Part 2 : Nanostuctures Module 3 : Electron Transport

Lecture 1 : Boltzmann Transport Equation (BTE)

Drude modelThe simplest model for charge carrier transport in metals and semiconductors is the Drude model. Inthe presence of an applied electric field , the charge current density current is

(1)

where is the density of the carriers of charge each, and is the average carrier drift velocity. Thedirection of is from (+)ve to (-)ve electrode; for electrons, , where is the magnitude of the

electronic charge, and the direction of is opposite to the latter being along .

If one neglects the thermal motion of the charge carriers, then the equation of motion of an averagecarrier of mass and charge is

(2)

The second term above represents a friction, which the carriers experience while drifting through thesample. The friction arises from scattering of the carriers by various sources, such as other carriers,impurities, defects, the vibration of the lattice etc. The main source of friction is usually the vibratingions/atoms in the lattice, which depends on the lattice temperature, and this is taken care of by atemperature dependent , the average momentum relaxation time. The name relaxation meansreturn to equilibrium; and it follows from the fact that, if the electric field is switched off at , theequation of motion for has the solution

(3)

which shows that the drift velocity relaxes exponentially to zero; is usually too short forobserving this directly.

In the steady state, the first term in Eqn.(2) vanishes, and the drift velocity is proportional to :

(4)

where the proportionality constant , called mobility, is then defined by

(5)

This leads to

(6)

where is the conductivity defined

(7)

This formula is known as Drude-conductivity formula. In semiconductors, is replaced by theeffective mass of the carrier. Clearly, the friction term leads to Ohm's Law, i.e., is independentof the electric field strength.In elementary derivations of Drude formula found in introductory text books of Solid State Physics, oneuses the concept of collision time . As the carriers drift through the solid, they occasionally collide, forinstance, with the vibrating atoms/ions in the lattice, and one would anticipate the existence of a meanfree time between the collisions or scattering. The average time interval between successive collisions isthe collision time , which is of the order of .

Note that is the resistivity (with unit cm = Volt cm/Amp) so that the mobility has the

unit of cm2/(Volt sec). To get an idea about the order of magnitude for , note that = 1.76

1015cm2/Volt and taking sec (inverse of a typical lattice vibrational frequency), one

gets 176 cm2/(Volt sec). In -type Germanium (Ge), , and

sec at room temperature, so that for this material at room temperature is 1760 cm2/(Volt sec),which agrees in order of magnitude with the observed mobility of 3900 cm2/(Volt sec).

Average momentum relaxation time:

In general, the momentum relaxation time depends on energy, and one may write

(8)

where is the constant of proportionality having the dimension of time, , with

denoting the Boltzmann constant and the absolute temperature; denotes the energy of the carrier.The power depends on the nature of scattering (for example, for acoustic deformation potentialscattering in semiconductors, , while for ionized impurity scattering, ), and so

does (for example, for acoustic deformation potential scattering, if the mean-free-path is , then, where the root mean square velocity in -dimension, and

itself depends on as , so that ; for ionized impurity scattering .

The fact that is in general energy dependent, requires a reformulation of the problem. One knowsthat there is a velocity distribution for the carriers. The drift for an applied electric field along -direction should be

(9)

Similarly, one might write,

(10)

which, as will be seen later, however is not the quantity that should enter the expression for themobility. The equation for the distribution function is

(11)

where the right hand side arises from collisions or scattering of the carriers, which drive the system toequilibrium once the external field is withdrawn. This is the so called Boltzmann Transport Equation(BTE) in the semi-classical form.

In general the distribution function depends on the position coordinate , the velocity

, and the time , so that using , , ,, and , the term df/dt can be

written as

(12)

The first four terms on the right hand side are zero for dc field, and if there is no temperature or carrierconcentration gradient in the sample. In addition, if , then the Boltzmann equation (11)above takes the form

(13)

Since is a function of velocity, it is natural to write

(14)

where is the velocity distribution of the carriers in thermal equilibrium in the absence of the

electric field . This then gives the simplest form of BTE in the relaxation time approximation, as

(15)

which leads to

(16)

Note that this is a nonlinear equation for . However, for small field ( ) one may retain terms

linear in only, , in , replace by , so that

(17)

In above, denotes the change in the distribution function to linear or first order in the appliedelectric field.

Using this linearized form described by Eqn.(14) in Eqn.(9), one gets

(18)

For a uniform isotropic sample (usually assumed), , and in that case, the termcontaining in the numerator vanishes (because this term is odd function of ) on integration,and one gets to linear order in the electric field,

(19)

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