Hydrogen Storage in Nanostructured Materials

Semiconductor Crystals - November 8, 2005After studying this section, you should be able to explain:

What are semiconductor crystals? How do I calculate the band gap in these crystals? What causes Excitations to form electron-hole pairs? What is Intrinsic Conduction: Electron-hole pair formation? What is Extrinsic Conduction: Doping? What is Recombination and Carrier Lifetime? What are Compound Semiconductors?__________________________________________________________________________

What are Semiconductor Crystals?Semiconductor crystals are characterized lower concentration of charge carriers (either electrons or holes) than metals. At room temperature, their electrical conductivity is ~10-5 to 105 S/m. They have band gaps much narrower than insulators (0.2-6.5eV).

Sources of Excitations of electron-hole pairs

1. Thermal Excitations Thermal energy (heat) causes electrons to cross the band-gap.2. Optical Excitations Photons having energy, h, large enough to excite electrons across the band-gap.3. Impurities Doping pure semiconductors with atoms having more or fewer electrons causes energy states associated with dopant to donate to the conduction band or accept from the valence band.

Intrinsic Semiconductors (No impurities states)Intrinsic conduction arises when there is a low impurity concentration in the semiconductor and so increase in temperature determines the electrical properties. Semiconductors showing intrinsic conductivity have small band gaps so that thermal excitations are enough to cause electrons to move from the valence band to the conduction band. When the band gap is large relative to temperature, there will be few intrinsic carriers and conductivity will drop.The band gap energy (Eg) is the energy difference between the lowest unoccupied band and the highest valence band.

Band Gap = Conduction Band Edge Valence Band Edge

Conductivity arises from electron mobility in the conduction band and hole (vacant sites left behind by electrons) mobility in the valence band.

Band Gap in Semiconductor CrystalsSemiconductor materials are characterized by a strong temperature dependence of conductivity. Typically, these materials have a lower concentration of charge carriers (either electrons or holes) than metals. At room temperature, their electrical resistivity is ~10-2 to 109 ohm-cm.

Semiconductor Type

III-V (trivalent-pentavalent)Al, Ga, InN, As, P

II-VI (divalent-hexavalent)Zn, Cd, HgS, Se, Te, O

IV-IV (Diamond Type)C, Si, Ge, SnC, Si, Ge, Sn

Measurement of Intrinsic Conductivity

Band gap is determined by Optical Absorption measurements of two types:

1. Direct Optical Absorption Measurements: Photons are absorbed at frequenciesg reveal the crystal to have a band gap of . Such measurements are made by scanning the range of photon frequencies until absorption occurs.

2. Indirect Optical Absorption Measurements: Indirect absorption are made when the band gap is too large for a single photon to be absorbed, then indirect absorption occurs. In this case, a phonon is emitted in the excitation process, so that, where is the phonon frequency. Typically, is greater than 3. Temperature dependence of Electrical Conductivity: The intrinsic conductivity and carrier concentration is controlled by the ratio of energy gap to temperature since the carrier concentration (n or p) is proportional to kbT / Eg. The exact equation is obtained by integrating the product of conduction band Density of States (called D()) with the Fermi-Dirac distribution (called f()).

n or p

The range on the integration is Ec to for n-type (electron carriers)

and - to Ev for p-type (hole carriers)

The bulk carrier concentration determined using Hall measurement (http://www.eeel.nist.gov/812/meas.htm). Temperature dependence of the carrier concentration can be used to calculate the band gap.

Calculation of Intrinsic Electron Charge Carrier Concentration

We must use the density of states and the Fermi-Dirac function to calculate the intrinsic charge carriers for Electrons.

Fermi Dirac Function, f():

The Fermi-Dirac Distribution gives the probability that an energy level, , will be occupied by an electron:

.

In the case of semiconductors, we wish to calculate the number of electrons excited to the conduction band at temperature, T. In this case, we call the Fermi Level (the highest occupied level of the valence band). In most cases, >>kBT, resulting in the Fermi-Dirac distribution taking the form of the Boltzman Distrubution,

Density of States Function, D():

The Density of States is defined as the number of orbitals per unit energy (or dN/d).

Intrinsic electron charge carrier concentration:The concentration of electrons in the conduction band then becomes, n, where

Integrating yields

Here, the density of states function uses the effective hole mass (which is different than the effective electron mass). In fact, the effective hole mass is larger due to the following:

1. Band curvature in E(k) near top and bottom of band2. Band curvature in E(k) leads to higher correction to electron/hole mass (always fractional)

3. Higher correction leads to lower effective mass.Density of occupied electron states vs. Density of electron states

Density of electron and hole states, D() assumes a parabolic shape near the energy gap given by the parabolic function (i.e. there are more energy bands as we move away from the energy gap. (However, fermi distribution actually causes a reduction of density of occupied electron states near top of conduction band.)We have examined the density of electron D() states. Not all states will be occupied by electrons. In order to determine the density of occupied electron states {D()f()}, we have employed the Fermi-Dirac Function!!!

Now, lets Calculate the density of occupied electron states, Ne

Calculation of Intrinsic Hole Charge Carrier Concentration

We will again use the density of states and the Boltzmann function to calculate the intrinsic charge carriers for Holes.

Fermi Dirac Function, f():

For holes, the distribution becomes 1-f()electrons.

Density of States Function, D():

The equation for the density of states is

Intrinsic electron charge carrier concentration:The concentration of holes in the conduction band then becomes, p, where

Resulting in

Calculating the Energy Gap from Electron and Hole carrier concentration Measurements

Multiplying n and p, we can have an equation which does not depend on the fermi energy (it cancels). The product only depends on the magnitude of Eg.if

then

with

Nv = and Nc =

Furthermore, for an intrinsic semiconductor, n=p. Thus,

Given the value of the Energy Gap, we can now calculate the Fermi Energy

The fermi energy is located within the energy gap, however, we are not certain where.We can determine the location of the fermi level for intrinsic semiconductors by the following:In intrinsic conduction, one electron excitation to the conduction level produces one hole in the valence level, therefore, n = p and

And

or

Since

Taking the ln of both sides, we have,

Simplification yields the Fermi energy,

However, Since the energy gap (e.g. Si = 1.1 eV) is much larger than kbT=0.025eV (near room temperature). We see that for Intrinsic Semiconductors, the Fermi level lies at the center of the Band Gap: !

Intrinsic Carrier Mobility, eor hThe intrinsic carrier mobility is defined as the drift velocity per unit electric field. The units of carrier mobility is [m/s/J] or [s/kg].

=|v|/E in units of [s/kg]

The electrical conductivity, , is defined as =ne2/m.

In terms of carrier mobility, this conductivity can be written as:

=(e + h)

=(nee + peh)

where n and p are the electron and hole concentrations, respectively.

Hence, the conductivity can be thought of as the sum of the contribution due to electrons and the contribution due to holes.

By comparison of the electrical conductivity based upon the drift velocity of electrons within the Fermi sphere, we see that the intrinsic carrier mobility becomes:

e=ee/me and h=e h/mhwhere e and h are the time between collisions with ion cores.

Temperature Dependence of mobility during Intrinsic conduction

There is a temperature dependence of carrier mobility. This temperature dependence is thought to be small compared with the temperature dependence of carrier concentration dependence of exp(-Eg/2kbT). In other words, the temperature dependence of intrinsic conduction is thought to be controlled by the formation of electron-hole pairs rather than the mobility of electrons and holes.

Impurity atoms enter the structure by either substitution of the lattice atoms or by fitting into interstitial sites (e.g. tetrahedral, octahedral, or cubic interstices). In general, atoms which have similar size to the host structure atoms will occupy substitutional sites rather than interstitial sites. We will confine our discussion of dopants in semiconductors to those which occupy substitutional sites.Types of ImpuritiesSince Si and Ge (Group 4 elements) are tetrahedrally coordinated and there are 4 valence electrons on each atom (Figure 1). If an impurity atom from Group 5 (e.g. As, P, Sb) is added as a dopant element, there will be an extra valence electron left unbonded and free to participate in conduction (Figure 2). Alternatively, if an impurity element from Group 3 (e.g. B, Al, Ga, In) is added as a dopant element, there will be an extra hole left over to participate in conduction (Figure 3).

Donors are substitutional atoms which give up an electron to the crystal (e.g. Group 5 elements). Acceptors are substitutional atoms which consume an electron (leaving behind a hole).

Donor State Electron Mobility

The donor electron is attracted by coulomb potential to the impurity ion. This potential is defined by

E=e/r

where is the static dielectric constant of the crystal after doping, r is the distance traversed by the electron in the presence of the electric field, and e is the electronic charge.

The donor ionization energy is given by, Ed=(me4)/(82h2)

A donor level (dashed line) is near the conduction band. Since electrons conduct, these are known as n-type conductors.

As we increase our donor dopant concentration, we have an increase in the fermi energy. From equations which show the exponential dependence of concentration on fermi level, we know that this is a logarithmic increase:

An acceptor level is near to the valence band. Ionization Energy and Bohr Radius of Donor States (pp 222)

Before the electron can participate in conduction, it must first be liberated from the donor (e.g. As in Figure 2). The energy required to ionize the donor impurity can be estimated as

in eV.

Notice that this is equivalent to the ionization energy of atomic hydrogen upon 2.

(Side note on effective mass, me pp 209-214. We note that effective mass is mass after we accounting for effects due to the energy band curvature. We may assume that me=0.2m in Si and me=0.1m in Ge).

The Bohr radius of the donor atom is :

in Angstroms

Significance of the Bohr radius arises when we attempt to understand conduction in doped semiconductors. The Bohr radius must be large enough to result in overlapping of impurity bands (~80Ang for very small effective electron massas in Si me*=0.2m and Ge me*=0.1m). Electron conduction, thus, occurs by movement of electrons from one donor impurity to the next. One final note is that electrons transport from one unoccupied donor site (e.g. As+) to another unoccupied donor site. If an electron already exists at the donor site (e.g. As0), then transport of the next electron will not occur.Acceptor State Electron Mobility (pp 224)

Acceptors will receive electrons from the neighboring atoms, leaving behind holes in the host lattice structure. Ionization energies of acceptors requires the input of energy (because the host lattice must form a hole).

Relative to Intrinsic ConductionIntrinsic conduction occurs in pure, undoped semiconductors. In this case, the number of electrons and holes are equal.

Thermal Ionization of Donors and AcceptorsDonor and acceptors may be ionized by increasing temperature. The equation governing this temperature effect is the same as the equation for thermal ionization of hydrogen,

where

where Nd is the donor concentration.

We may write ,

Nd + (Si/Ge)( Nd+ + e- + (Si/Ge) for the donor reaction with the host lattice

Na + (Si/Ge) ( Na- + h+ + (Si/Ge) for the acceptor reaction with the host lattice

As such, the law of chemical equilibrium suggests that the reaction equilibrium constant, k is

k=[ Nd+] [e-]/[ Nd ] for the donor reaction

k=[ Na-][h+] /[Nd ] for the acceptor reaction.

We may call the electron carrier concentration, n=[e-]. Therefore, n=[ Nd+] satisfies charge neutrality.

Likewise, we may call the hole carrier concentration, p= [h+] = [ Na-]

Recombination Lifetime

By law of mass action, product of n and p yields an expression which, at a given temperature T, is proportional to the rate at which an electron-hole pair will recombine in the reaction e+h=photon (called the recombination reaction). The emitted photon will have an energy equal to that of the band gap,

Eg = hThis quantity is useful if we consider that an incoming photon will generate excess electron-hole pairs above the concentration formed via thermal excitations and doping.

Excess Charge carrier produced by optical excitations:

Extrinsic: Without radiation, we have a balance between thermal excitation and recombination. In the presence of radiation, we increase the number of electron-hole pairs. With optical production, a much greater change is produced in the minority carrier concentration (10^4 holes increase to 10^14 holes increase by a factor of million while 10^16 electrons increase by 10^10 factor of 1 part in 1 million.

Intrinsic: Without radiation, equal number of electron-hole pairs. Increase in the number of electron hole pairs after optical excitation.

In extrinsic cases, after the light is removed, the decrease in carrier concentration depends on excess minority charge carrier (since it was most effected) in an equation of the form: exp(-t/ where is the excess minority carrier lifetime. The minority carrier lifetime depends on dopant concentration and other defects which may effect the recombination reaction.Si

Si

Si

Si

Figure 1. Tetrahedrally Coordinated Silicon

Si

Si

Si

As

Si

Si

Si

Si

+

Conduction e-

Figure 2. Tetrahedrally Coordinated Silicon doped with As (Group 5 element). A conduction electron is left over. The result is an n-type semiconductor.

Si

B

Si

Si

Si

Si

-

+

Conduction hole

Figure 3. Tetrahedrally Coordinated Silicon doped with B (Group 3 element). A conduction hole is left over. The result is a p-type semiconductor.

EMBED Equation.3

Conduction

Heat Input

(T)

Electrons excited from Valence to Conduction Band

{D(), f(), n}

Hole Left behind in the Valence Band (D(h), f(h), p)

Under applied field, , e and h move creating current (e, h)

Conduction Band

Valence Band

Energy Gap

0

Eg

Density of states D() for semiconductors in the vicinity of the energy gap is parabolic at the bottom of the conduction band and the top of the valence band.

Energy Gap

Conduction Band

Eg

Valence Band

0

Density of occupied states is D()f() decreases exponentially to approach zero assymptotically in the conduction band away from the energy gap.

Log Nd or Na

Materials having high dielectric constants will inhibit electron mobility in a given electric field. Therefore, insulating materials are known for having a high dielectric constant. Materials such as SiO2, ZrO2, and Al2O3 have high dielectric constants and are sometimes called dielectric materials. (An interesting sidenote: as the semiconductor industry moves toward smaller and smaller devicesMoores Law, the SiO2 gate dielectrics used to insulate silicon are becoming too thin to inhibit electron tunneling (e.g. current across the dielectric occurs)thereby leading to the study of materials having higher dielectric constants than SiO2 to substitute as gate dielectrics.)

Ef

-

excess electron

________________________

14Semiconductor Crystals

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