3 Electrons and Holes in Semiconductors

3.1. Introduction Semiconductors : optimum bandgap Excited carriers → thermalisation

Chap. 3 : Density of states, electron distribution function, doping, quasi thermal equilibrium electron and hole currents Chap. 4 : Charge carrier generation, recombination, transport equation

3.2. Basic Concepts 3.2.1. Bonds and bands in crystals Free electron approximation to band Ref. CLASSIC “Bonds and Bands in Semiconductors” J. C. Phillips, Academic 1973

Atomic orbitals → molecular orbitals → bands Valence band : HOMO Conduction band : LUMO 1

Semiconductors : 0.5  Eg  3 eV 

Semi-metals : 0  Eg  0.5 eV 

Si : sp3 orbital, diamond-like structure

3.2.2. Electrons, holes and conductivity Semiconductors At T  0 K, all electrons are in valence band – no conductivity At elevated temperature, some electrons in conduction band, and some holes in valence band

2 Conduction Band

Electrons

Valence Band a) a)’

Electron in CB

Hole in VB

b) b)’

Fig. 3.4. Electron in CB and Hole in VB

3.3. Electron States in Semiconductors 3.3.1. Band structure Electronic states in crystalline Solving Schrödinger equation in periodic potential (infinite) Bloch wavefunction

ikr  k, r  uik r e (3.1)

for crystal band i and a wavevector k, which is a good quantum number. The energy E against wavevector k is called by crystal band structure.

Typical band diagram plotted along the 3 major directions in the reciprocal space. 0 0 0, X 10 0, M110, L111

3  Point k  is called the Brillouin zone boundary. a  2   3 3      For zinc blende structure, 0 0 0, X  0 0, K 0, L   a   2a 2a   a a a 

3.3.2. Conduction band

At the vicinity of the minimum Ek in the conduction band k  k c0 , approximation

 2 k  k 2 c0 (3.2) Ek  Ec0  * 2mc

* where Ec0  Ek c0  and mc is a parameter with dimensions of mass, and defined by

1 1  2 E k c (3.3) *  2 2 mc  k

is called the parabolic band approximation.

* * * eg. for GaAs mc  0.067 m0 , for Si mc //  0.19 m0 mc  0.92 m0 ,

where m0 is the static electron mass.

is determined by the atomic potentials, should actually be a tensor,

but is usually treated as a scalar, and is called the effective mass of electron. Momentum of an electron and the applied force are related as dp dk F    (3.4) dt dt Following eq. (3.2), the velocity and momentum can be obtained as

1 k  k c0  v  k Ek  * (3.5)  mc

* p  mc v  k  k c0  (3.6)

3.3.3. Valence band

At the vicinity of the maximum in the valence band k  k v0

2 2  k  k v0 Ek  Ev0  * (3.7) 2mv

4 * where Ev0  Ek v0  and mv is the effective mass of hole.

1 k  k v0  v  k Ek   * (3.8)  mv

* p  mv v  k  k v0  (3.9)

1 1  2 E k v (3.10) *   2 2 mv  k

* * * eg. mv for GaAs mv  0.5 m0 , for Si mv  0.54 m0 ,

* mc is the curvature of the band.

3.3.4. Direct and indirect band gaps Whether or not minimum of conduction band and maximum of valence band occur at the same wavevector k  k c0  k v0

Reciprocal lattices for (a) fcc and (b) bcc crystals

5 When the band gap is indirect, transition requires involvement of phonons for conservation of momentum.

(Sze)

3.3.5. Density of states According to Pauli’s exclusion principle, each quantum state only supports one electron (two, considering spins). Also, from Heisenberg’s uncertainty principle, p  x ~ h ∴ k  x ~ h ∴ k  x ~ 2

Therefore, for a crystal of volume L L L 2 2 k ~  x L

3  L  ∴   states in this unit volume.  2 

Hence, the density of electron states per unit crystal volume considering spin is 2 gkd 3k  d 3k (3.11) 2 3 If the band structure Ek is spherically symmetric and isotropic around k  0

gkd 3k  gk 4k 2dk (3.12)

Therefore, dk gEdE  gk 4k 2  dE (3.13) dE 2 dk ∴ gE   4k 2  (3.14) 2 3 dE 6  2 k  k 2 From eq. (3.2) c0 Ek  Ec0  * 2mc

2m* ∴ k 2  c E  E  (3.15)  2 c0 Similarly, for the valence band (hole) 2m* ∴ k 2  v E  E (3.15’)  2 v0 Differentiating (3.15)

1 2 1 2  2m*  dk 1  2m*  k   c  E  E 1 2 ∴   c  E  E 1 2  2   c0   2   c0     dE 2   

Therefore, from (3.14) 1 2 2 dk 2 1  2m*  g E   4k 2    4k 2   c  E  E 1 2 c   3 3  2   c0  2  dE 2  2    1 2 2 2m* 1  2m*    4  c E  E   c  E  E 1 2 (3.16) 3 2  c0   2   c0  2   2    3 2 3 2 4  2m*  1  2m*     c  E  E 1 2   c  E  E 1 2 3  2   c0  2  2   c0  2     2    Similarly,

3 2 1  2m*  g E   v  E  E 1 2 (3.17) v   2  2   v0  2   

Question : Derive the DOS for 2D and 1D materials. (Box 3.3)

Excitons A bound state of a pair of electron and hole of same k . The wavefunction can be described as

ex  e k,r h k,r

Satisfying a hydrogenic effective mass equation

 2 q 2 2  *  ex  ex  Eex ex 2 4 s r  r

 * : reduced effective mass of e-h pair

 Eex : binding energy of the exciton

7  *  2 Ryd 0 Eex  2 2 m0  s l

3.3.6 Electron distribution function Fermi function in semiconductors 1 f E   E  EF  exp  1  k BT  This can be expressed in terms of f k, r although the energy is only dependent on the wavevector.

nr d 3k  gk f k, r d 3k

The total electron density is given by

3 nr  gc k f k, r d k (3.20) CB k

3 pr gv k1 f k, rd k (3.21) VB k

3.3.7 Electron and hole currents Using the above formalism

q 3 J n r   k gc k f k, r d k (3.22) * CB k mc

q 3 J p r  k gv k 1 f k, r d k (3.23) * VB k mv

3.4. Semiconductor in equilibrium 3.4.1. Fermi Dirac statistics Equilibrium : no net charge exchange. In this case, it is independent of position.

f k,r  f0 Ek, EF , T  (3.24)

1 (3.25) f 0 E, EF , T   EE  k T e F B 1

3.4.2. Electron and hole densities in equilibrium The number density of electrons nE with energy in the range between E and E  E

nEdE  gEf0 E, EF , T dE (3.26)

8 Then the total density of electrons in a conduction band of minimum energy Ec is

 n  gc Ef0 E, EF , T  dE (3.27) E c

and holes in a valence band with maximum (minimum) energy Ev is

Ev p  gv E1 f0 E, EF , T  dE (3.28) 

3.4.3. Boltzmann approximation

When the Fermi energy EF is located far enough from the band edge such that E  EF , in the conduction band, (3.25) turns into

1 E E  k T F B (3.29) f0 E, EF , T   EE  k T  e e F B 1

Similarly, in the valence band E  EF leading to

EF E  kBT 1 e 1 EE  k T F B 1 f0 E, EF , T   1 EE  k T  1 E E k T  E E  k T  e e F B 1 1 e F B 1 e F B (3.30) Using these results, (3.27), (3.28) can be integrated

* 3 2     1 2mc 1 2 EF E  kBT n  gc E f0 E, EF , T  dE    E  Ec0  e dE E E 2  2  c c 2    (3.31)  E  E   F c   Nc exp   kBT 

where 9 3 2  m*k T  N  2 c B  (3.32) c  2   2 

For holes in valence band

E v  Ev  EF  p  gv E1 f0 E, EF , T dE  Nv exp  (3.33)     kBT  where

3 2  m*k T  N  2 v B  (3.34) v  2   2 

And the product np is given as

Eg kBT np  Nc Nv e (3.35)

This is actually a constant and the intrinsic carrier density ni suffice the relation

2 Eg kBT ni  np  Nc Nv e (3.36)

Introducing intrinsic potential energy Ei , which is the Fermi level for intrinsic semiconductor,

 E  E   E  E   F i  (3.37)  i F  (3.38) n  ni exp  p  ni exp   k BT   k BT  where

1 1  N   c  Ei  Ec  Ev  k BT ln  2 2  N v  (3.39) 1 3  m*   E  E  k T ln c   c v  B  *  2 4  mv 

Electron affinity Least amount of energy required to remove an electron from solid. → Better determined as energy difference between vacuum level and CBM.

10

Ec  Evac  

Ev  Evac    Eg (3.40)

EF  Evac    k BT ln N c  Evac    Eg  k BT ln N v

can all be derived by statistical mechanics.

Validity of Boltzmann approximation The condition for the above relations to hold is

E  E E  E F v 1 and c F 1 k BT k BT

These relations hold regardless of Nc and Nv

→ basically, just statistical mechanical consideration.

3.4.4. Electron and hole currents in equilibrium

To find J n r and J p r in equilibrium, consider (3.22) ~ (3.24)

q 3 J n r   k gc k f k, r d k (3.22) * CB k mc

q 3 J p r  k gv k 1 f k, r d k (3.23) * VB k mv

f k,r  f0 Ek, EF , T  (3.24)

For parabolic band structure, i.e.,

2 2  k  k c0 Ek  Ec0  * (3.2) 2mc

Therefore, Ek obviously is an even function of k . Hence,

(3.24)

11 is also an even function of k and so is

gkd 3k  gk 4k 2dk (3.12)

Consequently,

k gc k f k, r

is an odd function of . Therefore, integration of in the entire -space is 0.

Therefore,

J n r  J p r  0

which means there is no net current in the semiconductor in equilibrium.

3.5. Impurities and Doping 3.5.1. Intrinsic semiconductors

Semiconductors with carrier densities of n (electrons) and p (holes), if the mobilities are  n

and  p , respectively, the conductivity is given by

  qn n  q p p (3.41)

In intrinsic semiconductors,

n  p  ni

where ni is a small value at room temperature or lower.

Both intrinsic carrier density and conductivity are determined as mere function of temperature Bandgap, location of Fermi level

10 3 6 1 1 At 300K for Si Eg 1.12 eV , ni 1.0210 cm ,   3.1610  cm 

In semiconductors, mobility increases as a function of temperature due to carriers thermally excited across the band gap. (ref. What happens with metals?) At 300K,

2 1 1 Ge : Eg  0.74 eV  ( 0.66 eV ?),   2.110  cm 

12 9 1 1 GaAs : Eg 1.42 eV ,   2.3810  cm 

(From “Data in Science and Technology, Semiconductors” Springer 1991) ======

Q. Estimate ni for Ge and GaAs at 300K

13 3 6 3 A. Ge : ni  2.3310 cm  GaAs : ni  2.110 cm 

======

3.5.2. n-type doping Electrons as majority carrier : By replacing the atoms by those that would readily ionize to provide free electron (leaving a positive ion). Such atoms are called donor atoms.

In case of Si, this can be accomplished by replacing some of the Si atoms by As, for example. Typically, this substitution is done on the order of ppm or less.

(Sze)

13 Ionisation energy approximated by hydrogenic bond is

m* 2 c 0 Vn  2 Ryd  m0 s

 s : dielectric function of the semiconductor (typically around 10)

Ryd : 13.6 [eV] 1 ∴ V  Ryd  0.136eV  or smaller n 100 Therefore, room temperature is sufficient to ionise these atoms. This is depicted in the right hand panel of Fig.3.10.

(Sze)

The density of carriers can be controlled by

1) density of dopants N d

2) donor level (choice of dopants) 3) temperature

Regarding 1), if Nd  ni and the donors are fully ionised at room temperature, then

n  N d (3.42) and 14 n 2 p  i (3.43) N d

since

2 Eg kBT ni  np  Nc Nv e (3.36)

at equilibrium. Here, obviously, n  p Hence, electrons are the majority carriers and holes the minority carriers.

Reference : Dielectric functions of matters

K. A. Mauritz, Univ. S. Mississippi, http://www.psrc.usm.edu/mauritz/dilect.html

Lorentz-Lorenz equation : oscillator n 2 1 4 2  N n  2 3

Kramers-Kronig relation

For complex function   1  i 2  or    i where  is a complex variable, analytic in the upper half plane and vanishes at

  

1    1       P 2 d     P 1 d 1    2         

3.5.3. p-type doping Holes as majority carrier : By replacing the atoms by those that would readily ionize to capture free electron (turning into a negative ion). Such atoms are called acceptor atoms. 15

Basically, the same argument that was used for n-type can be applied to p-type doping. In case of Si, this can be accomplished by replacing some of the Si atoms by B, for example.

When V p is small, the dopant atoms is ionised by removing a valence electron from another bond, leaving a hole.

Regarding doping density, if Na  ni and the acceptors are fully ionised at room temperature,

p  N a (3.44)

n 2 n  i (3.45) N a since

2 Eg kBT ni  np  Nc Nv e (3.36)

at equilibrium. Here, obviously, p  n . Hence, holes are the majority carriers and electrons the minority carriers.

16 The related energy levels are summarized in Fig.3.12.

3.5.4. Effects of heavy doping Increase in carrier density Adds tail to the CB and VB DOS : effectively reduces band gap- Increase defect states : acts as centers for carrier recombination and trapping

3.5.5. Imperfect and amorphous crystals Increase in carrier density Modify electronic structures in general Grain boundaries

17 3.6. Semiconductor under Bias Actual devices under operation : non-equilibrium condition

3.6.1. Quasi thermal equilibrium System disturbed from equilibrium In quasi thermal equilibrium, for electrons and holes,

f k, r  f E, E , T  : for electrons c 0 Fn n

fv k, r f0 E, EF , Tp  : for holes p

f E, E , T  and f0 E, EF , Tp  are those of equilibrium, implying there is no net current. 0 Fn n p

fc k, r and fv k, r are general distribution functions, both r and k dependent.

The Fermi levels defined for this quasi thermal equilibrium is different from those at equilibrium (and also)

E : electron quasi Fermi level Fn

E : hole quasi Fermi level Fp

Small correction factor introduced

f k, r f E, E , T  f k, r (3.46) c 0 Fn n A

1 E E  k T f E, E , T   e Fn B n (3.47) 0 Fn n EE  k T e Fn B n 1

f A k, r is the part that is antisymmetric in .

3.6.2. Electron and hole density under bias In 3.4.3, relations

 E  E   E  E   F i  (3.37)  i F  (3.38) n  ni exp  p  ni exp   k BT   k BT 

were introduced. These can be applied for the quasi Fermi levels

 EF  Ei   Ei  EF  n  n exp n  (3.48) p  n exp p  (3.49) i   i    k BT   k BT 

Also applying the results

18 * 3 2     1 2mc 1 2 EF E  kBT n  gc E f0 E, EF , T  dE    E  Ec0  e dE E E 2  2  c c 2    (3.31)  E  E   F c   Nc exp   kBT 

3 2  m*k T  N  2 c B  (3.32) c  2   2 

E c  Ev  EF  p  gv E1 f 0 E, EF , T dE  Nv exp  (3.33)     kBT 

3 2  m*k T  N  2 v B  (3.34) v  2   2  we can obtain

 EF  Ec   n  (3.50) n  Nc exp   kBTn 

 Ev  EF  p  N exp p  (3.51) v    k BTp 

where Tn and Tp are the electron and hole effective temperatures that could be different from ambient temperature T . For instance,

Tn  T ‘hot’ electrons

In the following we do not consider ‘hot’ carriers, and assume

Tn  Tp  T

(‘Hot’ carriers will be considered in Chap 10.)

19 One thing to be noted for semiconductor under bias is that the Fermi levels for electrons and holes are NOT identical, i.e.,

E  E  E Fn Fp F and the Fermi levels are split. The difference between the quasi Fermi levels

  E  E (3.52) Fn Fp

The relationship between carrier densities was defined by

2 Eg kBT ni  np  Nc Nv e (3.36)

Substituting (3.50) and (3.51) into (3.36) E  E  E  E   v F  Eg EF EF  kBT np  N exp Fn c N exp p   N N e n p c   v   c v  kBTn   kBTp  (3.53) E E k T Eg kBT  Fn Fp  B 2  kBT  Nc Nv e e  ni e

In general, these quasi Fermi levels E and E are not constant throughout the device and Fn Fp

are functions of position. We consider local quasi equilibrium at any point in the semiconductor, and local quasi Fermi levels.

3.6.3. Current densities under bias If the distribution function is symmetric as discussed previously, there would be no current.

Hence, for currents J n r and J p r to be non-zero, the distribution function f needs to be antisymmetric.

------Box 3.4 Boltzmann Transport eq. and relaxation time approximation ------

Let the antisymmetric part of the distribution function be f A .

Since f c is a function of k, r and t , by definition

df dr dk f c   f   f  c (3.54) dt dt r c dt k c t where dr v  (3.55) dt dk F   (3.56) dt 20 Define Ek, r as the energy with respect to the conduction band minimum (CBM)

E  Ec  Ek, r (3.57)

where

f k, r  f E, E , T  f k, r (3.46) c 0 Fn n A

We assume that

f k, r  f E, E , T  c 0 Fn n

From (3.47)

1 1 f E, E , T   f E  Ek, r, E , T    0 Fn 0 c Fn EE  k T Ek, rE E  k T e Fn B 1 e c Fn B 1

Ek, r k T Ec EF  kBT  e B e n

From physical consideration, we can assume that Ek, r varies as a function of k , but not much as function of r .

Therefore, for calculating r f c , let

E E  k T f k, r  f E, E , T  e c Fn B c 0 Fn n

E E  k T E E  k T Ec  EF f ∴  f   e c Fn B  e c Fn B  n   0  E  E  (3.58) r c r r r c Fn kBT kBT

For calculating  f , note that E  E is a function of but not of , and that k c c Fn varies as a function of , but not much as function of Hence, let be a function of , but not . Therefore, using,

Ek, r k T Ec EF  kBT f k, r  f E, E , T  e B e n c 0 Fn n

 E E k T  E E k T Ek, r kBT  c Fn  B  c Fn  B Ek, r kBT k f c  k e e  e k e

 E E k T E k, r f  c Fn  B Ek, r kBT   0  e e k    k Ek, r k BT k BT

1 k  k c0  From (3.5) v  k Ek  *  mc

Here, we can assume k c0  0 (the result will be the same even if this is not the case).

Hence, 21 f f 0 0 (3.59) k f c    k Ek, r   v k BT k BT dp dk Substituting (3.58) and (3.59) into (3.54) together with F    (3.4) dt dt

df dr dk f dr f dk f f c    f    f  c    0   E  E   0 v  c r c k c r c Fn dt dt dt t dt k BT dt k BT t (3.60)’  f dk  f  f  f   0 v   E  E   v  c   0 v   E  E  F  v  c  r c Fn   r c Fn  k BT dt  t k BT  t

Here, from physical consideration,

r Ec  F (gradient of conduction band is the force on the electron)

df f  f ∴ c  0 v  E  c (3.60) r Fn dt kBT t

To solve this for f c , the following assumptions are made.

1) Intraband relaxation is much more frequent than interband Intraband relaxation is dominantly phonon scattering with lattice and fast.

 f  f c  c t t collisions

2) The distribution relaxes exponentially towards the quasi equilibrium with time constant 

 f  f  f  c   c 0 (3.61) t collisions 

This is called the relaxation time approximation. Substituting (3.61) into (3.60),

df f  f  f  c  0 v  E  c 0 r Fn dt kBT 

Under steady state, this equals 0,

f  f  f  0 v  E  c 0 r Fn k BT 

   f  f 1 v  E  (3.62) c 0  r Fn   k BT  Therefore,  f  f v  E (3.63) A 0 r Fn kBT 22 (The sign might be wrong, but never mind : it is how the direction is considered)

Substituting for fc , the equation for current becomes

q 3 J n r   k g c k f k, r d k * CB k mc

q 3 q  3   k g c k f A k, r d k   k g c k f 0 v r EF d k * CB k * CB k n mc mc k BT  E  E (3.64) r Fn q 3 r Fn q k 3   k g c k f 0  v d k   k g c k f 0  d k * CB k * CB k * k BT mc k BT mc mc  2   q  k  3   r EF  g c k f 0    d k n k T CB k  m*   B  c  

Since the term in brackets is NOT a function of bias, we rewrite this as n n where

 n is the electron mobility

Then,

J r   n E (3.65) n n r Fn

Similarly

J r   p E (3.66) p p r Fp

------END of Box 3.4------The net current is given by

Jr  J r J r   n E   p E (3.67) n p n r Fn p r Fp

3.7. Drift and Diffusion 3.7.1. Current equations in terms of drift and difusion

 E  E   F c  (3.31) n  Nc exp   kBT 

By differentiating both sides

23  E  E   E  E   E  E   Fn c   Fn c   Fn c  r n  r Nc exp   exp  r Nc  Ncr exp   kBT   kBT   kBT 

n  EF  Ec   EF  Ec  n  EF  Ec   n   n   n   r Nc  Nc exp  r    r Nc  n r   Nc  kBT   kBT  Nc  kBT  n  n  ln N   E  E  r c r Fn c kBT n ∴  n  n  ln N   E  E  r r c r Fn c kBT k T k T ∴  E  E  B  n  n  ln N   B  n  k T  ln N r Fn c n r r c n r B r c k T ∴  E   E  k T  ln N  B  n (3.68) r Fn r c B r c n r Similarly, from (3.33) k T  E   E  k T  ln N  B  p (3.69) r Fp r v B r v p r

Note the relation (refer to Sze Fig. 4-32, below)

Ec  Evac  

Ev  Evac    Eg (3.40)

EF  Evac    k BT ln N c  Evac    Eg  k BT ln N v

The gradient in the conduction/valence band edge is

r Ec  qF  r  (3.70)

although, typically r   0 . Similarly

r Ev  qF  r   r Eg (3.71)

although, typically r Eg  0 .

24 (Sze)

The electrostatic field is defined by 1 F   E (3.72) q r vac Substituting (3.68), (3.70) into (3.65), (3.66)

J r   n E (3.65) n n r Fn

k T  E   E  k T  ln N  B  n (3.68) r Fn r c B r c n r

r Ec  qF  r  (3.70)

25  kBT  J n r  nnr Ec  kBT r ln Nc  r n  n 

 kBT  ∴  nnqF  r   kBT r ln Nc  r n (3.73)  n 

 nnqF  r   kBT r ln Nc  nkBT r n

 qDn r n  nnqF  r   kBT r ln Nc  where k T D   B (3.77)’ n n q Similarly

J p r  qDp r p   p pqF  r   r Eg  kBT r ln Nv  (3.74) where k T D   B (3.77)” p p q

As mentioned, typically r   0 , r Eg  0 , and N c , Nv (doping levels) are invariant.

Therefore,

J n r  qDn r n  qnFn (3.75)

J p r  qDp r p  q pF p (3.76)

The first terms are the diffusion current, and the second the drift.

Driving forces Drift : potential gradient Diffusion : concentration gradient

26 The total current for drift (without diffusion) is

Jr  J n r J p r  qnn   p pF (3.78)

The total current for diffusion (without drift) is

Jr  J n r J p r  qDn r n  Dn r p (3.79)

3.7.2. Validity of the drift-diffusion equations Assumptions

1) Electrons and hole populations each form quasi themal equilibrium with EF and T

2) T  Tn  Tp

3) Intraband phonon scattering dominant 4) Electron and hole states described by k

5) Boltzmann approximation E  E  k T , E  E  k T c Fn B Fp v B

6) Compositional invariance (uniformity of materials)

3.7.3. Current equations for non-crystalline solids “Non-crystalline” can mean a number of things. eg. defective crystals, amorphous, etc. In any case, high DOS within band gap.

1) n, p : sensitive to , F , illumination 27 2) n ,  p : T , F , n, p dependent

3) conduction due to localized states between band gap

(3.80) J  J n  J p   J i localized states i

(3.81)  J i  Ei  f Ei  gEi  localized states i i

 J i  Ei  f Ei  gEi  localized states i i

Ei  : mobility of carrier through localized states → mechanism dependent

f Ei  : (Fermi-Dirac) distribution function

gEi  : DOS of localized states

Discussed in Chap 8.

3.8. Summary Conduction band electrons : nearly free particles responsible for transport Valence band holes

Equilibrium and Fermi level determining occupancy probability Doping Quasi Fermi levels

28