MSE-630 Week 3

Conductivity, Energy Bands and Charge Carriers in Semiconductors

Objectives:• To understand conduction, valence energy

bands and how bandgaps are formed• To understand the effects of doping in

semiconductors• To use Fermi-Dirac statistics to calculate

conductivity and carrier concentrations• To understand carrier mobility and how it is

influenced by scattering• To introduce the idea of “effective mass”• To see how we can use Hall effect to determine

carrier concentration and mobility

ConductivityCharge carriers follow a random path unless an external field is applied. Then, they acquire a drift velocity that is dependent upon their mobility, n and the strength of the field,

Vd = -n

The average drift velocity, vav is dependentUpon the mean time between collisions, 2

Charge Flow and Current DensityCurrent density, J, is the rate at which charges, cross any plane perpendicular to the flow direction.

J = -nqvd = nqn

n is the number of charges, and

q is the charge (1.6 x 10-19

C)

OHM’s Law: V = IR

Resistance, R() is an extrinsic quantity. Resistivity, (m), is the corresponding intrinsic property.

= R*A/l

Conductivity, , is the reciprocal of resistivity: (m)-1 = 1/

The total current density depends upon the total charge carriers, which can be ions, electrons, or holes

J = q(nn + pp)

When we add carriers by doping, the number of additional carrers, Nd, far exceeds those in an intrinsic semiconductor, and we can treat

conductivity as = qNdd

In general, np=ni2, where ni is the intrinsic concentration of carriers at a

given temperature (ni = 1.5 x 1010 cm-3 in Si) In a doped semiconductor, charges balance, thus Na+n = Nd+p. For an n-type semiconductor, n ~ Nd, and p ~ ni

2/Nd

As the distance between atoms decreases, the energy of each orbital must split, since according to Quantum Mechanics we cannot have two orbitals with the same energy.

The splitting results in “bands” of electrons. The energy difference between the conduction and valence bands is the “gap energy” We must supply this much energy to elevate an electron from the valence band to the conduction band. If Eg is < 2eV, the material is a semiconductor.

In the ground state, at 0K, all the electrons have energy less than Ef, the Fermi energy. If we add energy we can boost an electron into the conduction band. If we add dopants, we can enhance the number of positive or negative carriers, vastly increasing conductivity

Adding a column 5 atom to Si (e.g., As) adds a 5th, loosely bound electron. The ionization energy is very low, and it can be boosted to the conduction band easily. This is an electron donor

If we add a material that is short one electron (e.g., B, valence 3) we will create an electron “sink” in the material. This results in “holes” that migrate in the valence band. Holes carry positive charge.

In an intrinsic semiconductor, elevating an electron to the conduction band leaves behind a hole in the valence band, or creates an “electron-hole pair”, or EHP

Fermi-Dirac statistics• Conductivity in semiconductors is

described by Fermi-Dirac statistics. In an intrinsic semiconductor, the conductivity is:

kTEg

oe 2

For an extrinsic semiconductor, corresponding probability of an electron is

kTE

oe

Where E = Ec-Ed for an n-type, and E = Ea-Ev for a p-type

At low temperatures, the conduction is provided by the dopants, and the slope is defined by E/k. When the available electrons/holes is depleted in the “exhaustion” range, the slope is constant. When the temperature becomes high enough, energies exceed Eg, and the semiconductor moves into the intrinsic conduction range, with a slope of Eg/2k.

Effective MassIn general, the curve of Energy vs. k is non-

linear, with E increasing as k increases. E = ½ mv2 = ½ p2/m = h2/4m k2

We can see that energy varies inversely with mass. Differentiating E wrt k twice, and solving for mass gives:

2

2

2*

2dkEd

hm

Effective mass is significant because it affects charge carrier mobility, and must be considered when calculating carrier concentrations or momentum

Effective mass and other semiconductor properties may be found in Appendix III

In reality, band structures are highly dependent upon crystal orientation. This image shows us that the lowest band gap in Si occurs along the [100] directions, whil for GaAs, it occurs in the [111]. This is why crystals are grown with specific orientations.

The diagram showing the constant energy surface (3.10 (b)), shows us that the effective mass varies with direction. We can calculate average effective mass from:

tln mmm21

311

*

Hall EffectWhen a magnet field is appplied

perpendicular to the direction in which a charged particle (electron or hole) is moving, the particle will be deflected as shown

The force on the particle will be F = q(+vXB)

In the x-direction, the force will be Fy = q(y+vxXBz)

To counter the flow of particles in the x-direction, we apply a field y=vxBz so that the net force is zero. The applied field is called the Hall effect, and the resulting voltage, VH=yd

Drift velocity for an electron in the x-direction is:<vy>=-Jy/qn

where J is the current density, n is the number of carriers and q is the charge

Defining the Hall coefficient, RH=1/qn, then y = vxBz = - Jx/qn Bz = RHJxBz

and

AB

zx

AB

zx

y

zx

H qtVI

wVqwt

I

qBJ

qRn

/

1

Measuring the resistance gives the resistivity:

(m)= Rwt/L = (Vcd/Ix)/(L/wt)

Since conductivity, = 1/ = qnn, the mobility is:

n = /qn = (1/)/q(1/qRH) = RH/