picture):

Interaction between the positively charged hole and thenegatively charged electron

bound electron-hole pair = exciton

Most simple description: treated like H-atom withand �r of the medium

h*

e*

eff mm111 +=

µ

GaAs: µeff very small and �r )) )) 13 EB )) )) 4 meV r1S )) )) 300 Å

overview - georgia institute of technologyweb.chemistry.gatech.edu/class/8873/semiconductor.pdf ·...

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