VII. Semiconducting Materials & Devices

VII. Semiconducting Materials & Devices • Band Structure and Terminology • Intrinsic Behavior • Optical Absorption by Semiconductors • Impurity Conductivity • Extrinsic Behavior • Hall Effect and Hall Mobility • The Diode: A Simple p-n Junction

conduction band Ec = conduction band edge Ev = valence band edge valence band Energy band diagram in k-space “Flat-band” diagram in real space A. Band Structure and Terminology Semiconductors and insulators have qualitatively similar band structures, with the quantitative distinction that the band gap Eg > 3.0 eV in insulators.

Fermi-Dirac Distribution Function The important external parameter that determines the properties of a semiconductor is the temperature T, which controls the excitation of electrons across the band gap in a pure (intrinsic) semiconductor. Fermi-Dirac distribution function The probability for an electron to be in an energy level  at temperature T  = chemical potential  EF for T << TF For nearly all T of interest: This is the Maxwell-Boltzmann (classical) limit of Fermi-Dirac statistics So we can approximate:

  as T  due to increased scattering ( ) low-T (extrinsic) region high-T (intrinsic) region “Intrinsic” means without impurities. Electrical conductivity is zero at T = 0, but for T > 0 some electrons are excited into the conduction band, also creating holes (H+) in the valence band. In general the conductivity can be written (using the nearly FEG model): B. Intrinsic Behavior The conductivity is controlled by the magnitude of n and p, which rise exponentially as T increases. The relaxation times are only dependent on 1/T, and this dependence is often ignored because the exponential behavior dominates. Experimentally we find that for a pure semiconductor:

Ec = Eg  It is relatively simple to calculate n(T) and p(T) for the intrinsic region, where the conductivity is caused by excitation of e- across the energy gap: Intrinsic Carrier Statistics e- in the conduction band: h+ in the valence band: Ev = 0 For parabolic bands the density of states are: Think: why are we justified in assuming parabolic E(k) here?

Which allows: Intrinsic Carrier Statistics, cont’d. We can write the density of states per unit volume: And now calculate the carrier concentration n(T): Now rearrange cleverly and pull out a factor of (kT)1/2:

The integral becomes: So finally: Intrinsic Carrier Statistics, cont’d. Now make a variable substitution: Whew! And next we can do the same calculation for holes to get p(T)!

And since we have Intrinsic Hole Carrier Statistics Now for holes in the valence band: Just as before, calculate the carrier concentration p(T): Now rearrange : Replace –E with E and flip limits due to minus sign: Pull out a factor of (kT)1/2:

Now make the variable substitution: The integral becomes: again! So finally: Intrinsic Hole Carrier Statistics, cont’d. So far these relations for n(T) and p(T) are true for any semiconductor, with or without impurities. It is very convenient to calculate the product np:

This gives an expression for (T): Now for an intrinsic semiconductor (or in the intrinsic region of a doped semiconductor) ni = pi, so: Intrinsic Carrier Statistics: Results And equating the earlier expressions for n and p: So the chemical potential, or Fermi level, has some dependence on T, but if mh and me are similar, then this is very small.

Definition of carrier mobility: Earlier FEG result: Now we can rewrite the total conductivity as: Experiment shows that  has a power-law temperature dependence: Thus the exponential temperature dependence of n and p dominates, and we can approximate the intrinsic conductivity The total conductivity, including both the electron and hole contributions, is: Carrier Mobility It is common to define a quantity that expresses the size of the drift speed for each type of carrier in an electric field E: (Note: the carrier mobility is directly related to the switching speed of a solid-state electronic device) So a plot of vs. 1/T gives a straight line with slope –Eg/2k. Conductivity measurements allow us to determine Eg!

Ec Ec Ev Ev Si GaAs C. Optical Absorption by Semiconductors Examine the following calculated 3-D band structures for semiconductors Si and GaAs. What difference(s) do you see? Indirect band gap (kgap 0) Direct band gap (kgap= 0)

So essentially we have: Optical Absorption and Conservation Laws Absorption of a photon by a semiconductor can promote an electron from the valence to the conduction band, but both energy and momentum must be conserved: For semiconductors Eg 1 eV so the photon wavevector can be estimated: But this is utterly tiny compared to a typical BZ dimension: A direct-gap (vertical) transition

But for indirect gap semiconductors it is clear that: To estimate a typical phonon energy, we know: Direct vs. Indirect Gap Semiconductors So for an indirect gap transition momentum can only be conserved by absorption or emission of a phonon (lattice vibration)

Experimental absorption coefficients () are measured to be: Optical Absorption: Experimental Results While for an indirect gap material with a direct transition at a slightly higher energy:

pure Si: (each line represents an e-) weakly bound extra electron Si:P missing electron Si:B Consider two types of substitutional impurities in Si: D. Impurity Conductivity in Semiconductors donor energy level (n-type material) acceptor energy level (p-type material)

-e, m Ed r +e We can estimate the ionization energy of a pentavalent donor impurity using the Bohr model: Donor Impurities in Semiconductors Bohr model for H: For an electron orbiting a positive ion inside a semiconductor, what changes must we make in the Bohr model equations? periodic potential (effective mass): dielectric medium: So in a semiconductor:

Assuming that the electron is initially in its lowest energy level, the donor ionization energy Ed is: Donor Ionization Energy For Si we can use representative values of the effective mass and dielectric constant to obtain: All within a factor of two of our rough estimate! Experimental data reveal ionization energies (in meV): The orbital radius is predicted to be: So this electron moves through a region that includes hundreds of atoms, which supports the use of the dielectric constant of the bulk semiconductor.

Ea What happens when the impurity atom is trivalent? Acceptor Impurities in Semiconductors At 0 K the acceptor level is empty, so a “hole” is bound to the impurity atom. However, the energy Ea is so small (50 meV in Si) that at room T electrons in the valence band bound to other Si atoms can be excited into the acceptor level, leaving behind a mobile hole in the valence band. Summary: Both donor (P, As) and acceptor (B, Ga) impurities provide an easy way to increase either n or p even at low T. When such impurity-related carriers dominate the electrical properties, the semiconductor material displays extrinsic behavior. Note: If impurity concentration is very large, the Bohr orbits (wavefunctions) of the donor electrons can overlap and form an “impurity band” that extends throughout the material. This leads to a so-called insulator-metal transition and causes an abrupt increase in the conductivity (see problem 10.5).

Nd = concentration of donor atoms and Eg Eg-Ed Ev = 0 Let’s consider donor impurities in a semiconductor (n-type): E. Extrinsic Behavior and Statistics Nd+ = concentration of ionized donor atoms Nd0 = concentration of neutral donor atoms Now in the presence of a large donor e- concentration, then n >> ni so p must decrease in order to keep the product np = constant. What physical process causes p to decrease? Essentially the large number of e- in the conduction band will be sufficient to fill most available holes in the valence band, so that:

Now solve for the electron concentration n: Extrinsic Carrier Statistics Now from our earlier treatment of intrinsic behavior: Equating the expressions for n: You can always use this exact master eqn. to solve for  and thus n, but you have to do it numerically.

Eg  Ev = 0  This discussion is relevant to several HW problems in Myers (see 10.4, 10.8, 10.9). It provides simple approximations for n and  corresponding to very small Nd and very large Nd. Limits of Low and High Impurity Concentrations We will develop approximations to simplify the solution of this eqn. Now consider two extreme limiting cases: • Nd << n0 In Si at 300K (high T limit) Since Eg >> 2kT at room temp, this means And solving for : Thus,

  We can neglect the “1” in the denominator here: Eg Eg-Ed Ev = 0 2. Nd >> n0 (low T limit) Limits of Low and High Impurity Concentrations And now solve for :  Does the low T limit make sense? Substituting into the above eqn. for n: Thus,

Summary of Impurity Semiconductor Behavior Now our schematic plot of ln  vs. 1/T is even easier to understand:

Na = concentration of acceptor atoms Eg Ea Ev = 0 Let’s consider acceptor impurities in a semiconductor: Extrinsic Behavior for p-Type Semiconductors Na- = concentration of occupied acceptor levels Na0 = concentration of neutral (unoccupied) acceptor levels Now in the presence of a large hole concentration, then p >> pi so n must decrease in order to keep the product np = constant. What physical process causes n to decrease? Essentially most of e- in the conduction band will fall down to fill up holes in the valence band, so that: and

Extrinsic p-Type Carrier Statistics Now from our earlier treatment of intrinsic behavior: Equating the expressions for p: Again, you can always use this exact master eqn. to solve for  and thus p, but you have to do it numerically.

Eg   Ev = 0 Thus, Here we provide simple approximations for p and  corresponding to very small Na and very large Na. Limits of Low and High Impurity Concentrations We will develop approximations to simplify the solution of this eqn. Now consider two extreme limiting cases: • Na << p0 (high T limit) As before, we argue that  is near Eg/2 as in the intrinsic case: Since Eg >> 2kT at room temp, this means And solving for :

Eg Ea  Ev = 0  We can neglect the “1” in the denominator here: 2. Na >> p0 (low T limit) Limits of Low and High Impurity Concentrations And now solve for :  Does the low T limit make sense? Substituting into the above eqn. for p: Thus,

where B I - + - + + - - + - + + - + - + - w The Hall effect is easier to measure in semiconductors than in metals, since the carrier concentration is smaller: F. Hall Effect and Mobility When one carrier dominates, we have a Hall coefficient: Hall measurements can tell us whether a semiconductor is n-type or p-type from the polarity of the Hall voltage: n-type p-type When one carrier dominates, we can write the conductivity: Measuring RH and  will thus give: sign, concentration, and mobility of carrier, So the mobility can be written:

For a semiconductor with significant concentrations of both types of carriers: General Form of Hall Coefficient So if holes predominate (ph > ne ), RH > 0 and the material is said to be p-type, while if RH < 0 (as for simple metals), the material is said to be n-type.

p n Na Nd - - - - + + + + p n depletion region E When p- and n-type materials are fabricated and brought together to form a junction, we can easily analyze its electronic properties. G. The Diode: A Simple p-n Junction Near the junction the free electrons and holes “diffuse” across the junction due to the concentration gradients there. As this happens, a contact potential  develops. The field E due to the contact potential inhibits further flow of electrons and holes toward the junction, and equilibrium is established at finite .

Ecn Ecp Evp Evn Ecp e Ecn Evp Evn We can also describe the situation in terms of flat band diagrams. Physics of a Simple p-n Junction Initially (before equilibrium) Finally (equilibrium established) Here we see “band bending” in equilibrium. This reflects the potential at the junction and the equalization of the chemical potential throughout the system.

Jng Jnr p-type Jpr n-type Jpg In this dynamic equilibrium two types of carrier fluxes are equal and opposite: Physics of a Simple p-n Junction 1. recombination flux: electrons in the n-type region and holes in the p-type region “climb” the barrier, cross the junction, and recombine with h+/e- on the other side. 2. generation flux: thermally-generated electrons in the p-type region and holes in the n-type region are “swept” across the junction by the built-in electric field there. We can picture the carrier fluxes (currents): In equilibrium at V=0:

e eV Now when an external voltage V is applied to the junction, there are two cases: A Simple p-n Junction With Applied Voltage 1. Forward bias: electrons in the n-type region are shifted upward in energy Generation currents are not affected since they depend on excitation across band gap: Recombination currents are increased by a Boltzmann factor, since they depend on carriers climbing the potential energy step at the junction (and Maxwell-Boltzmann statistics applies):

Is = “saturation current” e eV We can now calculate the net current density from both holes and electrons: Current-Voltage Relation for A Simple p-n Junction I > 0 so current flows from p  n 2. Reverse bias: electrons in the n-type region are shifted downward in energy Here the only difference is that recombination currents are decreased by a Boltzmann factor, which just changes the sign of V in the exponential terms. So the resulting current is: I < 0 so current flows from n  p (leakage current) We can express both cases in one “ideal diode equation” if we define forward bias to be V > 0 and reverse bias to be V < 0:

I Vbr V -Is What about in the real world? Current-Voltage Relation for A Real Diode The ideal diode equation is approximately correct, but we have made some assumptions that are not rigorously true, and have neglected other effects, so in a real diode we see behavior like this: Mechanisms for breakdown in reverse bias include: 1. Zener breakdown: a large reverse bias allows tunneling of electrons from valence band of p-type region to conduction band of n-type region, where they can carry current! 2. Avalanche breakdown: electrons generated in p-type region and swept across the junction acquire enough kinetic energy to generate other electrons, which in turn generate more, etc.

VII. Semiconducting Materials & Devices