Lecture #20 Objectives:

1. Last time: equilibrium statistics this time non-equilibrium behavior 2. Recombination-Generation processes 3. Carrier transport 4. p-n Junctions

Questions you should be able to answer by the end of the lecture:

1. Know the definition and different types of R-G processes 2. Understand the role of momentum in R-G processes 3. Know how to set-up rate equations for R-G statistics and how to derive the steady

state equations. 4. What is surface recombination and what is it’s significance. 5. What is the drift current and what are its sources 6. The role of mobility in carrier transport 7. How is the mobility effected by doping and temperature 8. Resistivity 9. Diffusion 10. What is the Einstein relationship 11. The equations of state 12. How are p-n junctions fabricated? 13. Qualitative description of p-n junction electrostatics 14. Understand what a depletion region is 15. What is the built-in potential and how is it effected by a forward or reverse bias 16. Ideal diode I-V response References: 1. Ashcroft and Mermin 2. Advanced Semiconductor Fundamentals from the Modular Series on Solid State

Devices (Pierrett) 3. The P-N Junction

Definitions: Recombination – a process whereby electrons and holes are annihilated Generation - a process whereby electrons and holes are created Recombination (Generation) processes (1) Band to band recombination – The electron which initially occupies a state in the

conduction band transitions to a state in the valence band. This process is called a direct thermal recombination where an electron from a conduction band “annihilates” with a hole in the valence band. Since the energy of the electron in the final state is lower than in the initial state the process is accompanied by a release of energy typically in the form of a photon (radiative process).

(2) R-G center recombination – defect states that are deep in the gap provide intermediate energy states for the electron which transitions from the conduction band to the deep level and then to the valence band. The two step transition involves the release of lower energies than those in the direct transition these energies are transferred to lattice vibrations (phonons).

(3) Recombination via shallow levels (4) Excitons – As we mentioned electrons and holes can be viewed as independent

particles. An exciton is an electron hole pair which are bound in an arrangement which is hydrogen-like. These bound particles are viewed as a particle with an energy of formation which is lower than the band gap thus forming “levels” near the edges of the conduction or valence bands.

(5) Auger recombination – in this process an electron in the conduction band absorbs the energy released during the direct or indirect recombination in a collision process. The electron makes a transition to a higher state in the conduction band and then loses the excess kinetic energy by lattice collisions.

Band structure and its relation to momentum considerations in determining the dominant recombination process All of the above mentioned processes are constantly occurring in a semiconductor (even at equilibrium). The question to consider is that of rate of occurrence specifically is there a particular mechanism that has a very high rate of occurrence thereby dominating the other processes? obviously it would be very useful to understand the factors that determine the rate. An important qualitative difference exists between the dominant R-G processes in direct (GaAs) and indirect (Si and Ge) bandgap SC’s:

When considering the probability or rate of a particular transition it is imperative to look at not only the energy change but also the required momentum change. The momentum of a photon is given by:

2p k

πλ

= =h h

the wavelength corresponding to a 1.24eV gap is 1 micron:

gap

hcE

λ=

which implies that the wavevector (and hence the momentum) is very close to k=0 since:

band edge photon 0k kaπ π

λ= >> = ;

thus in a direct band semiconductor the k values of the electrons and holes are near k=0 little change in momentum is needed in a recombination process thus a radiative direct transition is favorabel. In an indirect SC the transition necessitates a large change in momentum which cannot be facilitated by the emitted photon thus this process requires a transfer of momentum to a phonon. This additional requirement which is needed in order to conserve momentum leads to diminished rates of recombination in indirect SC. Direct recombination is thus negligible in indirect SC’s when compared with R-G center processes.

R-G statistics In order to calculate the rate of change of the concentration of charge carriers (be they electrons or holes) due to recombination-generation processes we define the following quantities:

nt

∂∂

- the rate of electron concentration change due both R and G processes.

pt

∂∂

- the rate of hole concentration change due both R and G processes.

Tn - the number density of R-G center levels that are occupied by electrons

Tp - the number density of R-G center levels that are unoccupied by electrons NT T Tp n= +

Comments: 1. The T label denotes a particular R-G mechanism which we called R-G centers –

or the recombination via deep Traps. 2. We have assumed one type of R-G center levels.

The (a) and (b) transitions are the only ones that effect the number of electrons in the conduction band

( ) ( )a b

n n nt t t

∂ ∂ ∂= +

∂ ∂ ∂

( ) ( )c d

p p pt t t

∂ ∂ ∂= +

∂ ∂ ∂

We assume that the probability of a transition is proportional to the number (density) of electrons in the conduction band and the number of empty trap levels (these assumptions hold for non-degenerate semiconductors)

( )n T

a

nc p n

t∂

= −∂

with - nc being the constant of proportionality sometimes called the electron capture coefficient which is a positive number. This process is assumed to be a first order process and can be likened to the rate equation that is used to describe the rate of a chemical reaction

( ) ( ) e empty R G center filled R G center+ − → − The electron emission process can likewise be related to the concentration of occupied trap levels and to the number of vacant levels in the conduction band (which is included in the constant of proportionality - ne )

with - ne being the constant of proportionality sometimes called the electron emission coefficient which is a positive number. In the same way expressions for the rate of hole change are obtained

( )p T

c

pc n p

t∂

= −∂

and

( )p T

d

pe p

t∂

=∂

substituting these relations into the rate equations

n T n T

nc p n e n

t∂

= − +∂

p T p T

pc n p e p

t∂

= − +∂

The basic question is how do we calculate or fined the capture and emission coefficients? For that we begin with calculating the relations between these coefficients under equilibrium conditions: Assuming that the electron capture and emission and hole capture and emission are independent processes we can write that at equilibrium

0n T n T

nc p n e n

t∂

= − + =∂

0p T p T

pc n p e p

t∂

= − + =∂

( )n T

b

ne n

t∂

=∂

we can thus relate the emission and capture coefficients for the electrons and holes

T

n

T

eq eqeq eqn eq

p ne c

n=

123

this coefficient can be explicitly calculated ( )

1i

T T T B

T T T T

eq eq eq eqT k Teq eq T

ieq eq eq eq

p n p N n Nn n n e

n n n n

µ µ−− −

= = = −

where iµ represents the position of the chemical potential in an intrinsic SC

3ln

2 4pc v

i Bn

mE Ek T

mµ

∗

∗

+= +

The expression for the occupation number of the trap levels is:

( )1

1

T

T

B

eq

ET k T

n

Ne

µ′ −=

+

where the temperature corrected energy of the trap level is { {

acceptor-like traps # of states allowed- donor-like traps per trap level 2

lnT T TE E kT g+

≈

′ = ±

The analysis performed in many cases on devices assumes that the device is operated under steady state conditions which means that all the macroscopic quantities are constant with time but the individual processes are not balanced (which would be the case at equilibrium assuming that the processes were independent)

0T

R G R G

n n pt t t− −

∂ ∂ ∂= − + =

∂ ∂ ∂

using the relations derived above ( )

( )( ) ( )( )/

/ /

i T

i T i T

E kTn T p T i

T E kT E kTn i p i

c N n c N n en

c n n e c p n e

µ

µ µ

′−

′ ′− − −

+=

+ + +

The net steady state recombination rate:

{( )( )

{( )( )

2

/ /

the hole the electronminority carrier minority carrier lifetimelifetime

1 1i T i T

pn

i

E kT E kTi i

p T n T

np nR

n n e p n ec N c N

µ µ

ττ

′ ′− − −

−−

−=

+ + +

where the lifetimes ( and n pτ τ ) represent the length of time an excess minority carrier will live in a band populated by majority carriers. For instance upon illumination of a semiconductor minority carriers will be excited the lifetime nτ will represent the time it will take for the minority carriers to decay in a p-type semiconductor. Note that the lifetimes depend explicitly on the number of R-G centers TN . Controlling lifetimes via impurity engineering The R-G centers are typically impurities that were unintentionally incorporated into the lattice during growth or processing. It is possible though to intentionally increase or decrease the lifetimes by either adding impurities (Au to Si) or gettering impurities for example one can diffuse phosphorous into the back side of an Si device in order to trap imurities and increase the lifetimes. Typical lifetime ranges in Si

Carrier transport This section is concerned with the motion of carriers under the influence of external fields and gradients. While the R-G processes focused on vertical transitions in the band diagram transport deals with horizontal movement. The two major processes that give rise to carrier motion are drift and diffusion. Drift Is defined as the motion of a charged particle in response to an applied electric field as depicted in the figure

while the microscopic details of the motion of individual charge carriers are obviously complicated and involve collisions one can assign an average velocity which is called the drift velocity vd. The drift current (for the case of holes in a p-type semiconductor) is:

p

p

driftd

driftd

I qpv A

J qpv

=

=

The drift velocity is related to the applied field and for small to moderate field values is taken to be proportional to it

{ {external field

hole mobility

d pv µ ε=

thus relating the drift current density to the mobility and carrier concentrations

p

driftpJ qpµ ε=

The total drift current density is given by

( )Total

- conductivity

driftn pJ q n p

σ

µ µ ε≡

= +1442443

Example of n-type SC drift calculation: In a non-degenerate donor-doped SC maintained in the extrinsic temperature regime

2

and p iD i D

D

nN n n N n

N>> → <<; ;

Total

n-typedriftD nJ qN µ ε;

typical values for carrier drift velocities in SC

Mobility Mobility is a measure of the ease of carrier motion within a material. The mobility can be represented as

qmτ

µ ∗=

where τ is the mean free time between collisions and m is the effective mass. In device semiconductors the dominant scattering mechanisms that determine τ are phonon scattering and scattering by ionized impurities. The calculation of mobilities are done typically using many approximations and empirical rules one of these is that the total mobility is given by the following sum:

1 1 1 1

lattice ionization otherµ µ µ µ= + +

Another widely used empirical rule relates the mobility to the temperature and doping levels

( ) ( )

( )1

c

b Ta T

Nc T

µ = +

+

where all the T dependences are simple power laws….

Diffusion Diffusion is a macroscopic motion due to carrier concentration differences. The diffusion motion and resulting current are described by diffusion equations:

diffusionp p

dpJ qD

dx= −

diffusionn n

dnJ qD

dx=

where D’s are the diffusion coefficients with units of cm^2/s The Einstein Relations For a non-uniformly doped SC

Note: in the notation we used in class the EF level was called the chemical potential µ which is the term that has thermodynamic significance… Under equilibrium conditions the chemical potential across the sample is constant (by definition of equilibrium). You also saw in the previous lecture that in an n-doped SC the chemical potential moves closer to the conduction band. Consequently in non-uniformly doped SC we expect to see band bending and thus a non-zero electric field is formed in the material. The current under equilibrium conditions is zero everywhere:

n0diffusion drift

n n n

dnJ J qD qn

dxµ ε+ = + =

Here we have used nµ to denote the mobility of the electrons in the conduction band not to be confused with the chemical potential.

However the internal field can be simply related to the gradient of the energy band 1 cdEq dx

ε =

The density gradient can be calculated by the expression for the number of electrons in the conduction band at temperature T which is equal approximately to:

( ) ( )( )3/ 2

2

214

c

c

En kT

cE

m kTn g E f E dE e

µ−∞ ∗ − =

∫ ; h

( )3/ 2

2

21 14

cEc n kT

c c

n

dE m kTdn dn dnq e q

dx dE dx dE kT

µ

ε ε−∗ −

=

= = = − h144424443

n

10diffusion drift

n n n

n

n

J J qD qn qnkT

D kTq

ε µ ε

µ

+ = − + =

→ =

This last relation is called the Einstein relation. For a typical non-degenerate semiconductor at room temperature:

2 2

0.026

1000 26n n

kTV

q

cm cmD

V s sµ →

⋅

;

; ;

Continuity equations

drift diffusion R-G other

n n n n nt t t t t

∂ ∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ ∂

drift diffusion R-G other

p p p p pt t t t t

∂ ∂ ∂ ∂ ∂= + + +

∂ ∂ ∂ ∂ ∂

p-n Junctions The p-n junction is the most fundamental of semiconductor devices the vast majority of SC devices employ some form of a p-n junction. For example: the bi-polar transistor is just two very closely spaced p-n junctions. A solar cell is a p-n junction built to absorb sunlight and convert it to an electrical current. p-n junctions are formed by a variety of fabrication techniques such as: 1. Alloying - 2. Epitaxial growth – SiCl4 and hydrogen are flowed over an n-type wafer at elevated temperatures impurity atoms such as borane (B2H6 to form p doping) can be added to the carrier gas. This process forms a fairly abrupt p-n junction. 3. Thermal diffusion – a gas containing impurities is passed over the substrate at elevated temperatures causing the diffusion of impurities into the substrate and the subsequent formation of a graded junction. 4. Ion implantation – impurity atoms are ionized and then accelerated and shot into the substrate. This process offers precise placement of the junction and can be done at low temperatures. The rest of this lecture will be from powerpoint slides…..