Chap 2. Carrier Motion - jupiter.math.nctu.edu.tw

MOS Device Physics and Design Chap. 2

Instructor: Pei-Wen LiDept. of E. E. NCU

1

Chap 2. Carrier Motion

u Carrier Driftu Carrier Diffusionu Graded Impurity Distributionu Hall Effectu Nonequilibrium excess carriers in semiconductoru Carrier Generation and Recombinationu Continuity Equationu Ambipolar Transportu Quasi-Fermi Energy Levelsu Excess-Carrier Lifetimeu Surface Effects



2

Carrier Action

u Three primary types of carrier actions in semiconductor– Drift– Diffusion– Recombination-Generation



3

Carrier Drift

u When an E-field (force) applied to a semiconductor, electrons and holes will experience a net acceleration and net movement, if there are available energy states in the conduction band and valence band. The net movement of charge due to an electric field (force) is called “drift”.

u Mobility: the acceleration of a hole due to an E-field is related by

If we assume the effective mass and E-field are constants, the we can obtain the drift velocity of the hole by

where vi is the initial velocity (e.g. thermal velocity) of the hole and t is the acceleration time.

qEdtdvmF p == *

EtvmeEtv i

pd ,* ∝+=



4

Mobility

E = 0

u In semiconductors, holes/electrons are involved in collisions with ionized impurity atoms and with thermally vibration lattice atoms. As the hole accelerates in a crystal due to the E-field, the velocity/kinetic energy increases. When it collides with an atomin the crystal, it lose s most of its energy. The hole will again accelerate/gain energy until is again involved in a scattering process.



5

Mobility

u If the mean time between collisions is denoted by τcp, then the average drift velocity between collisions is

where µp (cm2/V-sec) is called the hole mobility which is an important parameter of the semiconductor since it describes how well a particle will move due to an E-field.

u Two collision mechanisms dominate in a semiconductor:– Phonon or lattice scattering: related to the thermal motion of atoms; µL ∝T-3/2

– Ionized impurity scattering: coulomb interaction between the electron/hole and the ionized impurities; µI ∝T3/2/NI., : total ionized impurity conc.↑, µI ↓

If T↑, the thermal velocity of hole/electron ↑⇒carrier spends less time in the vicinity of the impurity. ⇒ less scattering effect ⇒ µI ↑

EEme

v pp

cpd µ

τ≡

= * *

p

cpdpp m

eE

v τµ ==

−+ += adI NNN



6

Mobility

Electron mobility Hole mobility



7

Drift Current Density

u If the volume charge density of holes, qp, moves at an average drift velocity vdp, the drift current density is given by

Jdrfp = (ep) vdp = eµppE.Similarly, the drift current density due to electrons is given by

Jdrfn = (-en) vdp = (-en)(-µnE)=eµnnEu The total drift current density is given by Jdrf = e(µnn+µpp) E



8

Conductivity

u The conductivity σ of a semiconductor material is defined by Jdrf≡ σ E, so σ= e(µnn+µpp) in units of (ohm-cm)-1

u The resistivity ρ of a semiconductor is defined by ρ ≡ 1/ σ



9

Resistivity Measurement

u Four-point probe measurement

factor correction: ;2 cc FFIVsπρ =



10

Velocity Saturation

u So far we assumed that mobility is indep. of E-field, that is the drift velocity is in proportion with the E-field. This holds for low E-filed. In reality, the drift velocity saturates at ~107 cm/sec at an E-field ~30 kV/cm. So the drift current density will also saturateand becomes indep. of the applied E-field.



11

Velocity Saturation of GaAs

u For GaAs, the electron drift velocity reaches a peak and then decreases as the E-field increases. ⇒negative differential mobility/resistivity, which could be used in the design of oscillators.

u This could be understood by considering the E-k diagram of GaAs.



12

Velocity Saturation of GaAs

u In the lower valley, the density of state effective mass of the electron mn

* = 0.067mo. The small effective mass leads to a large mobility. As the E-field increases, the energy of the electron increases and can be scattered into the upper valley, where the density of states effective mass is 0.55mo. The large effective mass yields a smaller mobility.

u The intervalley transfer mechanism results in a decreasing average drift velocity of electrons with E-field, or the negative differential mobility characteristic.



13

Carrier Diffusion

u Diffusion is the process whereby particles flow from a region ofhigh concentration toward a region of low concentration. The netflow of charge would result in a diffusion current.



14

Diffusion Current Density

u The electron diffusion current density is given by Jndif = eDndn/dx,

where Dn is called the electron diffusion coefficient, has units of cm2/s.

u The hole diffusion current density is given by Jpdif = -eDpdp/dx,where Dp is called the hole diffusion coefficient, has units of cm2/s.

u The total current density composed of the drift and the diffusion current density.

1-D

or 3-D

dxdpeD

dxdneDEepEenJ pnxpxn −++= µµ

peDneDEepEenJ pnxpxn ∇−∇++= µµ



15

Graded Impurity Distribution

u In some cases, a semiconductors is not doped uniformly. If the semiconductor reaches thermal equilibrium, the Fermi level is constant through the crystal so the energy-band diagram may qualitatively look like:

u

u Since the doping concentration decreases as x increases, there will be a diffusion of majority carrier electrons in the +x direction.

u The flow of electrons leave behind positive donor ions. The separation of positive ions and negative electrons induces an E-field in +x direction to oppose the diffusion process.



16

Induced E-Field

u The induced E-field is defined asthat is, if the intrinsic Fermi level changes as a function of distance through a semiconductor in thermal equilibrium, an E-field exists.

u If we assume a quasi-neutrality condition in which the electron concentration is almost equal to the donor impurity concentration, then

u So an E-field is induced due to the nonuniform doping.

dxdE

edxeEd

dxdE FiFi

x1))/((

=−

−=−=φ

dxxdN

xNekTE

dxxdN

xNkT

dxEd

dxEEd

nxNkTEExN

kTEEnn

d

dx

d

d

iiF

i

diFd

iFio

)()(

1

)()(

)()(

)(ln)(exp

−=⇒

=−

=−

⇒

=−⇒≈

−

≈



17

Einstein Relation

u Assuming there are no electrical connections between the nonuniformly doped semiconducotr, so that the semiconductor is in thermal equilibrium, then the individual electron and holecurrents must be zero.

u Assuming quasi-neutrality so that n ≈ Nd(x) and

u Similarly, the hole current Jp = 0

dxdneDEenJ nxnn +==⇒ µ0

relation-Einstein ---

)()()(

1)(0

)()(0

ekTD

dxxdNeD

dxxdN

xNekTxNe

dxxdNeDExeNJ

n

n

dn

d

ddnn

dnxndn

=⇒

+

−=⇒

+==

µ

µ

µ

ekTD

p

p =⇒µ



18

Einstein Relation

u Einstein relation says that the diffusion coefficient and mobility are not independent parameters.

Typical mobility and diffusion coefficient values at T=300K(µ = cm2/V-sec and D = cm2/sec)

µn Dn µp Dp

Silicon 1350 35 480 12.4GaAs 8500 220 400 10.4Germaium 3900 101 1900 49.2



19

Hall Effect

u The hall effect is a consequence of the forces that are exerted on moving charges by electric and magnetic fields.

u We can use Hall measurement to – Distinguish whether a semiconductor is n or p type– To measure the majority carrier concentration– To measure the majority carrier mobility



20

Hall Effect

u A semiconductor is electrically connected to Vx and in turn a current Ix flows through. If a magnetic field Bz is applied, the electrons/holes flowing in the semiconductor will experience a force F = q vx x Bz in the (-y) direction.

u If this semiconductor is p-type/n-type, there will be a buildup of positive/negative charge on the y = 0 surface. The net charge will induce an E-field EH in the +y-direction for p-type and -y-direction for n-type. EH is called the Hall field.

u In steady state, the magnetic force will be exactly balanced by the induced E-field force. F = q[E + v x B] = 0 ⇒ EH = vx Bz and the Hall voltage across the semiconductor is VH = EHW

u VH >0 ⇒ p-type, VH < 0 ⇒ n-type



21

Hall Effect

u VH = vx W Bz, for a p-type semiconductor, the drift velocity of hole is

u for a n-type,

u Once the majority carrier concentration has been determined, we can calculate the low-field majority carrier mobility.

u For a p-semiconductor, Jx = epµpEx.

u For a n-semiconductor,

( )( ) H

zxzxH

xxdx edV

BIpepd

BIVWdep

IepJv =⇒=⇒==

H

zx

edVBIn −=

WdepVLI

x

xp =⇒ µ

WdenVLI

x

xn =⇒ µ



22

Hall Effect

Hall-bar with “ear” van deer Parw configuration



23

Nonequilibrium

u When a voltage is applied or a current exists in a semiconductordevice, the semiconductor is operating under nonequilibriumconditions.

u Excess electrons/holes in the conduction/valence bands may be generated and recombined in addition to the thermal equilibrium concentrations if an external excitation is applied to the semiconductor.

u Examples: 1. A sudden increase in temperature will increase the thermal generation rate of electrons and holes so that their concentration will change with time until new equilibrium reaches.2. A light illumination on the semiconductor (a flux of photons)can also generate electron-hole pairs, creating a nonequilibriumcondition.



24

Generation and Recombination

u In thermal equilibrium, the electrons are continually being thermal generated from the valence band (hereby holes are generated) to conduction band by the random thermal process.

u At the same time, electrons moving randomly through the crystal may come in close proximity to holes and recombine. The rate of generation and recombination of electrons/holes are equal so the net electron and hole concentrations are constant (independent of time).



25

Excess Carrier Generation and Recombination

u When high-energy photons are incident on a semiconductor, electron-hole pairs are generated (excess electrons/holes) ⇒ the concentration of electrons in the conduction band and of holes in the valence band increase above their thermal-equilibrium value. n= no +δn, p = po+ δp where no/po are thermal–equilibrium concentrations, and δn/δp are the excess electron/hole concentrations. np ≠ nopo = ni

2 ( nonequilibrium)

u For the direct band-to-band generation, the generation rates (in the unit of #/cm3-sec) of electrons and holes are equal; gn’ = gp’ (may be functions of the space coordinates and time)



26


u An electron in conduction band may “fall down” into the valence band and leads to the excess electron-hole recombination process.

u Since the excess electrons and holes recombine in pairs so the recombination rates for excess electrons and holes are equal, Rn’ = Rp’. (in the unit of #/cm3-sec). ⇒ δn(t) = δp(t)

u The direct band-to-band recombination is spontaneous, thus the probability of an electron and hole recombination is constant with time.

u Rn’ = Rp’ ∝ the electron and hole concentration.



27

Recombination Process

u Band-to-Band: direct thermal recombination.This process is typically radiative, with the excess energy releasedduring the process going into the production of a photon (light)

u R-G Center: Induced by certain impurity atoms or crystal defects. Electron and hole are attractedto the R-G center and lead to the annihilationof the electron-hole pair.Or a carrier is first captured at the R-G site and then makes an annihilating transition to the opposite carrier band.This process is indirect thermal recombination (nonradiative). Thermal energy (heat) is released during the process (lattice vibrations, phonons are produced)



28


u Recombination via Shallow Levels:—induced by donor or acceptor sites. At RT, if an electron is captured at a donor site,however, it has a high probability of being re-emitted into the conduction band before completing the recombination process. Therefore, the probability of recombination via shallow levels is quite low at RT. It should be noted that the probability of observing shallow-level processes increases with decreasing system temperature.

u Recombination involving Excitons:It is possible for an electron and a hole to become boundtogether into a hydrogen-atom-like arrangement which moves as a unit in response to applied forces. This coupled e-h pair is called an “exciton”. The formation of an excitoncan be viewed as introducing a temporary level into thebandgap slightly above or below the band edge.



29


u Recombination involving Excitons: Recombination involving excitons is a very important mechanism at low temperatures and is the major light-producing mechanism in LED’s.

u Auger Recombinations:In a Auger process, band-to-band recombinationat a bandgap center occurs simultaneously with the collision between two like carriers. The energy released by the recombination or trappingsubprocess is transferred during the collision to the surviving carrier. Subsequently, this high energetic carrier “thermalizes”-loses energy through collisions with the semiconductor lattice.Auger recombination increases with carrier concentration, becoming very important at high carrier concentration. Therefore, Auger recombination mmust be considered in treating degenerately doped regions (like solar cell, junction lasers, and LED’s)



30

Generation Process

u Band-to-Band generation:

u R-G center generation:

u Photoemission from band gap centers:



31

Generation Process

u Impact-Ionization:An e-h pair is produced as a result of the energy released when a highly energetic carrier collides with the crystal lattice. The generation of carriers through impact ionizationroutinely occurs in the high e-filed regions of devices and is responsible for the avalanche breakdown in pn junctions.



32

Momentum Consideration

u In a direct semiconductor where the k-values of electrons and holes are all bunched near k = 0, little change is required for the recombination process to proceed. The conservation of both energy and crystal momentum is readilymet by the emission of a photon.

u In a indirect semiconductor, there isa large change in crystal momentumassociated with the recombination

process. The emission of a photon will conserve energy but cannot simultaneously conserve momentum. Thus for band-to-band recombination to proceed in an indirect semiconductor a phonon must be emitted coincident with the emission of a photon.



33


u Low-level injection: the excess carrier concentration is much less than the thermal equilibrium majority carrier concentration, e.g., for a n-type semiconductor, δn = δp << no.

u High-level injection: δn ≈ no or δn >> no

u For a p-type material (po >> no) under low-level injection, the excess carrier will decay from the initial excess concentration with time;

where τn0 is referred to as the excess minority carrier lifetime (τn0∝1/p0) , and the recombination rate of excess carriers Rn’ = Rp’=

u For a n-type material (no >> po) under low-level injection, Rn’ = Rp’=

0

)(

n

tnτ

δ

0/)0()( pntetptp τδδ −==

0

)(

p

tpτ

δ



34

Continuity Equations

u Consider a differential volume element in which a 1-D hole flux, Fp

+ (# of holes/cm2-sec), is entering this element at x and is leaving at x+dx.

u So the net change in hole concentration per unit time is

----continuity equation for holes

u Similarly, the continuity equation for electron flux ispt

pp pg

xF

tp

τ−+

∂

∂−=

∂∂ +

ntn

n ngx

Ftn

τ−+

∂∂

−=∂∂ −



35

Quasi-Fermi Levels

u At thermal-equilibrium, the electron and hole concentrations are functions of the Fermi level by

u Under nonequilibrium conditions, excess carriers are created in a semiconductor, the Fermi energy is strictly no longer defined. We may define a quasi-Fermi level, EFn, for electrons and a quasi-Fermi level, EFp, for holes that apply for nonequilibrium. So that the total electron and hole concentrations are functionsof the quasi-Fermi levels.

exp and exp

−

=

−

=kT

EEnpkT

EEnn FFiio

FiFio

exp and exp

−

=+

−

=+kT

EEnppkT

EEnnn FFiio

FiFnio δ



36

Quasi-Fermi Levels

u For a n-type semiconductor under thermal equilibrium, the band diagram is

u Under low-level injection, excess carriers are created and the quasi-Fermi level for holes (minority), EFp, is significantly different from EF.



37

Excess-Carrier Lifetime

u An allowed energy state, also called a trap, within the forbidden bandgap may act as a recombination center, capturing both electrons and holes with almost equal probability. (it means that the capture cross sections for electrons and holes are approximately equal)

u Acceptor-type trap: – it is negatively charged when it contains an electron and it is neutrall when it

does not contain an electron.

u Donor-type trap:– it is positively charged when empty and neutral when filled with an electron



38

Shockley-Read-Hall Theory of Recombination

uAssume that a single recombination center exists at an energy Et within the bandgap. And there are four basic processes that may occur at this single trap.

u Process 1: electron from the conduction band captured by an initially neutral empty trap.

u Process 2: electron emission from atrap into the conduction band.

u Process 3: capture of a hole fromthe valence band by a trapcontaining an electron.

u Process 4: emission of a holefrom a neutral trap into thevalence band.



39


u In Process 1: the electron capture rate (#/cm3-sec): Rcn = CnNt(1-fF(Et))n

Cn=constant proportional to electron-capture cross sectionNt = total concentration in the conduction bandn = electron concentration in the conduction band

fF(Et)= Fermi function at the trap energy

u For Process 2: the electron emission rate (#/cm3-sec): Ren = EnNtfF(Et)

En=constant proportional to electron-capture cross section Cn

u In thermal equilibrium, Rcn = Ren, using the Boltzmannapproximation for the Fermi function,

u In nonequilibrium, excess electrons exist,

( )n

tccnn C

kTEENCnE

−−

== exp'

( )[ ])()(1 'tFtFtnencnn EfnEfnNCRRR −−=−=



40


u In Process 3 and 4, the net rate at which holes are captured from the valence band is given by

u In semiconductor, if the trap density is not too large, the excess electron and hole concentrations are equal and the recombination rates of electrons and holes are equal.

u In thermal equilibrium, np = ni2 ⇒ Rn = Rp = 0

[ ]))(1()( 'tFtFtpp EfpEpfNCR −−=

( )

−−

=kT

EENp vtv exp'

RppCnnC

nnpNCCRR

ppCnnCpCnC

Ef

pn

itpnpn

pn

pntF

≡+++

−==

++++

=⇒

)'()'()(

and

)'()'('

)(

2



41

Surface Effects

u Surface states are functionally equivalent to R-G centers localized at the surface of a material. However, the surface states (or interfacial traps) are typically found to be continuously distributed in energy throughout the semiconductor bandgap.



42

Surface Recombination Velocity

u As the excess concentration at the surface becomes smaller than that in the bulk, excess carriers from the bulk region diffuse toward the surface where they recombine, and the surface recombination velocity increases.

u An infinite surface recombination velocity implies that the excess minority carrier concentration and lifetime are zero.



43

Carrier Action Equation Summary

Equation of State

Current and R-G Relationships

processother

GRthermalP

processother

GRthermal

N

tp

tpJ

qtp

tn

tnJ

qtn

∂∂

+∂∂

+•∇=∂∂

∂∂

+∂∂

+•∇=∂∂

−

−

1

1

Ln

nnP

n

Ln

ppN

p

Gpx

pDtp

Gn

xn

Dtn

+∆

−∂∆∂

=∂∆∂

+∆

−∂

∆∂=

∂

∆∂

τ

τ

2

2

2

2

PN

PpPtPP

NnNtNN

JJJpqDpEqJJJ

nqDnEqJPJJ

+=

∇−=+=

∇+=+=

µ

µ

diff|drif|

diff|drif|

diffusiondrift cc

n

n

ptp

ntn

τ

τ

∆−=

∂∂

∆−=

∂∂

G-Rthermal-i

G-Rthermal-i



44


u Key Parametric Relationships

u Resistivity and Electrostatic Relationships

pPP

nNN

DL

DL

τ

τ

≡

≡

qkTDq

kTD

p

P

n

N

=

=

µ

µ

Tpp

Tnn

Nc

Nc1

1

=

=

τ

τ

( )pnq pn µµρ

+=

1

torsemiconduc type-p

torsemiconduc type-n

... 1

... 1

Ap

Dn

Nq

Nq

µρ

µρ

=

=

( )ref1 111 EEq

VdxdE

qdxdE

qdxdE

qE C

iVC −−====



45


u Quasi-Fermi Level Relationships

PpPi

iP

NnNi

iN

FpJnpEF

FnJnnEF

∇=

−≡

∇=

+≡

µ

µ

ln

ln

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Chap 2. Carrier Motion - jupiter.math.nctu.edu.tw