semiconductor physics: energy bands - scu · donor and acceptor levels s. saha ho #2: elen 251 -...

Semiconductor Physics: Energy Bands

HO #2: ELEN 251 - Semiconductor Physics S. Saha

• The e− in a solid exist in certain bands of allowed energy:– valence e− exist in the valence band– conduction e− or free electrons exist in the conduction band

– energy gap, Eg = Ec − Ev @ T :110810x02.7160.1)(

24

+−=

−

TTTEg (1)

Ec

Eg

Ev

Conduction Band

Valence Band

Kinetic Energy of holes

Kinetic Energy of electrons

Distance

Elec

tron

Ener

gy• When an e− in the conduction band gains energy, it moves up to an E > Ec.

• Similarly, when a hole in the valence band gains energy, it moves down to an E < Ev.

Hol

e En

ergy

Intrinsic Semiconductors


• In a pure single crystal semiconductor, all the e− in the conduction band are thermally excited from the valence band. Then at any temperature, T n = p = ni

where n = free e- (cm-3)p = free holes (cm-3)ni = intrinsic carrier (cm-3).

• Note:– T↑ ⇒ ni↑– Eg↑ ⇒ ni↓

Intrinsic Carrier Concentration


The product of the number of e− and holes is given by:

∴ ni ≅ 1.45x1010 cm-3 for silicon @ T = 300oK.

If T0 = known reference temperature (generally, 300oK), then at any temperature T, we can write:

where, Eg(T) is given by (1). Eq. (3) is used in SPICE to calculate ni at any temperature.

kTTE

i

kTTE

i

g

g

eTTn

eCTTnnp

2)(

23

16

)(32

10x9.3)( or,

)( −

−

=

==

(2)

0

023

2)(

2)(

00ii )()( kT

TEkT

TE gg

eTTTnTn

+−

⎟⎟⎠

⎞⎜⎜⎝

⎛= (3)

Extrinsic Semiconductors


• To increase and control the conductivity of semiconductors we add controlled amounts of “dopants” such as:– N type: P, As, and Sb each has five valence electrons– P type: B, Al, and Ga each has three valence electrons.

• These atoms occupy substitutional lattice sites and the extra e- or hole is very loosely bound, i.e. can easily move to the conduction or valence bands, respectively. In terms of band diagrams this can be represented as:

Ec

Ev

ED

1.1eV ≈0.05eV

Donor

Ec

Ev

EA

1.1eV ≈0.05eV

Acceptor

Donor and Acceptor Levels


Measured donor and acceptor levels for various impurities in silicon

Si

1.12

eV

Sb P As0.039 0.045 0.054

B Al Ga In

0.045 0.067 0.072 0.16

Ec

Ev

• Acceptor levels are below the gap center and measured from the top of the valance band.

• Donor levels are above the gap center and measured from the bottom of the conduction band band.

Fermi Level


Ev Ec

Den

sity

of S

tate

sZero in forbidden band

• The probability that an allowed state is occupied is described by the Fermi-Dirac distribution function:

f(E) = 1/[1 + e(E − EF)/kT] (4)EF = Fermi Level ≡ energy at which the probability of occupancy = 1/2

k = Boltzmann constant; T = temperature.

• Again, the probability of a state not being occupied is:1 − f(E) = 1/[1 + e(EF − E)/kT] (5)

= Probability a hole exists.

Fermi Level: Temperature Effect


-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.0 0.5 1.0f(E)

E - E

F (e

V)

T = 0 KT = 100 KT = 200 KT = 300 KT = 400 KT = 500 KT = 600 K

At T = 0oK• all states < EF

are occupied.

• all states > EFare empty.

At T > 0oK• some e− have

E > EF.

• some holes exist E < EF.

The Boltzmann Approximation


0.0

0.5

1.0

-0.3 0.0 0.3

E - EF

f(E)

F-D functionM-B function

• Equation (5) represents Maxwell−Boltzmann function.

• This approximation is quite good for many device applications, as we will see later.

• However, for heavily doped materials Fermi function (4) must be used.

For energies > 3kT (≅ 75 meV @ room temp) above EF, f(E) can be approximated as:

f(E) ≅ e−(E − EF)/kT] (6)

Approximation

Summary of Equilibrium Carrier Distribution


• In an intrinsic or undoped semiconductor: n = p = ni = 3.9x1016T3/2e−Eg/2kT (2') EF ≅ (1/2)(Ec + Ev) ≡ Ei (7)

• Fermi level EF in doped as well as undoped materials is given by: n = nie(EF − Ei)/kT = nieq(φ − φF)/kT (8) p = nie(Ei − EF)/kT = nieq(φ

F− φ)/kT (9)

where φ = − Ei/q and φF = − EF/q.

• If both n- and p-type dopants are added in intrinsic semiconductor:n + NA = p + ND (10)

Then using np = ni2, we can show for an n-type semiconductor:

[ ][ ] ( )12 /4)2/1(

(11) 4)2/1(222

22

DiDiDn

DDiDn

NnNnNp

NNnNn

≅−+=

≅++=

Carrier Transport in Semiconductors


• In thermal equilibrium, mobile (conduction band) e- will be in random thermal motion. – avg. velocity of thermal motion, vth ≅ 107 cm/sec @ 300oK.

• In an applied ε field:

– motion of e− is in the direction opposite to ε.

– this process is called drift and causes a net current flow.

Si Crystal

− +

ε

The average e− velocity called the drift velocity is given by:

carrier of mass effective collision;between timefreemean */mobility where

(13)

====

≡

m*mq

vd

ττµ

µε

Drift - ε Field Driven Motion


• vd = µε linear relationship holds only for small fields.

• Mobility depends on parameters like:– material– doping – temperature– scattering

mechanisms.

Scattering Mechanisms


• Phonon scattering. Lattice atoms vibrate, and from Quantum Mechanics we know that discrete allowed vibrational states exist. This can be modeled as discrete particles called phonons. Transfer of energy from electrons to the lattice is called phonon scattering.

• Ionized impurity atom scattering - important at high dopant concentrations.

• Neutral impurity atom scattering - usually negligible.

• Carrier-carrier (e− - e− & e− - hole) scattering - important at high carrier concentrations.

• Surface scattering effects - important in MOSFET devices.

• Crystal defects - can be minimized by good crystals.

µn Vs. Temperature


Ionized impurity scattering is more important at low T.

Assuming that the different scattering mechanisms operate independently:

...111

++=

IL µµ

µ

Drift - ε Field Driven Motion


• At high fields, vd becomes comparable to vth which is an approximate upper bound of vd. Thus, vd saturates and is called:

vsat ≡ scattering limited velocity≅ 1 x 107 cm/sec for e− in Si≅ 8 x 106 cm/sec for holes in Si.

• The general expression for vd is: (14)ββεµ

εµ1

1⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+

=

sat

o

od

v

v

• The critical field at which vsat occurs is: ~ 2x104 V/cm = 2 V/µm.

This is an important physical limit in very small semiconductor devices like MOSFETs.

Drift - Current


The drift of carriers under an applied ε results in a current flow: I = −qAnvd (15)

where A = area; n = number of e− per unit volume.

Substituting for vd = µε and considering both e− and holes: I/A = (nqµn + pqµp)ε (16)

Usually, the majority carrier-term in (16) is important. Since for a semiconductor of length L under applied bias V, ε = V/L, then:

1/(qµnn + qµpp) = (V/L)/(I/A) = (V/I)(A/L) = R(A/L) = ρ where R = ρ(L/A) = resistance of the semiconductor:

(17)

where ρ = resistivity in ohm.cm; σ = conductivity in (ohm.cm)-1

σµµρ 11

=+

=∴pqnq pn

Resistivity Vs. Dopant Density @ T = 300°K


Diffusion - Gradient Driven Motion


Diffusion occurs from high concentration regions towards low concentration regions.

Carrierconcentration

Flux ofdiffusioncurrent

Flux ofdiffusioncurrent

t1

t2

t3

t1< t2 < t3

0 x−x The diffusion flux obeys Fick’s first law: F = −D(dn/dx) (18)

where F = flux of carriers D = diffusion constant n = carrier density.

Transport Equations


The diffusion current through a cross-sectional area is: I = qADn(dn/dx) − qADp(dp/dx) (19)

where Dn = e− diffusion constant = 38 cm2/sec in Si Dp = hole diffusion constant = 13 cm2/sec in Si

Dn and Dp are related to their respective carrier mobilities by the Einstein relationship:

Dp/µp = Dn/µn = kT/q (20)

If both drift and diffusion are important, In = qA[µnnε + Dn(dn/dx)] (21) Ip = qA[µppε − Dp(dp/dx)] (22)

We will use these basic equations in deriving device characteristics.

Generation-Recombination


• Thermal equilibrium is defined by the condition np = ni2. In

thermal equilibrium, the equations discussed earlier may be used to calculate EF, n, p etc.

• However, the equilibrium condition may be disturbed by the introduction or removal of free carriers.

φ

-+

N P

• Injection: Extra carriers are generated by optical or electrical means, and np >ni

2.

• Extraction: Here, carriers are removed electrically. And, as a result np < ni

2.

Injection of Carriers


If we shine light on the semiconductor with E = hν > Eg so that e− can be excited into the conduction band. n = ni + nL

p = ni + pL

n & p > ni

INJECTION

Eg

Ec

Ev

hν > Eg

ν = frequencyh = Planck’s constant

Low Level and High Level Injection


Now, if we shine light on an extrinsic semiconductor with:ND = 1015 cm-3.

1020

1018

1016

1014

1012

1010

1008

1006

1004

1002

1000

nno

pno

nn

pn

nnpn

ND = 1015 cm-3

ni for Si@ 300 oK

Equilibrium Lowlevel

Highlevel

In this case: n ≅ ND + nL

p ≅ ni2/ND + pL

Low level: nL << ND

High level: nL ≥ ND

In Fig., a low level injection of excess carrier = 1013 is shown.

Since the mathematics for high level injection are complex, we will consider only low level injection.

Direct (Band to Band) Recombination


If we generate excess carriers (∆n, ∆p) at a rate GL due to incident light. Then for low level injection we can show that: ∆p = ∆n = Uτ = GLτ

Where ∆p = p − po

∆n = n − no

p and n are total concentrations due to generation no and po are equilibrium concentrations U = net recombination rate = GL

τ = excess carrier lifetime = avg. time excess carriers remain free.

Thus, τ = ∆p/GL = ∆p/U = ∆n/U (23) For band to band recombination, τn = τp as the single phenomena

annihilates e− and hole simultaneously.

Ec

Ev

n electrons

p holes

Direct Recombination


• Example: Net recombination rate U = 1018 carriers cm-3 sec-1

Minority-carrier lifetime τ = 100 µsec, Then from (23), excess-carrier density:

∆p = ∆n = (1018) (100x10-6) = 1014 cm-3.

• Summary: For a semiconductor with equilibrium carrier concentrations no and

po the steady state carrier concentrations under light are: n = no + τGL nopo= ni

2

p = po + τGL np > ni2 (24)

Indirect Recombination (Trap-assisted)


In Si and Ge this is often the dominant mechanism since they are “indirect band gap semiconductors.” Physically this means that the minimum energy gap between Ec and Ev does not occur at the same point in the momentum space.

Since momentum (k) must be conserved in any energy level transition, GaAs can easily make direct transitions which makes it useful for LEDs. Ep = hν = Eg. While Si and Ge tend to recombine through intermediate trapping levels.

Egk

E

“Direct band gap Semiconductor” GaAs

Egk

E

“Indirect band gap Semiconductor” Si, Ge

Ec

Ev

Ec

Ev

Indirect Recombination


Let us consider the following example where an impurity like Au is introduced that provides a “trapping level” or a set of allowed states at energy Et.

Ec

Ev

Et

The level Et is assumed to act like an acceptor, that is, it can be neutral or negatively charged. Recombination is accomplished by trapping an e- and a hole. (The analysis can be easily extended to the case where the trap acts like a donor, i.e. + and neutral charge states).

“1” = e- capture“2” = e- emission“3” = hole capture“4” = hole emission1 + 3 or 3 +1 ≡ recombination

Ec

Ev

Et Eio

1 2

3 4EF

Indirect Recombination - SRH


• Shockley, Read, and Hall showed that for low level injection, the net recombination rate is given by:

where vth = carrier thermal velocity (≈ 107 cm/sec) σ = carrier capture cross-section (≈10-15 cm2)

• From Eq (25) we observe:– the “driving force” or the rate of recombination is ∝ np − ni

2, (that is, the deviation from the equilibrium condition).

– U = 0 when np = ni2.

– U is maximum when Et = Ei. That is, trapping levels near the mid-band are the most efficient recombination centers

( )

⎥⎦⎤

⎢⎣⎡ −

++

−=

kTEEnpn

npnNvUit

i

itth

cosh2

2σ(25)

Indirect Recombination - SRH


• Eq (25) shows that for Et → Ec, U↓ because:⇒ e− capture is more probable while hole capture is less probable.⇒ e− emission back to conduction band is highly probably.

• When Et = Ei , from (25) we get: U = vthσNt(pn - ni

2)/[n + p + 2ni] (26)

a) N-type: n >> p and n >> ni, then from (26)∴ τp = 1/(vthpσpNt) (27)

In N-type material, lots of e− are around for capture. Therefore, the limiting factor in recombination is the minority carrier lifetime τp.

b) P-type: p >> n and p >> ni

∴ τn = 1/(vthnσnNt) (28) Again, the limiting factor is the minority carrier lifetime.

Quasi-Fermi Levels


Under thermal equilibrium, we have: n = nie(EF − Ei)/kT

p = nie−(EF − Ei)/kT

In non-equilibrium conditions such as (1) injection (np > ni2) or

(2) extraction (np < ni2), we cannot use these relationships.

Here, we define two new quantities called quasi-Fermi levels such that the relationships of the type shown above hold and:

n = nie(EFn − Ei)/kT (29) p = nie−(EFp − Ei)/kT (30)

where EFn ≡ e− Quasi-Fermi level; EFp ≡ hole Quasi-Fermi level. EFn and EFp are the mathematical tools; their values are chosen

so that the “correct” carrier concentrations are given in non equilibrium situations.

In general, EFn ≠ EFp

Quasi-Fermi Levels: Example


Let us consider a p-type semiconductor with NA = 1016 cm-3, τ= 10 µsec, and we shine light on it such that GL = 1018 carriers cm-3 sec-1.

Given, po = NA = 1016 cm-3

τGL = (10x10-6 sec)(1018 carriers cm-3 sec-1) = 1013 carriers cm-3

∴ n = no + τGL = ni2/NA + τGL ≅ 1013 cm-3

p = po + τGL = (1016 + 1013) ≅ 1016 cm-3

Then from (29) and (30) we get:EFn = Ei + kTln(n/ni)

≅ 0.18 eV above Ei

EFp = Ei − kTln(p/ni)

≅ 0.36 eV below Ei

Ec

Ev

EFn

EFp

Ei

p ≅ 1016 cm-3

n ≅ 1013 cm-3

Quasi-Fermi Levels


Ec

Ev

EFn

EFp

Ei

p ≅ 1016 cm-3

n ≅ 1013 cm-3 In example, since p ≅ po, EFp is essentially unchanged while EFn moves up to reflect n >> no

enpn kTEE

iFpFn −

= 2 (31)Here,

In equilibrium: EFn = EFp = EF ⇒ pn = ni2.

However, under injection: EFn = moves up towards the CB. EFp = moves down towards the VB.

Thus, |EFn − EFp| is a measure of the degree of non-equilibrium in the semiconductor.

We will use this concept of Quasi-Fermi levels in our discussion of PN junctions.

Non-Uniform Doping Distributions


In our discussion of semiconductor statistics, we have assumed spatially uniform doping levels. Often this is not the case as we can see for the Bipolar Transistors.

xjBE xjBC

N+

N−P

Bipolar TransistorN

Depth

1 The impurity profile we have drawn represents the fixed positions of ionized donors and acceptors, i.e. ND

+, NA−.

2 The carriers (e− and holes) are mobile and attempt to diffuse away from regions of high concentrations.

Non-Uniform Doping: Built-in Electric Field


Now, consider a simpler situation where we only have N-type material (no junctions) but ND ≠ constant. Here,N

x

1018

1014

1010ND

+

1 diffusion flux is given by: F = Dn(dn/dx).

2 as e− move (drift) away, they leave behind + charged ions (ND

+) which try to pull e− back.

• Equilibrium is established when diffusion = drift. Here n(x) ≠ND(x) and a “built-in” electric field that exists is given by: Ex = − (kT/q)(1/n)[dn/dx] (32)

Similarly for holes, Ex = (kT/q)(1/p)[dp/dx] (33)

This built-in Ε−filed accelerates the minority carrier e− injected into the base from the emitter in their travel across the base or p-region.

Basic Semiconductor Equations


• To this point we have discussed:– Carrier transport − drift and diffusion– Carrier generation– Carrier recombination.

• The mechanism which ties all of these together is the condition of continuity of current.

Jp(x + ∆x)Jp(x)

∆x

U Rate of change ofnumber of holes

in ∆x

holesrecombining

Jp = hole currentU = recombination rate

From the conservation of charge,

net holesflowing

out of ∆x= +

Continuity Equations


p

oppUτ−

=

)(1)(1 xJq

xxJq

pp −∆+=

xdxdJ

qp

∆=1

From our discussion of recombination,

xpopxUp

∆−

=∆τ

∴ holes recombining =

Net holes flowing out of ∆x

Continuity Equations


Finally, the rate of change of holes within ∆x

(decrease due to recombination)xdtdp

∆−=

xppxdxdJ

qx

dtdp

p

op∆

−+∆=∆−∴

τ1

∴ dtdppp

dxdJ

q p

op−=

−+

τ1

(34)

Similarly,dtdnnn

dxdJ

q n

on−=

−+−

τ1

(35)

Eq. (34) & (35), called the continuity equations, are fundamentally important in the operation of semiconductor devices.

Drift-Diffusion Equations


The following equations that we previously derived are the transport equations (21) and (22) and are also fundamentally important in device operation.

)(dxdnDnqJ nnn += εµ

(36)

(37)

)(dxdnDpqJ ppp −= εµ

Poisson’s Equation


One final equation is needed to describe device operation and can be obtained from Poisson’s equation:

Where ε = electric field ρ = space charge density (charges/cm3) K = dielectric constant εo = permittivity of free space = 8.854x10-14 F/cm

Therefore,

If we know the charge density as f(x) we could find the electric field.

oKdxd

ερε

=

∫=oK

dxε

ρε

Poisson’s Equation


In general, the net charge density in a semiconductor is given by:

ρ = q[p+ + ND+ − n − NA]

(34) − (38) constitute a complete set of equations to describe carrier, current, and field distributions. Given appropriate boundary conditions, we can solve them for an arbitrary device structure. Generally, we will be able to simplify them based on physical approximations.

dxdV

−=ε e,Furthermor

( ) ( )[ ]DAo

NNpnKq

xdVd −+−=

ε22

that so (38)

semiconductor physics: energy bands - scu · donor and acceptor levels s. saha ho #2: elen 251 -...

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