PHYSICS OF SEMICONDUCTOR DEVICES There are two classes of materials (solids): conductors and insulators. Conductor: The small D.C. conductivity at 0K is non-zero. Insulator: The small D.C. conductivity at 0K is zero. All insulators have finite conductivity at T > 0K. Semiconductors are those insulators with conductivity that is not too small at room temperature (300K).

Block Theorem In a crystalline solid, atoms are arranged in a periodic fashion. Silicon, for example, is a diamond structure, whereas GaAs is a zinc-­blende structure. With single electron approximation, time independent S.E. becomes: with V(r) a periodic function corresponding to the lattice. The solution of such S.E. is called the Block function, and has the form:

The solution of the Schrodinger Equation of a crystalline semiconductor gives the following results: The allowed electronic states are lumped together to form bands. There is a bandgap. At finite temperature, there are electrons and holes in the semiconductor.

The most important aspect of the block functions is that Ek formed into bands: In the case of insulator (semiconductor), there is a bandgap between the valence band and the conduction band.

Brief Summary of Block Theorem For a single electron in a “rigid” lattice, the time independent S.E. again is again: Let TR be an operator that moves the wave function by , a lattice constant. i.e., Now

since i.e., In other word, the two operators commute. A general QM theory states that there is a set of common eigenfunctions for two commuting operators.

If

Write

periodic in the lattice In general, with i.e., periodic in the lattice

Born-Von Karman boundary condition (Block’s theorem) But i.e.,

number of unit cells in the crystal allowable vectors in a primitive cell number of unit cells in the crystal

Consequence of the Block Theorem there is an infinite solution for each indexed by n.

( = a reciprocal vector) with proper choice of the index is periodic with K

Wave Packet Using block-waves, we construct a wave packet to localize the electrons. This is necessary since the Block waves are extended states of the stationary Schrodinger Equation. In order to treat a wave packet, we require time-dependent perturbation theory. The results are that, with a wave packet centered at k,

Where (1/m)ij is the (1/effective mass) tensor. and In reality we have imperfection in the crystal and lattice vibration. Thus there will be scattering, and the above equation will only be followed between scattering events. In general, the transport is governed by the Boltzmann transport equation.

Fermi-Dirac Distribution Boltzmann Equation Before going into Boltzmann Equation, we need to discuss f. Fermi-Dirac Distribution In Equilibrium where m is the electro-chemical potential, defined as

m is also called the Fermi-energy EF . In a solid with En(k) The total number of electrons in a band is therefore

Converting S into integration Since i.e.,

Splitting the integration over surfaces of constant energy, and then integrate over all energy in the band, we have: That is, the total number of electrons between E(k) and E(k) +dE is

i.e., Near the conduction band minimum, the constant energy surface is approximated by an ellipsoid, and it is very simple to obtain N(E). Let n(E) = total number of states inside an ellipsoid with constant energy surface of E

But for an ellipsoid with the proper choice of principal axes, and the principal radii are: i.e.,

Since i.e., where MC is the number of equivalent minimum valleys (6 for silicon).

For electron density in the conduction band, we have: if NC is the effective density of states of conduction band.

In general Defining which is Fermi integral of order j

What about valence bands? Key: a full band does NOT conduct (velocity cancelled pair-wise and therefore average v=0) Conduction of a current happens when some electron in the valence band is missing. Thus, we introduce an accounting principle, called a “hole” º an electron state that is empty. i.e., and

where and For silicon, we actually have two valence bands (+ a split-off band):

Ignoring the split off band, To make sure everything is consistent, we have hole transformation:

In summary, we have Now, let’s look at the Boltzmann Equation. Under the following two assumptions: relaxation time approximation (1) (2) f is essentially the same as f0

Then the Boltzmann Equation is equivalent to the device equations: Actually, if the system is not far from equilibrium,

For the purpose of this course, it is sufficient to use these equations except when the electric field is high. In that case, the carrier distributions are no longer Maxwellian, and velocity overshoot can happen; or when dE/dx is larger, we must abandon even Boltzmann transport. It is important to distinguish the effective masses. Conduction effective mass is the arithmetic mean of the “masses” For silicon, it turns out that for conduction minimum, it is a circular ellipsoid.

For silicon conduction band, with

Extrinsic Semiconductor To change the conductivity (as well as the dominant carrier type), we introduce into the seminconductor donors and acceptors. NA º acceptors concentration ND º Donors concentration

Under charge neutrality What about the interaction between the electrons and the holes. Since they are from orthogonal block waves, their interaction is small! Concept of quasi-fermi energies: Efn, Efp i.e., the electrons in the conduction band are in quasi-equilibrium with an electro-chemical potential mfn, while the holes in the valence band are in equilibrium with mfp (although mfn¹ mfp).

This concept makes sense when the relaxation time between the electrons and holes is large and is in general true. Recall ni2 is the pn product of non-degenerate semiconductor sample at equilibrium.

ED-EF > 3kT, EF -EA > 3kT In such case, the semiconductor is non-degenerate (i.e. pn=ni2), and

But with

Generation and Recombination of Excess Carriers Electrons and holes are not at equilibrium (with each other). Processes to restore equilibrium Note: Electrons are in equilibrium among themselves with relaxation time in the order of ~ femto-second.

(a) Band-to-Band (radiative recombination) · o EC EV hν

At equilibrium i.e., Net recombination: Assume quasi-charge neutrality

For low level injection (i.e. ) We can write: with i.e., t ↓as dopant concentration ↑. For direct band-to-band recombination, t is small (~1-100 pS)

B. Auger recombination: This is a process involving three particles. Since three particles involved Þ C is small. In heavily doped material, say N+ material, n is large Þ Auger can be important

C. Shockley-Read-Hall (SRH) recombination (“trap-assisted” recombination) For indirect gap materials , say silicon, to the first order, band-to-band recombination is not possible. hν Impossible!

(a) e- capture; (b) e- emission; (c) hole capture; (d) hole emission. Ev EC Single trap level recombination process (a) e- capture; (b) e- emission; (c) hole capture; (d) hole emission.

(b) is number of filled traps: Similarly

Net rate of recombination of e- Net rate of recombination of p

Rate of change of the number of e- in the traps:

By detailed balance, at equilibrium: i.e., At equilibrium

gives

with Similarly, with we have

\We have In non-equilibrium, these two equations are still correct, but we do not know fT. Let us look at the special case of steady-state:

With we have

If we write and is the capture cross-section

Since with

Defining and we have

Quasi-neutral region recombination Quasi-neutrality: and Assume can ignore its charge contribution

Define t : (i.e., if the excitation is removed) Since we have

n- type; low level injection (Dn <<no). (Note: no>>po) Assume (i.e., ET is not too close to the band edges)

Similarly, for p-type material we have

Einstein Relationship We have and

That is Comparing with

For non-degenerate n-type semiconductor, Similarly, for p-type semiconductor,

Carriers’ Drift Mobility m is determined by the scattering lifetime t(E): Two types of carrier scattering: 1. phonon scattering 2. impurity scattering (ionized impurity) (Matthiessen’s rule)

Typically, it was found that for silicon: and

m versus Temperature

m versus doping concentration

Irvin’s curves

To summarize: and True for e not too large.

The rate that e- gain energy from the E-field is: With very high E, the effective T of e- is high. The rate e- loss energy to the lattice can be drived to be: where Te is the “temperature” of the electron gas, is speed of sound in the semiconductor and is the mean free path between collision.

tm : mean time between scatterings In steady-state, energy gain=energy loss, with tm : mean time between scatterings But and

Accounting for the angle of scattering: With for phonon scattering:

i.e., and

If If

Basic Semiconductor Device Equations Electron (hole) conduction eqn. Electron (hole) continuity eqn. Poisson’s eqn.

Also, we have and

Simple Example n-type Also Creating excess carriers Dn=Dp=Dpn(0) at x=0. Also

In steady state, i.e.,

Minority carriers are much simplier to deal with!

So, the cont. eqn for p becomes: Boundary conditions for very long sample Solution for very long sample: where

Solution for small W (W<Lp): In general Boundary conditions

Surface recombination s is called surface recombination velocity. Typically <1 cm/sec for well passivated surface Consider minority carriers Þ Diffusion only

We have in the steady state: Boundary conditions i.e., at x=0

General solution (with the B.C. for is: Applying B.C. for x = 0

as expected