PETER PAZMANY CATHOLIC UNIVERSITYConsortiummembersSEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

**Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben

***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg.

Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework**

Consortium leader

PETER PAZMANY CATHOLIC UNIVERSITY

Consortium members

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

The Project has been realised with the support of the European Union and has been co-financed by the European Social Fund ***

PHYSICS FOR NANOBIO-TECHNOLOGY

Principles of Physics for Bionic Engineering

Chapter 10.

Heuristic Models for Semiconductors

(A nanobio-technológia fizikai alapjai )

(Heurisztikus modellek félvezetőkhöz)

Árpád I. CSURGAY,

Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ

Table of Contents

10. Heuristic Models for Semiconductors 1. Semiconductor Materials

2. Semiconductors in Thermal Equilibrium 3. Contact Potential

4. Carrier Transport in Semiconductors

1. Semiconductor materials

Semiconductor materials – high purity (109 :1, 1010 :1) crystals composed of

(i) elements (Si, Ge), (ii) IV-IV compounds

(SiC, SiGe),

(iii) III-V compounds

(GaAs, GaP, GaSb, InP, AlP) (iv) II-VI compounds

(Zn Se, ZnTe, Cd S),

(v) alloys (Al_xGa_1–xAs,GaAs_1–xP_x, Al_xGa_1–xAs_ySb_1–y, Ga In As P ).

II III IV V VI

4

Be

5

B

6

C

7

N

8

O

12

Mg

13

Al

14

Si

15

P

16

S

30

Zn

31

Ga

32

Ge

33

As

34

Se

48

Cd

49

In

50

Sn

51

Sb

52

Te

80

Hg

81

Tl

82

Pb

83

Bi

84

Po

In semiconductors there are no electrons in the conduction band at T = 0, and at T > 0 every electron in the conduction band left behind a ‘hole’ in the valence band.

The large number of electrons in the conduction band and the

large number of holes in the valence band behave similar to the free electrons in the conduction band of the metals.

Holes are virtual particles with positive charge and positive effective mass.

2. Semiconductors in thermal equilibrium

The electronic macro-state in thermal equilibrium is

where is the density of admitted states (band structure), is the Fermi function. Density of admitted states for electrons in

the conduction band

Density of admitted states for holes in the valence band

     

dN   E dE  g E f E d ,E

 

g E ^{f E}

 

   

0 0

( ) 3 2 eff

n 3 C C

4π 2 h .

n

n

V m

g E  E  E  K  E  E

  



0 0

p 3 2 eff

p 3 V p V

4π 2

h . V m

g E  E  E K  E  E

Electron density in the conduction band

Probability that an energy level is populated by holes (is equal to the probability that the level is not populated by electrons)

Hole density in the valence band

   

C0 C0 B

n n C

k

d 1 d .

e 1

E E

E E T

n g E f E E K E E E

 

   

 

F F

B B

k k

1 1

1 .

e 1 e 1

E E E E

T T

 

 

 

V

F B

p V

k

1 d .

e 1

E

E E T

p K E E _ E



 



Thermodynamic equilibrium of intrinsic semiconductors

The Fermi level of an intrinsic semiconductor is determined by the charge neutrality principle: the number of electrons in the

conduction band, n, is equal to the number of holes in the valence band:





C0 B F

C0 B

n 3 2 k eff

0 C 3

k

2 k

1 d , 2

e 1 h

EF E T

n C E E C

E T

m T

n K E E E N e N



 

    







V V

B F

B

p 3 2 k eff

V V 3

k

2 k

1 d , 2

e 1 h

E EF

E

T

p E E V

T

m T

p K E E E N e N









    



i i

n  p

F C0 V F

B B

k k

C e V e ,

E E E E

T T

i i

n N N p

 

  

 

0 0  

p

V C V V C eff

F B B n

C eff

1 3

k ln k ln .

2 2 2 4

E E N E E m

E T T

N m

 

   





F C0 B

n 3 2 k eff

C C 3

2π k

e , 2 .

h

E E T i

m T

n N N



 





V F

B

p 3 2 k eff

V V 3

2π k

, 2 .

h

E E T i

m T

p N e N



 

2kB

i C V e ,

E T

n pi N N

 

 

N_C

N_V

EF

EV C0

E

E

Conductors (metals)

Insulators (dielectrics)

Semiconductors (intrinsic)

Charge neutrality

C0 0 E 

F0

E EB

B F0

EW  E  E

0

2 2/3 F

3

8 π

h n

E m

 

  

C0

E

EV

No electrons

No holes

C0 V

F ,

2

E E

E 



F0

E

C0

E

EV

Electrons in the conduction band

Holes in the valence band

2k

i i C V e .

W

n p N N T

 

 

i i

n  p

F0

E

n – type (donor) E_d, N_d p – type (acceptor) E_a, N_a Thermodynamic equilibrium of doped semiconductors n – type

and p – type semiconductors Intrisic semiconductor:

Doping:

Number of donor atoms

Number of acceptor atoms

F ?

E  EV

C0

E

E E N^d, ^d

EV C0

E

E

a, a

E N

0 0

C , V, C , V

N N E E

C0

N NC0

NV N_V

Charge neutrality: n n _a  p N_d n_d  E_F

 ^C₀ ^F

kB

C e ,

E E

n N T

 



a F

B

a a

k

,

e 1

E E T

n  _N



 _{F V}

kB

V e .

E E

p N T

 



d F

B

d d

k

.

e 1

E E T

n  _N



 ^C0 ^F  F V

B B

a F d F

B B

k a k

C V d d

k k

e e 1 .

e 1 e 1

E E E E

T T

E E E E

T T

N N N N N

   

 

   

 

p – type semiconductor

Minority carriers Majority carriers Majority  Minority carriers

n d

n  N

 ^C₀ ^F

B

0

k d

C d F C B

C

e k ln .

E E

T N

N N E E T

N

 

   

 F V

B B

k C V k

n V

d n

e e .

E E E

T N N T n pi i

p N

N n

   

  

2

n n i i i .

n p  n p  n

n

n i i , p n p

 n

n – type semiconductor

Minority carriers Majority carriers Majority  Minority carriers

p a

p  N

i i p

p

n p , n  p

2

p p i i i .

p n  n p  n

Conductors (metals)

Insulators (dielectrics)

Semiconductors (intrinsic)

Charge neutrality

C0 0 E 

F0

E EB

B F0

EW  E  E

0

2 2/3 F

3

8 π

h n

E m

 

  

C0

E

EV

No electrons

No holes

C0 V

F ,

2

E E

E 



F0

E

C0

E

EV

Electrons in the conduction band

Holes in the valence band

2k

i i C V e .

W

n p N N T

 

 

i i

n  p

F0

E E

E E

n – type semiconductors

p – type

semiconductors

donor

E EF

acceptor

E

nn

pn

Majority carriers

Majority carriers Minority

carriers

Minority carriers

2 i

n d n

n

, n

n N p

  n

2 i

p a p

p

, n

p N n

  p

Charge neutrality





n n

n  p  N n

n

 n

 p

C0

E

EV

EF

np

pp C0

E

EV

Example 1: Intrinsic semiconductor

The energy band gap in silicon is 1.1 eV. The average electron effective mass is 0.31m, where m is the free electron mass. Calculate the electron

concentration in the conduction band of silicon at room temperature.

Assume that the Fermi level is at the middle of the band gap.

Intrinsic semiconductor:

C0 F

B V C0

k

i C e F

2

E E

T E E

n N E

  

 





C 3 i C C

2π k

2 , e e

h

E E E E

T T

m T

N n N N

 

 

  

     

 

 

31 23

24 3

C 3

34 2

2π 0.31 9.1 10 kg 1.38 10 J/K 300 K

2 4.35 10 m

6.625 10 Ws N

 





       

 

  

  

 

C0 V

15 3

2k

i C e 2.6 10 electron/m

E E

n N T

 

  

Example 2. A sample of Si is doped with phosphorus.

The donor impurity level lies 0.045 eV below the bottom of the conduction band. At 300 K, E_F is 0.010 eV above the donor level. Calculate

(a) the impurity concentration,

(b) the number of ionized impurities, (c) the free electron concentration, and (d) the hole concentration.

For Si

n – type semiconductor

n p

eff eff

E 1.1 eV, m 0.31 m, m 0.38 m.

     

 _k^C0_B ^F





n C C 3

2π k

e , 2 4.35 10 m ,

h

E E

T m T

n N N

 

    

 ^F_k ^V





n V V 3

2π k

e , 2 5.95 10 m .

h

E E

T m T

p N N

 

    

C0 F 0.045 0.010 0.035 eV,

E E   

F V 1.10 0.035 1.065 eV, E E   

23

B 19

1.38 10 300 Ws

k 0.026 eV,

1.6 10 As T





 

 



 ^C0 ^F

kB 24 3

n C e 1.07 10 m ,

E E

n N T

 

   

 F V

kB 6 3

n V e 9.52 10 m ,

E E

p N T

 

   

24 6 24 3

d 0.010 eV d

0.026 eV

1.07 10 9.52 10 1 1 2.61 10 m .

e 1

N N ^



 

 

         



 

Donor level

0.045 eV 0.010 eV

Fermi level

1.1 eV

the majority carrier the minority carrier

electrons holes

From examples 1 and 2 we have learned, that order of magnitude of carriers in silicon (at room temperature) are

 in intrinsic material

 in doped n-type material

Note that indeed

15 3

i i ~ 10 electron and hole/m n  p

24 3

n ~ 10 electron/m

n p_n 10 hole/m^{6 3}

2 n n ~ i

n p n

3. Contact potential

Two electrically conductive pieces of matter are brought into

thermal equilibrium with each other through physical contact.

When the work functions are different, an electrostatic potential emerges between the two pieces:

This potential difference is called contact voltage.

Mobile charges (electrons or holes) migrate from one sample to the other until the charge separation grows to a value sufficient to stop further motion of carriers.

In equilibrium the Fermi levels are equal.

2 1

contact

eV  E_W  E_W

F1

E

W1

E

F1

E

W1

E

F2

E

W2

E

F2

E

W2

E

' E

EAB

A

A

B

1 2

F F

'

E  E  E

1 2

AB W W

E  E  E

1 2

W W

AB e

E E

V 



A B

1 2

Most of the electronic and photonic devices are composed of contacted ‘blocks’, (samples) of

 metallic conductors,

 dielectric insulators,

 intrinsic semiconductors,

 n-type and

 p-type semiconductors.

In equilibrium contact potentials appear between neighboring blocks. Applying static voltages between the blocks (direct current, DC voltage), we can insert potential difference

between the Fermi levels of individual blocks.

p – n

‘diode’

p – n – p

‘transistor’

Bipolar p – n junction diode

p – type n – type

0

p

EC

p

EV

F0

E

0

n

EC

n

EV

p – type n – type

E E

0

p

EC

p

EV

F0

E

0

n

EC

n

EV

E E

0

p

EC

p

EV

0

n

EC

n

EV

E E

0

p

EC

p

EV

E

4. Carrier transport in semiconductors

In a semiconductor subject to external electric field ‘current’

emerges as an average ‘drift’ superposed on the random thermal motion of the electrons and holes. The average drift velocity of holes flows in the direction of the external field, but the average drift velocity of electrons flows opposite to the direction of the external field.

The current density is proportional to the drift velocity, to the carrier density and to the charge of the electron.

conductivity:

is called ‘mobility’

n e e

J  n v _n Jⁿ e v^e E n E

  

e /

v E _n  v_e / E

Drift of the electrons:

Drift of holes:

Conductivity of a semiconductor:

intrinsic semiconductor

n-type

semiconductor

p-type

semiconductor

n en n Jn en nE

    

p ep p Jp ep pE

    





e n p

    

 

i eni n p

    

n enn n

   _p  ep_p_p

Transport dynamics in semiconductors

Why does the charge carrier density change in a semiconductor ?

 

i eni n p ,

     ^B

3 2 eff B 2k

i i 2

2π k

2 e ,

h

E

m T T

p n

 

 

   

 

 

 

i i n p 2 n p

2π k

e 2e e .

h

E

m T T

 n    

 

 

     

 

 

 

n  n r t p  p r t

/ , / ?

n t p t

   

(i) Drift currents

(ii) Diffusion currents (iii) Relaxation

(iv) Generation and recombination

Carrier densities change in time, (i) because of drift and (ii)

diffusion currents of electrons and holes, (iii) tendency to relax into equilibrium, and (iv) electron and hole generation and

recombination.

 

 

n  n r t p  p r t

n e n

J  n E J_p  ep _pE

n  eDn gradn J





 





 

p

p g

t

 



n

n g

t

 



p  eDp grad p J

Einstein relation:

0

n n

n

1 div e

n n

n g

t 



    

 J _p ⁰ _p

p

1 div e

p p

p g

t 

     

 J





div  e p n N N divgradU U

 

         

E

n p B

n p

k e

D D T

 ^  ^ gradU

  E