Classical and Quantum Statistics - 1 | Physics for IIT JAM, UGC - NET, CSIR NET PDF Download

Introduction

Classical Maxwell–Boltzmann statistics and quantum mechanical Fermi–Dirac statistics are introduced to calculate the occupancy of states. Special attention is given to analytic approximations of the Fermi–Dirac integral and to its approximate solutions in the nondegenerate and the highly degenerate regime. In addition, some numerical approximations to the Fermi–Dirac integral are summarized.

Semiconductor statistics includes both classical statistics and quantum statistics. Classical or Maxwell–Boltzmann statistics is derived on the basis of purely classical physics arguments. In contrast, quantum statistics takes into account two results of quantum mechanics, namely (i) the Pauli exclusion principle which limits the number of electrons occupying a state of energy E and (ii) the finiteness of the number of states in an energy interval E and E + dE. The finiteness of states is a result of the Schrödinger equation. In this section, the basic concepts of classical statistics and quantum statistics are derived. The fundamentals of ideal gases and statistical distributions are summarized as well since they are the basis of semiconductor statistics. 

Probability and distribution functions 

Consider a large number N of free classical particles such as atoms, molecules or electrons which are kept at a constant temperature T, and which interact only weakly with one another. The energy of a single particle consists of kinetic energy due to translatory motion and an internal energy for example due to rotations, vibrations, or orbital motions of the particle. In the following we consider particles with only kinetic energy due to translatory motion. The particles of the system can assume an energy E, where E can be either a discrete or a continuous variable. If N_i particles out of N particles have an energy between E_i and E_i + dE, the probability of any particle having any energy within the interval E_i and E_i + dE is given by 

 (13.1)

where f(E) is the energy distribution function of a particle system. In statistics, f(E) is frequently called the probability density function. The total number of particles is given by 

     (13.2)

where the sum is over all possible energy intervals. Thus, the integral over the energy distribution function is 

  (13.3)

In other words, the probability of any particle having an energy between zero and infinity is unity. Distribution functions which obey 

  (13.4)

are called normalized distribution functions. The average energy or mean energy  of a single particle is obtained by calculating the total energy and dividing by the number of particles, that is

(13.5) 
In addition to energy distribution functions, velocity distribution functions are valuable. Since only the kinetic translatory motion (no rotational motion) is considered, the velocity and energy are related by 

(13.6) 
The average velocity and the average energy are related by 

   (13.7)

  (13.8)

and is the velocity corresponding to the average energy 

    (13.9)  

In analogy to the energy distribution we assume that N_i particles have a velocity within the interval v_i and v_i + d_v. Thus, 

   (13.10)  

where f(v) is the normalized velocity distribution. Knowing f(v), the following relations allow one to calculate the mean velocity, the mean square velocity, and the root-mean-square velocity

  (13.11)  

  (13.12)  

  (13.13)  

Up to now we have considered the velocity as a scalar. A more specific description of the velocity distribution is obtained by considering each component of the velocity v = (v_x, v_y, v_z). If N_i particles out of N particles have a velocity in the ‘volume’ element v_x + dv_x, v_y + dv_y, and v_z + dv_z, the distribution function is given by 

 (13.14)  

Since Σ_i N_i  =  N, the velocity distribution function is normalized, i. e.

  (13.15)

The average of a specific propagation direction, for example v_x is evaluated in analogy to Eqs. (13.11 – 13). One obtains 

 (13.16)  

  (13.17)

In a closed system the mean velocities are zero, that is  . However, the mean square velocities are, just as the energy, not equal to zero.

  Ideal gases of atoms and electrons

The basis of classical semiconductor statistics is ideal gas theory. It is therefore necessary to make a small excursion into this theory. The individual particles in such ideal gases are assumed to interact weakly, that is collisions between atoms or molecules are a relatively seldom event. It is further assumed that there is no interaction between the particles of the gas (such as electrostatic interaction), unless the particles collide. The collisions are assumed to be (i) elastic (i. e. total energy and momentum of the two particles involved in a collision are preserved) and (ii) of very short duration.

Ideal gases follow the universal gas equation  

P V   =   R T (13.19) 
where P is the pressure, V the volume of the gas, T its temperature, and R is the universal gas constant. This constant is independent of the species of the gas particles and has a value of R = 8.314 J K^–1 mol^–1. 

Next, the pressure P and the kinetic energy of an individual particle of the gas will be calculated. For the calculation it is assumed that the gas is confined to a cube of volume V, as shown in Fig. 13.1. The quantity of the gas is assumed to be 1 mole, that is the number of atoms or molecules is given by Avogadro’s number, N_Avo = 6.023 × 10²³ particles per mole. Each side of the cube is assumed to have an area A = V ^2/3. If a particle of mass m and momentum m v_x (along the x-direction) is elastically reflected from the wall, it provides a momentum 2 m v_x  to reverse the particle momentum. If the duration of the collision with the wall is dt, then the force acting on the wall during the time dt is given by 

(13.20) 
where the momentum change is dp = 2 m v_x. The pressure P on the wall during the collision with one particle is given by 

(13.21) 
where A is the area of the cube’s walls. Next we calculate the total pressure P experienced by the wall if a number of N_Avo particles are within the volume V. For this purpose we first determine the number of collisions with the wall during the time dt. If the particles have a velocity v_x, then the number of particles hitting the wall during dt is (N_Avo / V) A v_x dt. The fraction of particles 

having a velocity v_x is obtained from the velocity distribution function and is given by. Consequently, the total pressure is obtained by integration over all positive velocities in the x-direction

Since the velocity distribution is symmetric with respect to positive and negative x-direction, the integration can be expanded from – ∞ to + ∞. 

Since the velocity distribution is isotropic, the mean square velocity is given by 

The pressure on the wall is then given by 

  (13.25)  

Using the universal gas equation, Eq. (13.19), one obtains 

 (13.26)  

The average kinetic energy of one mole of the ideal gas can then be written as 

  (13.27)  

The average kinetic energy of one single particle is obtained by division by the number of particles, i. e.

  (13.28)

where k = R / N_Avo is the Boltzmann constant. The preceding calculation has been carried out for a three-dimensional space. In a one-dimensional space (one degree of freedom), the average velocity is  and the resulting kinetic energy is given by

   (per degree of freedom) . (13.29)

Thus the kinetic energy of an atom or molecule is given by (1/2) kT. Equation (13.29) is called the equipartition law, which states that each ‘degree of freedom’ contributes (1/2) kT to the total kinetic energy.

Maxwell velocity distribution

The Maxwell velocity distribution describes the distribution of velocities of the particles of an ideal gas. It will be shown that the Maxwell velocity distribution is of the form 

 (13.30)

 where (1/2) m v² is the kinetic energy of the particles. If the energy of the particles is purely kinetic, the Maxwell distribution can be written as 

(13.31) The proof of the Maxwell distribution of Eq. (13.30) is conveniently done in two steps. In the first step, the exponential factor is demonstrated, i. e. f_M(E) = A exp (– α E). In the second step it is shown that α = 1 / (kT). .

In the theory of ideal gases it is assumed that collisions between particles are elastic. The total energy of two electrons before and after a collision remains the same, that is  

E₁ + E₂ = E₁^′+ E₂^′ (13.32)  
where E₁ and E₂ are the electron energies before the collision and E₁^′ and E₂^′ are the energies after the collision. The probability of a collision of an electron with energy E₁ and of an electron with energy E₂ is proportional to the probability that there is an electron of energy E₁ and a second electron with energy E₂. If the probability of such a collision is p, then 

  (13.33)

 where B is a constant. The same consideration is valid for particles with energies E₁^′ and E₂^′. Thus, the probability that two electrons with energies E₁^′ and E₂^′ collide is given by 

(13.34) If the change in energy before and after the collision is ∆E, then ∆E = E₁^′ – E₁ and ∆E = E₂ – E₂^′. Furthermore, if the electron gas is in equilibrium, then p = p′ and one obtains 

    (13.35)  

Only the exponential function satisfies this condition, that is 

  (13.36)  

where α is a positive yet undetermined constant. The exponent is chosen negative to assure that the occupation probability decreases with higher energies. It will become obvious that α is a universal constant and applies to all carrier systems such as electron-, heavy- or light-hole systems.

Next, the constant α will be determined. It will be shown that α = 1 / kT using the results of the ideal gas theory. The energy of an electron in an ideal gas is given by 

  (13.37)

The exponential energy distribution of Eq. ( 13.36) and the normalization condition of Eq. (13.15) yield the normalized velocity distribution 

  (13.38)  

The average energy of an electron is obtained by (first) calculating the mean square velocities,   from the distribution and (second) using Eq. (13.37) to calculate E from the mean square velocities. One obtains

E = (3 / 2) α ⁻¹ . (13.39)  

We now use the result from classic gas theory which states according to Eq. (13.28) that the kinetic energy equals E = (3/2) kT. Comparison with Eq. (13.39) yields

  (13.40)

 which concludes the proof of the Maxwell distribution of Eqs. (13.30) and (13.31).

Having determined the value of α, the explicit form of the normalized maxwellian velocity distribution in cartesian coordinates is 

Due to the spherical symmetry of the maxwellian velocity distribution, it is useful to express the distribution in spherical coordinates. For the coordinate transformation we note that , and that a volume element  in spherical coordinates. The maxwellian velocity distribution in spherical coordinates is then given by

The maxwellian velocity distribution is shown in Fig. 13.2. The peak of the distribution, that is the most likely velocity, is . The mean velocity is given by . The root-mean-square velocity can only be obtained by numerical integration.

The Boltzmann factor 

The maxwellian velocity distribution can be changed to an energy distribution by using the substitution E = (1/2) m v². Noting that the energy interval and the velocity interval are related by dE = m v dv and that the number of electrons in the velocity interval, f_M(v) dv, is the same as the number of electrons in the energy interval, f_MB(E) dE, then the energy distribution is given by 

(13.43)  
which is the Maxwell–Boltzmann distribution.

For large energies, the exponential term in the Maxwell–Boltzmann distribution essentially determines the energy dependence. Therefore, the high-energy approximation of the Maxwell– Boltzmann distribution is 

 (13.44)  

which is the Boltzmann distribution. The exponential factor of the distribution, exp (– E / kT), is called the Boltzmann factor or Boltzmann tail. The Boltzmann distribution does not take into account the quantum mechanical properties of an electron gas. The applicability of the distribution is therefore limited to the classical regime, i. e. for E >> kT. 

The Fermi–Dirac distribution

In contrast to classical Boltzmann statistics, the quantum mechanical characteristics of an electron gas are taken into account in Fermi–Dirac statistics. The quantum properties which are explicitly taken into account are

• The wave character of electrons. Due to the wave character of electrons the Schrödinger equation has only a finite number of solutions in the energy interval E and E + dE.

• The Pauli principle which states that an eigenstate can be occupied by only two electrons of opposite spin. 

Since the Pauli principle strongly restricts the number of carriers per energy level, higher states are populated even at zero temperature. This situation is illustrated in Fig. 13.3, where two electron distributions are illustrated at zero temperature. The distribution in Fig. 13.3(a) does not take into account the Pauli principle while that in Fig. 13.3(b) does. 

The first restriction imposed by quantum mechanics is the finiteness of states within an energy interval E and E + dE. Recall that the finiteness of states played a role in the derivation of the density of states. The density of states in an isotropic semiconductor was shown to be

   (13.45)

where E is the kinetic energy. Note that for the derivation of the density of states the Pauli principle has been taken into account. Therefore, the states given by Eq. (13.45) can be occupied only by one electron. Since the number of states per velocity-interval will be of interest, Eq. (13.45) is modified using E = (1/2) mv² and dE = mv dv. Note that the number of states per energy interval dE is the same as the number of states per velocity interval dv, i. e. ρ_DOS(E) dE = ρ_DOS(v) dv. The number of states per velocity interval (and per unit volume) is then given by 

(13.46)  
for an isotropic semiconductor. The Fermi–Dirac distribution, also called the Fermi distribution, gives the probability that a state of energy E is occupied. Since the Pauli principle has been taken into account in the density of states given by Eq. (13.45), each state can be occupied by at most one electron. The Fermi distribution is given by 

  (13.47)  

where E_F is called the Fermi energy. At E = E_F the Fermi distribution has a value of 1/2. For small energies the Fermi distribution approaches 1; thus low-energy states are very likely to be populated by electrons. For high energies the Fermi distribution decreases exponentially; states of high energy are less likely to the populated. Particles which follow a Fermi distribution are called fermions. Electrons and holes in semiconductors are such fermions. A system of particles which obey Fermi statistics are called a Fermi gas. Electrons and holes constitute such Fermi gases. 

 An approximate formula for the Fermi distribution can be obtained for high energies. One obtains for E >> E_F

  (13.48)

This distribution coincides with the Boltzmann distribution. Thus the (quantum-mechanical) Fermi distribution and the (classical) Boltzmann distribution coincide for high energies, i. e. in the classical regime.

Next we prove the Fermi distribution of Eq. (13.47) by considering a collision between two electrons. For simplification we assume that one of the electrons has such a high energy that it belongs to the classical regime of semiconductor statistics. Quantum statistics applies to the other low-energy electron. During the collision of the two electrons, the energy is conserved 

(13.49)  
where, as before (Eq. 13.32), E₁ and E₂ are electron energies before the collision and E₁^′ and E₂^′ are the energies after the collision. The probability for the transition (E₁, E₂) → (E₁^′, E₂^′) is given by

 (13.50)  

where it is assumed that E₂ and E₂^′ are relatively large energies and the corresponding electron can be properly described by the Boltzmann distribution. The terms [ 1 – f_F(E₁^′) ] and [ 1 – f_B(E₂^′) ] describe the probability that the states of energies E₁^′ and E₂^′ are empty, and are available for the electron after the collision. Further simplification is obtained by considering that E₂^′ is large and therefore . Equation (13.50) then simplifies to

  (13.51)  

The same considerations are valid for the transition . The probability of this transition is given by

  (13.52)  

Under equilibrium conditions both transition probabilities are the same, i. e. p = p^′ . Equating Eqs. (13.51) and (13.52), inserting the Boltzmann distribution for f_B(E), and dividing by  f_F(E₁)  yields

(13.53)  
which must hold for all E₁ and E₁^′. This condition requires that 

 (13.54)

where A is a constant. If the value of the constant is taken to be A = exp (– E_F / kT) one obtains the Fermi–Dirac distribution

(13.54)  
which proves Eq. (13.47). 

The Fermi–Dirac distribution is shown for different temperatures in Fig. 13.4. At the energy E = E_F the probability of a state being populated has always a value of ½ independent of temperature. At higher temperatures, states of higher energies become populated. Note that the Fermi–Dirac distribution is symmetric with respect to E_F, that is 

 (13.55)  

where ∆E is any energy measured with respect to the Fermi energy.

The Fermi–Dirac velocity distribution of the particles in a Fermi gas is obtained by multiplication of Eq. (13.46) with Eq. (13.47) 

 
where we have used the fact that the energy of the Fermi gas is purely kinetic, i. e. E = (1/2) mv². Note that g(v) is the number of carriers per velocity interval v and v + dv and per unit volume. If the velocity v is expressed in terms of its components, then the spherical volume element, 4π v² dv, is modified to a volume element in rectangular coordinates, dv_x dv_y dv_z. Thus, using g(v) dv = g(v_x, v_y, v_z) dv_x dv_y dv_z, one obtains 

which is the Fermi velocity distribution (per unit volume) in cartesian coordinates. The Fermi distribution of energies of an ideal gas is obtained by multiplication of Eq. (13.45) with Eq. (13.47) and is given by 

when g(E) is the number of particles in the energy interval E and E + dE and per unit volume.