dq
e
2
exp 2
d . (5.189)
2
1/ 2 q()
2 q 2 ()
1/2
The last integral74 equals 1/2/2, so that taking into account that for this stationary system centered at the coordinate origin, q() = 0, we finally get a very simple result, Correlation
T
~ ~
K ( ) q( t) q( t ) q( t) q( t ) q (5.190) function:
q
2 q( t) q( t )
e
.
linear
relaxator
As a sanity check, for = 0 it yields Kq(0) q 2 = T/, in accordance with Eq. (185). As is increased the correlation function decreases monotonically – see the solid-line sketch in Fig. 8.
So, the solution of this very simple problem has required straightforward but somewhat bulky calculations. On the other hand, the same result may be obtained literally in one line using the Langevin formalism – namely, as the Fourier transform (59) of the spectral density (68) in the corresponding limit m << , with S () given by Eq. (73a):75
F
T
1
T
cos
T
K ( ) 2 S () cos d 2
cos d 2
d
(5.191)
q
q
e
2
2
( )
0
0
.
2
2
0
This example illustrates the fact that for linear systems (and small fluctuations in nonlinear systems) the Langevin approach is usually much simpler than the one based on the Fokker-Planck or Smoluchowski equations. However, again, the latter approach is indispensable for the analysis of fluctuations of arbitrary intensity in nonlinear systems.
To conclude this chapter, I have to emphasize again that the Fokker-Planck and Smoluchowski equations give a quantitative description of the time evolution of nonlinear Brownian systems with dissipation in the classical limit. The description of the corresponding properties of such dissipative (“open”) and nonlinear quantum systems is more complex,76 and only a few simple problems of their theory have been solved analytically so far,77 typically using a particular model of the environment, e.g., 74 See, e.g., MA Eq. (6.9c).
75 The involved table integral may be found, e.g., in MA Eq. (6.11).
76 See, e.g., QM Sec. 7.6.
77 See, e.g., the solutions of the 1D Kramers problem for quantum systems with low damping by A. Caldeira and A. Leggett, Phys. Rev. Lett. 46, 211 (1981), and with high damping by A. Larkin and Yu. Ovchinnikov, JETP
Lett. 37, 382 (1983).
Chapter 5
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as a large set of harmonic oscillators with different statistical distributions of their parameters, leading to different frequency dependences of the generalized susceptibility ().
5.10. Exercise problems
5.1. Treating the first 30 digits of number = 3.1415… as a statistical ensemble of integers k (equal to 3, 1, 4, 1, 5,…), calculate the average k and the r.m.s. fluctuation k. Compare the results with those for the ensemble of completely random decimal integers 0, 1, 2,..,9, and comment.
5.2. Calculate the variance of fluctuations of a magnetic moment m placed into an external magnetic field H, within the same two models as in Problem 2.4:
(i) a spin-½ with a gyromagnetic ratio , and
(ii) a classical magnetic moment m, of a fixed magnitude m0, but an arbitrary orientation, both in thermal equilibrium at temperature T. Discuss and compare the results.78
Hint: Mind all three Cartesian components of the vector m.
5.3. For a field-free, two-site Ising system with energy values Em = – Js 1 s 2, in thermal equilibrium at temperature T, calculate the variance of energy fluctuations. Explore the low-temperature and high-temperature limits of the result.
5.4. For a uniform, three-site Ising ring with ferromagnetic coupling (and no external field), calculate the correlation coefficients Ks sksk' for both k = k' and k k' .
5.5.* For a field-free 1D Ising system of N >> 1 “spins”, in thermal equilibrium at temperature T, calculate the correlation coefficient Ks slsl+n, where l and ( l + n) are the numbers of two specific spins in the chain.
Hint: You may like to start with the calculation of the statistical sum for an open-ended chain with arbitrary N > 1 and arbitrary coupling coefficients Jk, and then consider its mixed partial derivative over a part of these parameters.
5.6. Within the framework of the Weiss molecular-field theory, calculate the variance of spin fluctuations in the d-dimensional Ising model. Use the result to derive the conditions of its validity.
5.7. Calculate the variance of energy fluctuations in a quantum harmonic oscillator with frequency , in thermal equilibrium at temperature T, and express it via the average value of the energy.
5.8. The spontaneous electromagnetic field inside a closed volume V is in thermal equilibrium at temperature T. Assuming that V is sufficiently large, calculate the variance of fluctuations of the total 78 Note that these two cases may be considered as the non-interacting limits of, respectively, the Ising model (4.23) and the classical limit of the Heisenberg model (4.21), whose analysis within the Weiss approximation was the subject of Problem 4.18.
Chapter 5
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energy of the field, and express the result via its average energy and temperature. How large should the volume V be for your results to be quantitatively valid? Evaluate this limitation for room temperature.
5.9. Express the r.m.s. uncertainty of the occupancy Nk of a certain energy level k by non-interacting:
(i) classical particles,
(ii) fermions, and
(iii) bosons,
in thermodynamic equilibrium, via the level’s average occupancy Nk, and compare the results.
~
5.10. Express the variance of the number of particles, 2
N V, T,, of a single-phase system in
equilibrium, via its isothermal compressibility 1/ V
V
P
.
T
T, N
5.11.* Starting from the Maxwell distribution of velocities, calculate the low-frequency spectral density of fluctuations of the pressure P( t) of an ideal gas of N classical particles, in thermal equilibrium at temperature T, and estimate their variance. Compare the former result with the solution of Problem 3.2.
Hints: You may consider a cylindrically-shaped container of volume
V = LA (see the figure on the right), calculate fluctuations of the force A
F( t)
N , T
exerted by the confined particles on its plane lid of area A, approximating it
F ( t)
as a delta-correlated process (62), and then re-calculate the fluctuations into
those of pressure P F / A.
L
5.12. Calculate the low-frequency spectral density of fluctuations of the electric q
current I( t) due to the random passage of charged particles between two conducting electrodes – see the figure on the right. Assume that the particles are emitted, at random times, by one of the electrodes, and are fully absorbed by the counterpart electrode. Can your result be mapped on some aspect of the electromagnetic blackbody radiation?
Hint: For the current I( t), use the same delta-correlated-process approximation as for the force F( t) in the previous problem.
5.13.79 A very long, uniform string, of mass per unit length, is attached F t
to a firm support, and stretched with a constant force (“tension”) T – see the T
figure on the right. Calculate the spectral density of the random force F( t)
exerted by the string on the support point, within the plane normal to its length,
in thermal equilibrium at temperature T.
Hint: You may assume that the string is so long that a transverse wave, propagating along it from the support point, never comes back.
79 This problem, conceptually important for the quantum mechanics of open systems, was given in Chapter 7 of the QM part of this series, and is repeated here for the benefit of the readers who, by any reason, skipped that course.
Chapter 5
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5.14.80 Each of two 3D harmonic oscillators, with mass m, resonance frequency 0, and damping
> 0, has electric dipole moment d = qs, where s is the vector of oscillator’s displacement from its equilibrium position. Use the Langevin formalism to calculate the average potential of electrostatic interaction of these two oscillators (a particular case of the so-called London dispersion force), separated by distance r >> ( T/ m)1/2/0, in thermal equilibrium at temperature T >> 0. Also, explain why the approach used to solve a very similar Problem 2.15 is not directly applicable to this case.
1
2
Hint: You may like to use the following integral:
2
.
2
2
2
0 1
d
a
4 a
5.15.* Within the van der Pol approximation,81 calculate major statistical properties of fluctuations of classical self-oscillations, at:
(i) the free (“autonomous”) run of the oscillator, and
(ii) their phase been locked by an external sinusoidal force,
assuming that the fluctuations are caused by a weak external noise with a smooth spectral density Sf().
In particular, calculate the self-oscillation linewidth.
5.16. Calculate the correlation function of the coordinate of a 1D harmonic oscillator with small Ohmic damping at thermal equilibrium. Compare the result with that for the autonomous self-oscillator (the subject of the previous problem).
5.17. Consider a very long, uniform, two-wire transmission line (see the
figure on the right) with wave impedance Z, which allows propagation of TEM
I
V
electromagnetic waves with negligible attenuation, in thermal equilibrium at
I
temperature T. Calculate the variance V2 of the voltage V between the wires within a small interval of cyclic frequencies.
Hint: As an E&M reminder,82 in the absence of dispersive materials, TEM waves propagate with a frequency-independent velocity (equal to the speed c of light, if the wires are in free space), with the voltage V and the current I (see Fig. above) related as V ( x,t)/ I( x,t) = Z, where Z is line’s wave impedance.
5.18. Now consider a similar long transmission line but terminated, at one end, with an impedance-matching Ohmic resistor R = Z. Calculate the variance V2 of the voltage across the 80 This problem, for the case of arbitrary temperature, was the subject of QM Problem 7.6, with Problem 5.15 of that course serving as the background. However, the method used in the model solutions of those problems requires one to prescribe, to the oscillators, different frequencies 1 and 2 at first, and only after this more general problem has been solved, pursue the limit 1 2, while neglecting dissipation altogether. The goal of this problem is to show that the result of that solution is valid even at non-zero damping.
81 See, e.g., CM Secs. 5.2-5.5. Note that in quantum mechanics, a similar approach is called the rotating-wave approximation (RWA) – see, e.g., QM Secs. 6.5, 7.6, 9.2, and 9.4.
82 See, e.g., EM Sec. 7.6.
Chapter 5
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resistor, and discuss the relation between the result and the Nyquist formula (81b), including numerical factors.
Hint: A termination with resistance R = Z absorbs incident TEM waves without reflection.
U ( q)
5.19. An overdamped classical 1D particle escapes from a potential well
I
with a smooth bottom, but a sharp top of the barrier – see the figure on the right.
q
Perform the necessary modification of the Kramers formula (139).
0
q
q
q
1
2
5.20. Perhaps the simplest model of the diffusion is the 1D discrete random walk: each time interval , a particle leaps, with equal probability, to any of two adjacent sites of a 1D lattice with spatial period a. Prove that the particle’s displacement during a time interval t >> obeys Eq. (77), and calculate the corresponding diffusion coefficient D.
5.21. A classical particle may occupy any of N similar sites. Its weak interaction with the environment induces random, incoherent jumps from the occupied site to any other site, with the same time-independent rate . Calculate the correlation function and the spectral density of fluctuations of the instant occupancy n( t) (equal to either 1 or 0) of a site.
Chapter 5
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Chapter 6. Elements of Kinetics
This chapter gives a brief introduction to the basic notions of physical kinetics. Its main focus is on the Boltzmann transport equation, especially within the simple relaxation-time approximation (RTA), which allows an approximate but reasonable and simple description of transport phenomena (such as the electric current and thermoelectric effects) in gases, including electron gases in metals and semiconductors.
6.1. The Liouville theorem and the Boltzmann equation
Physical kinetics (not to be confused with “kinematics”!) is the branch of statistical physics that deals with systems out of thermodynamic equilibrium. Major effects addressed by kinetics include: (i) for autonomous systems (those out of external fields): the transient processes ( relaxation), that lead from an arbitrary initial state of a system to its thermodynamic equilibrium;
(ii) for systems in time-dependent (say, sinusoidal) external fields: the field-induced periodic oscillations of the system’s variables; and
(iii) for systems in time-independent (“dc”) external fields: dc transport.
In the last case, we are dealing with stationary (/ t = 0 everywhere), but non-equilibrium situations, in which the effect of an external field, continuously driving the system out of equilibrium, is balanced by the simultaneous relaxation – the trend back to equilibrium. Perhaps the most important effect of this class is the dc current in conductors and semiconductors,1 which alone justifies the inclusion of the basic notions of kinetics into any set of core physics courses.
The reader who has reached this point of the notes already has some taste of physical kinetics, because the subject of the last part of Chapter 5 was the kinetics of a “Brownian particle”, i.e. of a
“heavy” system interacting with an environment consisting of many “lighter” components. Indeed, the equations discussed in that part – whether the Smoluchowski equation (5.122) or the Fokker-Planck equation (5.149) – are valid if the environment is in thermodynamic equilibrium, but the system of our interest is not necessarily so. As a result, we could use those equations to discuss such non-equilibrium phenomena as the Kramers problem of the metastable state’s lifetime.
In contrast, this chapter is devoted to the more traditional subject of kinetics: systems of many similar particles – generally, interacting with each other, but not too strongly, so that the energy of the system still may be partitioned into a sum of single-particle components, with the interparticle interactions considered as a weak perturbation. Actually, we have already started the job of describing such a system at the beginning of Sec. 5.7. Indeed, in the absence of particle interactions (i.e. when it is unimportant whether the particle of our interest is “light” or “heavy”), the probability current densities in the coordinate and momentum spaces are given, respectively, by Eq. (5.142) and the first form of Eq.
(5.143a), so that the continuity equation (5.140) takes the form
w
.
(6.1)
q
wq
p
wp 0
t
1 This topic was briefly addressed in EM Chapter 4, carefully avoiding the aspects related to the thermal effects.
© K. Likharev
SM: Statistical Mechanics
If similar particles do not interact, this equation for the single-particle probability density w(q, p, t) is valid for each of them, and the result of its solution may be used to calculate any ensemble-average characteristic of the system as a whole.
Let us rewrite Eq. (1) in the Cartesian-component form,
w
wq
wp
,
(6.2)
j
j 0
t
j q
p
j
j
where the index j lists all degrees of freedom of the particle under consideration, and assume that its motion (perhaps in an external, time-dependent field) may be described by a Hamiltonian function H ( qj, pj, t). Plugging into Eq. (2) the Hamiltonian equations of motion:2
H
q
,
p H ,
(6.3)
j
j
p
q
j
j
we get
w
H
w
w H 0.
(6.4)
t
j
q
p
p
q
j
j
j
j
After differentiation of both parentheses by parts, the equal mixed terms w2H/ qj pj and w2H/ pj qj cancel, and using Eq. (3) again, we get the so-called Liouville theorem 3
w
w
w
q
p
0
Liouville
.
(6.5)
j
j
t
theorem
j q
p
j
j
Since the left-hand side of this equation is just the full derivative of the probability density w considered as a function of the generalized coordinates qj( t) of a particle, its generalized momenta components pj( t), and (possibly) time t,4 the Liouville theorem (5) may be represented in a surprisingly simple form:
(
dw q, p, t)
0 .
(6.6)
dt
Physically this means that the elementary probability dW = wd 3 qd 3 p to find a Hamiltonian particle in a small volume of the coordinate-momentum space [q, p], with its center moving in accordance to the deterministic law (3), does not change with time – see Fig. 1.
q( t), p( t)
Fig. 6.1. The Liouville
t' t
theorem’s
interpretation:
probability’s conservation
t
at its flow through the [q, p]
d 3 qd 3 p
space.
2 See, e.g., CM Sec. 10.1.
3 Actually, this is just one of several theorems bearing the name of Joseph Liouville (1809-1882).
4 See, e.g., MA Eq. (4.2).
Chapter 6
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At the first glance, this may not look surprising because according to the fundamental Einstein relation (5.78), one needs non-Hamiltonian forces (such as the kinematic friction) to have diffusion. On the other hand, it is striking that the Liouville theorem is valid even for (Hamiltonian) systems with deterministic chaos,5 in which the deterministic trajectories corresponding to slightly different initial conditions become increasingly mixed with time.
For an ideal gas of 3D particles, we may use the ordinary Cartesian coordinates rj (with j = 1, 2, 3) for the generalized coordinates qj, so that pj become the Cartesian components mvj of the usual (linear) momentum, and the elementary volume is just d 3 rd 3 p – see Fig. 1. In this case, Eqs. (3) are just p
j
r
v , p ,
(6.7)
j
j
j
F j
m
where F is the force exerted on the particle, so that the Liouville theorem may be rewritten as 3
w
w
w
v
F
0
,
(6.8)
j
j
t
j
r
1
p
j
j
and conveniently represented in the vector form
w
v w F w 0 .
(6.9)
t
r
p
Of course, the situation becomes much more complex if the particles interact. Generally, a system of N similar particles in 3D space has to be described by the probability density being a function of 6 N + 1 arguments (3 N Cartesian coordinates, plus 3 N momentum components, plus time). An analytical or numerical solution of any equation describing the time evolution of such a function for a typical system of N ~ 1023 particles is evidently a hopeless task. Hence, any theory of realistic systems’
kinetics has to rely on making reasonable approximations that would simplify the situation.
One of the most useful approximations (sometimes called Stosszahlansatz – German for the
“collision-number assumption”) was suggested by Ludwig Boltzmann for gas of particles that move freely most of the time but interact during short time intervals, when a particle comes close to either an immobile scattering center (say, an impurity in a conductor’s crystal lattice) or to another particle of the gas. Such brief scattering events may change the particle’s momentum. Boltzmann argued that they may be still approximately described Eq. (9), with the addition of a special term (called the scattering integral) to its right-hand side:
w
w
Boltzmann
v w F w
.
(6.10)
r
p
scattering
equation
t
t
This is the Boltzmann equation, also called the “Boltzmann transport equation”. As will be discussed below, it may give a very reasonable description of not only classical but also quantum particles, though it evidently neglects the quantum-mechanical coherence/entanglement effects6 – besides those that may be hidden inside the scattering integral.
5 See, e.g., CM Sec. 9.3.
6 Indeed, the quantum state coherence is described by off-diagonal elements of the density matrix, while the classical probability w represents only the diagonal elements of that matrix. However, at least for the ensembles close to thermal equilibrium, this is a reasonable approximation – see the discussion in Sec. 2.1.
Chapter 6
Page 3 of 38
Essential Graduate Physics
SM: Statistical Mechanics
The concrete form of the scattering integral depends on the type of particle scattering. If the scattering centers do not belong to the ensemble under consideration (an example is given, again, by impurity atoms in a conductor), then the scattering integral may be expressed as an evident generalization of the master equation (4.100):
w
3
d p'
w r p ' t
w r p t ,
(6.11)
g
scatteerin
( , , )
( , , )
p ' p
p p
'
t
where the physical sense of pp ’ is the rate (i.e. the probability per unit time) for the particle to be scattered from the state with the momentum p into the state with the momentum p ’ – see Fig. 2.
scattering
center
p '
Fig. 6.2. A single-particle scattering event.
p
Most elastic interactions are reciprocal, i.e. obey the following relation (closely related to the reversibility of time in Hamiltonian systems): pp ’ = p ’p, so that Eq. (11) may be rewritten as7
w
3
d p'
w r p ' t w r
t
g
scatteerin
( , , ) ( , , )
.
(6.12)
t
p p
'
p
With such scattering integral, Eq. (10) stays linear in w but becomes an integro-differential equation, typically harder to solve analytically than differential equations.
The equation becomes even more complex if the scattering is due to the mutual interaction of the particle members of the system – see Fig. 3.
p ' '
interaction
p '
region
p'
Fig. 6.3. A particle-particle scattering event.
p
In this case, the probability of a scattering event scales as a product of two single-particle probabilities, and the simplest reasonable form of the scattering integral is8
7 One may wonder whether this approximation may work for Fermi particles, such as electrons, for whom the Pauli principle forbids scattering into the already occupied state, so that for the scattering p p ’, the term w(r, p, t) in Eq. (12) has to be multiplied by the probability [1 – w(r, p ’, t)] that the final state is available. This is a valid argument, but one should notice that if this modification has been done with both terms of Eq. (12), it becomes w
3
d p'
w r p ' t 1 w r p t w r p t 1 w r ' t g
scatteerin
( , , )
( , , )
( , ,
)
( , , )
.
t
p p
'
p
Opening both square brackets, we see that the probability density products cancel, bringing us back to Eq. (12).
8 This was the approximation used by L. Boltzmann to prove the famous H-theorem, stating that entropy of the gas described by Eq. (13) may only grow (or stay constant) in time, dS/ dt 0. Since the model is very approximate, that result does not seem too fundamental nowadays, despite all its historic significance.
Chapter 6
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SM: Statistical Mechanics
(
w r, p ' , t) (
w r, p ' , t)
p
p
'
' , '
w
3
3
'
p
'
p
d p' d p
.
(6.13)
g
scatteerin
'
t
(
w r, p, t) (
w r, p , t)
pp ' ,
'
'
'
p
'
p
The integration dimensionality in Eq. (13) takes into account the fact that due to the conservation of the total momentum at scattering,
p p p ' p ' ,
(6.14)
'
'
one of the momenta is not an independent argument, so that the integration in Eq. (13) may be restricted to a 6D p-space rather than the 9D one. For the reciprocal interaction, Eq. (13) may also be a bit simplified, but it still keeps Eq. (10) a nonlinear integro-differential transport equation, excluding such powerful solution methods as the Fourier expansion – which hinges on the linear superposition principle.
This is why most useful results based on the Boltzmann transport equation depend on its further simplifications, most notably the relaxation-time approximation – RTA for short.9 This approximation is based on the fact that in the absence of spatial gradients ( = 0), and external forces (F = 0), in at the thermal equilibrium, Eq. (10) yields
w
w
,
(6.15)
scattering
t
t
so that the equilibrium probability distribution w 0(r, p, t) has to turn any scattering integral to zero.
Hence at a small deviation from the equilibrium,
~
w(r, p, t) (
w r, p, t) w (r, p, t) 0 , (6.16)
0
the scattering integral should be proportional to the deviation w
~ , and its simplest reasonable model is
Relaxation-
w
w
~
time
,
(6.17)
approximation
t
g
scatteerin
(RTA)
where is a phenomenological constant (which, according to Eq. (15), has to be positive for the system’s stability) called the relaxation time. Its physical meaning will be more clear in the next section.
The relaxation-time approximation is quite reasonable if the angular distribution of the scattering rate is dominated by small angles between vectors p and p ’ – as it is, for example, for the Rutherford scattering by a Coulomb center.10 Indeed, in this case the two values of the function w, participating in Eq. (12), are close to each other for most scattering events so that the loss of the second momentum argument (p ’) is not too essential. However, using the Boltzmann-RTA equation that results from combining Eqs. (10) and (17),
w
w
~
Boltzmann-
v w F w ,
(6.18)
RTA
t
r
p
equation
we should always remember that this is just a phenomenological model, sometimes giving completely wrong results. For example, it prescribes the same time scale () to the relaxation of the net momentum 9 Sometimes this approximation is called the “BGK model”, after P. Bhatnager, E. Gross, and M. Krook who suggested it in 1954. (The same year, a similar model was considered by P. Welander.)
10 See, e.g., CM Sec. 3.7.
Chapter 6
Page 5 of 38
SM: Statistical Mechanics
of the system, and to its energy relaxation, while in many real systems the latter process (that results from inelastic collisions) may be substantially longer. Naturally, in the following sections, I will describe only those applications of the Boltzmann-RTA equation that give a reasonable description of physical reality.
6.2. The Ohm law and the Drude formula
Despite its shortcomings, Eq. (18) is adequate for quite a few applications. Perhaps the most important of them is deriving the Ohm law for dc current in a “nearly-ideal” gas of charged particles, whose only important deviation from ideality is the rare scattering effects described by Eq. (17). As a result, in equilibrium it is described by the stationary probability w 0 of an ideal gas (see Sec. 3.1): g
w (r, p, t)
N
(6.19)
0
,
2
3
where g is the internal degeneracy factor (say, g = 2 for electrons due to their spin), and N() is the average occupancy of a quantum state with momentum p, that obeys either the Fermi-Dirac or the Bose-Einstein distribution:
1
N
.
(6.20)
exp
,
/ T
p
1
(The following calculations will be valid, up to a point, for both statistics and hence, in the limit / T
–, for a classical gas as well.)
Now let a uniform dc electric field E be applied to the gas of particles with electric charge q, exerting force F = q E on each of them. Then the stationary solution to Eq. (18), with / t = 0, should also be stationary and spatially-uniform ( r = 0), so that this equation is reduced to w
~
q E w .
(6.21)
p
Let us require the electric field to be relatively low, so that the perturbation w
~ it produces is relatively
small, as required by our basic assumption (16).11 Then on the left-hand side of Eq. (21), we can neglect that perturbation, by replacing w with w 0, because that side already has a small factor (E). As a result, this equation yields
w
~
w
q E w
q E
,
(6.22)
p
p 0
0
where the second step implies isotropy of the parameters and T, i.e. their independence of the direction of the particle’s momentum p. But the gradient p is nothing else than the particle’s velocity 11 Since the scale of the fastest change of w 0 in the momentum space is of the order of w 0/ p = ( w 0/)( d/ dp) ~
(1/ T) v, where v is the scale of particle’s speed, the necessary condition of the linear approximation (22) is e E << T/ v, i.e. if e E l << T, where l v has the meaning of the effective mean-free path. Since the left-hand side of the last inequality is just the average energy given to the particle by the electric field between two scattering events, the condition may be interpreted as the smallness of the gas’ “overheating” by the applied field. However, another condition is also necessary – see the last paragraph of this section.
Chapter 6
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v – for a quantum particle, its group velocity.12 (This fact is easy to verify for the isotropic and parabolic dispersion law, pertinent to classical particles moving in free space,
p 2
p 2 p 2 p 2
p
1
2
3
.
(6.23)
2 m
2 m
Indeed, in this case, the j th Cartesian components of the vector p is
p
v ,
(6.24)
p
j
j
j
p
m
j
so that p = v.) Hence, Eq. (22) may be rewritten as
w
~
w
q E
0
v
.
(6.25)
Let us use this result to calculate the electric current density j. The contribution of each particle to the current density is qv so that the total density is
j q wd 3
v
p q
w w~
v
.
(6.26)
0
d 3 p
Since in the equilibrium state (with w = w 0), the current has to be zero, the integral of the first term in the parentheses has to vanish. For the integral of the second term, plugging in Eq. (25), and then using Eq. (19), we get
2
Sommerfeld
w
gq
N
theory’s
2
j q vE v
0
3
d p
v E v
2
result
d p dp
(6.27)
,
2
3
where d 2 p is the elementary area of the constant energy surface in the momentum space, while dpis the momentum differential’s component normal to that surface. The real power of this result13 is that it is valid even for particles with an arbitrary dispersion law (p) (which may be rather complicated, for example, for particles moving in space-periodic potentials14), and gives, in particular, a fair description of conductivity’s anisotropy in crystals.
For free particles whose dispersion law is isotropic and parabolic, as in Eq. (23), the constant energy surface is a sphere of radius p, so that d 2 p = p 2 d = p 2 sin d d, while dp = dp. In the spherical coordinates, with the polar axis directed along the electric field vector E, we get (Ev) = E
v cos. Now separating the vector v outside the parentheses into the component v cos directed along the vector E, and two perpendicular components, v sincos and v sinsin, we see that the integrals of the last two components over the angle give zero. Hence, as we could expect, in the isotropic case the net current is directed along the electric field and obeys the linear Ohm law, Ohm
j
law
E ,
(6.28)
12 See, e.g., QM Sec. 2.1.
13 It was obtained by Arnold Sommerfeld in 1927.
14 See, e.g., QM Secs. 2.7, 2.8, and 3.4. (In this case, p should be understood as the quasimomentum rather than the genuine momentum.)
Chapter 6
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with a field-independent, scalar15 electric conductivity
2
2
gq
N
2
2
2
d
sin d cos θ p dp v
.
(6.29)
2 3
0
0
0
(Note that is proportional to q 2 and hence does not depend on the particle charge sign.16) Since sin d is just – d(cos), the integral over equals (2/3). The integral over d is of course just 2, while that over p may be readily transformed to one over the particle’s energy (p) = p 2/2 m: p 2 =
2 m, v 2 = 2/ m, p = (2 m)1/2, so that dp = ( m/2)1/2 d, and p 2 dpv 2 = (2 m)( m/2)1/2 d (2/ m)
(8 m3)1/2 d. As a result, the conductivity equals
2
gq 4
N
1/ 2
8
m
d .
(6.30)
3
3
2
3
0
Now we may work out the integral in Eq. (30) by parts, first rewriting [- N()/] d as – d[ N()]. Due to the fast (exponential) decay of the factor N() at , its product by the factor (8 m3)1/2 vanishes at both integration limits, and we get
2
gq 4
N d
gq
m
m
N
d
3
2
8
3 1/2
4 8 1/2
3 1/2
2 3
0
2 3 3
2
0
(6.31)
2
3 / 2
q
gm
N 1/ 2
d .
m
2 2 3
0
But according to Eq. (3.40), the last factor in this expression (after the sign) is just the particle density n N/ V, so that the Sommerfeld’s result is reduced, for arbitrary temperature, and any particle statistics, to the very simple Drude formula,17
q
2
n ,
(6.32) Drude
m
formula
which should be well familiar to the reader from an undergraduate physics course.
As a reminder, here is its simple classical derivation.18 Let 2 be the average time interval between two sequential scattering events that cause a particle to lose the deterministic component of its velocity, vdrift, provided by the electric field E on the top of particle’s random thermal motion – which does not contribute to the net current. Using the 2nd Newton law to describe particle’s acceleration by 15 As Eq. (27) shows, if the dispersion law (p) is anisotropic, the current density direction may be different from that of the electric field. In this case, conductivity should be described by a tensor jj’, rather than a scalar.
However, in most important conducting materials, the anisotropy is rather small – see, e.g., EM Table 4.1.
16 This is why to determine the dominating type of charge carriers in semiconductors (electrons or holes, see Sec.
4 below), the Hall effect, which lacks such ambivalence (see, e.g., QM 3.2), is frequently used.
17 It was derived in 1900 by Paul Drude. Note that Drude also used the same arguments to derive a very simple (and very reasonable) approximation for the complex electric conductivity in the ac field of frequency : () =
(0)/(1 – i), with (0) given by Eq. (32); sometimes the name “Drude formula” is used for this expression rather than for Eq. (32). Let me leave its derivation, from the Boltzmann-RTA equation, for the reader’s exercise.
18 See also EM Sec. 4.2.
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the field, dvdrift/ dt = q E/ m, we get vdrift = q E/ m. Multiplying this result by the particle’s charge q and density n N/ V, we get the Ohm law j = E, with given by Eq. (32).
Sommerfeld’s derivation of the Drude formula poses an important conceptual question. The structure of Eq. (30) implies that the only quantum states contributing to the electric conductivity are those whose derivative [- N()/] is significant. For the Fermi particles such as electrons, in the limit T << F, these are the states at the very surface of the Fermi sphere. On the other hand, Eq. (32) and the whole Drude reasoning, involves the density n of all electrons. So, what exactly electrons are responsible for the conductivity: all of them, or only those at the Fermi surface? For the resolution of this paradox, let us return to Eq. (22) and analyze the physical meaning of that result. Let us compare it with the following model distribution:
~
w
w (r, p p, t) ,
(6.33)
model
0
where p~ is some constant, small vector, which describes a small shift of the unperturbed distribution w 0
as a whole, in the momentum space. Performing the Taylor expansion of Eq. (33) in this small parameter, and keeping only two leading terms, we get
~
~
~
w
w (r, p, t) w
,
with w
p w (r, p, t) .
(6.34)
model
0
model
model
p
0
Comparing the last expression with the first form of Eq. (22), we see that they coincide if p~ E
q τ F .
(6.35)
This means that Eq. (22) describes a small shift of the equilibrium distribution of all particles (in the momentum space) by q E along the electric field’s direction, justifying the cartoon shown in Fig. 4.
p
(a)
p
(b)
2
2
F q E
Fig. 6.4. Filling of momentum states by
0
p
p
a degenerate electron gas: (a) in the
1
1
absence and (b) in the presence of an
external electric field E. Arrows show
representative scattering events.
p~ F
At E = 0, the system is in equilibrium, so that the quantum states inside the Fermi sphere ( p < p F), are occupied, while those outside of it are empty – see Fig. 4a. Electron scattering events may happen only between states within a very thin layer ( p 2/2 m – F ~ T) at the Fermi surface, because only in this layer the states are partially occupied, so that both components of the product w(r, p, t)[1 – w(r, p ’, t)], mentioned in Sec. 1, do not vanish. These scattering events, on average, do not change the equilibrium probability distribution, because they are uniformly spread over the Fermi surface.
Now let the electric field be turned on instantly. Immediately it starts accelerating all electrons in its direction, i.e. the whole Fermi sphere starts moving in the momentum space, along the field’s direction in the real space. For elastic scattering events (with p ’ = p ), this creates an addition of occupied states at the leading edge of the accelerating sphere and an addition of free states on its trailing Chapter 6
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edge (Fig. 4b). As a result, now there are more scattering events bringing electrons from the leading edge to the trailing edge of the sphere than in the opposite direction. This creates the average backflow of the state occupancy in the momentum space. These two trends eventually cancel each other, and the Fermi sphere approaches a stationary (though not a thermal-equilibrium!) state, with the shift (35) relatively to its thermal-equilibrium position.
Now Fig. 4b may be used to answer the question of which of the two different interpretations of the Drude formula is correct, and the answer is: either. On one hand, we can look at the electric current as a result of the shift (35) of all electrons in the momentum space. On the other hand, each filled quantum state deep inside the sphere gives exactly the same contribution to the net current density as it did without the field. All these internal contributions to the net current cancel each other so that the applied field changes the situation only at the Fermi surface. Thus it is equally legitimate to say that only the surface states are responsible for the non-zero net current.19
Let me also mention another paradox related to the Drude formula, which is often misunderstood (not only by students :-). As was emphasized above, is finite even at elastic scattering – that by itself does not change the total energy of the gas. The question is how can such scattering be responsible for the Ohmic resistivity 1/, and hence for the Joule heat production, with the power density p = jE =
j 2?20 The answer is that the Drude/Sommerfeld formulas describe just the “bottleneck” of the Joule heat formation. In the scattering picture (Fig. 4b) the states filled by elastically scattered electrons are located above the (shifted) Fermi surface, and these electrons eventually need to relax onto it via some inelastic process, which releases their excessive energy in the form of heat (in solid state, described by phonons – see Sec. 2.6). The rate and other features of these inelastic phenomena do not participate in the Drude formula directly, but for keeping the theory valid (in particular, keeping the probability distribution w close to its equilibrium value w 0), their intensity has to be sufficient to avoid gas overheating by the applied field. In some poorly conducting materials, charge carrier overheating effects, resulting in deviations from the Ohm law, i.e. from the linear relation (28) between j and E, may be observed already at rather practicable electric fields.
One final comment is that the Sommerfeld theory of the Ohmic conductivity works very well for the electron gas in most conductors. The scheme shown in Fig. 4 helps to understand why: for degenerate Fermi gases the energies of all particles whose scattering contributes to transport properties, are close ( F) and prescribing them all the same relaxation time is very reasonable. In contrast, in classical gases, with their relatively broad distribution of , some results given by the Boltzmann-RTA equation (18) are valid only by the order of magnitude.
6.3. Electrochemical potential and the drift-diffusion equation
Now let us generalize our calculation to the case when the particle transport takes place in the presence of a time-independent spatial gradient of the probability distribution, rw 0, caused for example by that of the particle concentration n = N/ V (and hence, according to Eq. (3.40), of the 19 So here, as it frequently happens in physics, formulas (or graphical sketches, such as Fig. 4b) give a more clear and unambiguous description of reality than words – the privilege lacked by many “scientific” disciplines, rich with unending, shallow verbal debates. Note also that, as frequently happens in physics, the dual interpretation of
is expressed by two different but equal integrals (30) and (31), related by the integration-by-parts rule.
20 This formula is probably self-evident, but if you need you may revisit EM Sec. 4.4.
Chapter 6
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chemical potential ), while still assuming that temperature T is constant. For this generalization, we should keep the second term on the left-hand side of Eq. (18). If the gradient of w is sufficiently small, we can repeat the arguments of the last section and replace w with w 0 in this term as well. With the applied electric field E represented as (–),21 where is the electrostatic potential, Eq. (25) becomes
~
w 0
w v
q w
.
(6.36)
0
Since in any of the equilibrium distributions (20), N() is a function of and only in the combination ( – ), it obeys the following relation:
N
N
.
(6.37)
Using this relation, the gradient of w 0 N() may be represented as22
w
0
w
,
for T const ,
(6.38)
0
so that Eq. (36) becomes
w
w
~
0
w
v q
0
v '
,
(6.39)
where the following sum,
Electro-
chemical
'
q ,
(6.40)
potential
is called the electrochemical potential. Now repeating the calculation of the electric current, carried out in the last section, we get the following generalization of the Ohm law (28):
j ' / q E ,
(6.41)
where the effective electric field E is proportional to the gradient of the electrochemical potential, rather of the electrostatic potential:
μ'
Effective
E
E
.
(6.42)
electric
q
q
field
The physics of this extremely important and general result23 may be explained in two ways.
First, let us have a look at the energy spectrum of a degenerate Fermi-gas confined in a volume of finite size, but otherwise free. To ensure such a confinement we need a piecewise-constant potential U(r) – a
“hard-wall, flat-bottom potential well” – see Fig. 5a. (For conduction electrons in a metal, such profile is 21 Since we will not encounter p in the balance of this chapter, from this point on, the subscript r of the operator
r is dropped for the notation brevity.
22 Since we consider w 0 as a function of two independent arguments r and p, taking its gradient, i.e. the differentiation of this function over r, does not involve its differentiation over the kinetic energy – which is a function of p only.
23 Note that Eq. (42) does not include the phenomenological parameter of the relaxation-time approximation, signaling that it is much more general than the RTA. Indeed, this equality is based entirely on the relation between the second and third terms on the left-hand side of the general Boltzmann equation (10), rather than on any details of the scattering integral on its right-hand side.
Chapter 6
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Essential Graduate Physics
SM: Statistical Mechanics
provided by the positively charged ions of the crystal lattice.) The well should be of a sufficient depth U 0 > F T = 0 to provide the confinement of the overwhelming majority of the particles, with energies below and somewhat above the Fermi level F. This means that there should be a substantial energy gap,
U T ,
(6.43)
0
between the Fermi energy of a particle inside the well, and its potential energy U 0 outside the well. (The latter value of energy is usually called the vacuum level. ) The difference defined by Eq. (43) is called the workfunction;24 for most metals, it is between 4 and 5 eV, so that the relation >> T is well fulfilled for room temperatures ( T ~ 0.025 eV) – and actually for all temperatures up to the metal’s evaporation point.
(a)
(b)
(c)
vacuum
d
level
2
1
1
2
U 0
Fig. 6.5. Potential profiles of (a) a single conductor and (b, c) a system of
two closely located conductors, for two different biasing situations: (b) zero
electrostatic field (the “flat-band condition”), and (c) zero voltage ’.
Now let us consider two conductors, with different values of , separated by a small spatial gap d – see Figs. 5b,c. Panel (b) shows the case when the electric field E = – in the free-space gap between the conductors equals zero, i.e. their electrostatic potentials are equal.25 If there is an opportunity for particles to cross the gap (e.g., by either the thermally-activated hopping over the potential barrier, discussed in Secs. 5.6-5.7, or the quantum-mechanical tunneling through it), there will be an average flow of particles from the conductor with the higher Fermi level to that with the lower Fermi level,26 because the chemical equilibrium requires their equality – see Secs. 1.5 and 2.7. If the particles have an electric charge (as electrons do), the equilibrium will be automatically achieved by them recharging the effective capacitor formed by the conductors, until the electrostatic energy difference q reaches the value reproducing that of the workfunctions (Fig. 5c). So for the equilibrium potential difference27 we may write
q
.
(6.44)
At this equilibrium, the electric field in the gap between the conductors is
24 Sometimes it is also called the “electron affinity”, though this term is mostly used for atoms and molecules.
25 In semiconductor physics and engineering, the situation shown in Fig. 5b is called the flat-band condition, because any electric field applied normally to a surface of a semiconductor leads to the so-called energy band bending – see the next section.
26 As measured from a common reference value, for example from the vacuum level – rather than from the bottom of an individual potential well as in Fig. 5a.
27 In physics literature, it is usually called the contact potential difference, while in electrochemistry (for which it is one of the key notions), the term Volta potential is more common.
Chapter 6
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Δ
Δ
E
n
n
;
(6.45)
d
qd
q
in Fig. 5c this field is clearly visible as the tilt of the electric potential profile. Comparing Eq. (45) with the definition (42) of the effective electric field E, we see that the equilibrium, i.e. the absence of current through the potential barrier, is achieved exactly when E = 0, in accordance with Eq. (41).
The electric field dichotomy, E E, raises a natural question: which of these fields we are speaking about in the everyday and laboratory practice? Upon some contemplation, the reader should agree that most of our electric field measurements are done indirectly, by measuring corresponding voltages – with voltmeters. A vast majority of these instruments belong to the so-called electrodynamic variety, which is based on the measurement of a small current flowing through the voltmeter.28 As Eq.
(41) shows, such electrodynamic voltmeters measure the electrochemical potential difference ’/ q.
However, there exists a rare breed of electrostatic voltmeters (also called “electrometers”) that measure the electrostatic potential difference between two conductors. One way to implement such an instrument is to use an ordinary, electrodynamic voltmeter, but with the reference point set at the flat-band condition (Fig. 5b) between the conductors. (This condition may be detected by vanishing electric charge on the adjacent surfaces of the conductors, and hence by the absence of its modulation in time if the distance between the surfaces is periodically modulated.)
Now let me return to Eq. (41) and make two very important remarks. First, it says that in the presence of an electric field, the current vanishes only if ’ = 0, i.e. that the electrochemical potential
’, rather than the chemical potential , has to be position-independent in a system in thermodynamic (thermal, chemical, and electric) equilibrium of a conducting system. This result by no means contradicts the fundamental thermodynamic relations for discussed in Sec. 1.5, or the statistical relations involving , which were discussed in Sec. 2.7 and beyond. Indeed, according to Eq. (40), ’(r) is “merely” the chemical potential referred to the local value of the electrostatic energy q(r), and in all previous parts of the course, this energy was assumed to be constant through the system.
Second, note another interpretation of Eq. (41), which may be achieved by modifying Eq. (38) for the particular case of the classical gas. Indeed, the local density n N/ V of the gas obeys Eq. (3.32), which may be rewritten as
(r)
n(r) const exp
.
(6.46)
T
Taking the spatial gradient of both sides of this relation (still at constant T), we get 1
n
n const exp
,
(6.47)
T
T
T
so that = ( T/ n) n, and Eq. (41), with given by Eq. (32), may be recast as
'
q
2
1
j
n q
nq E T n
.
(6.48)
q
m
q
m
28 The devices for such measurement may be based on the interaction between the measured current and a permanent magnet, as pioneered by A.-M. Ampère in the 1820s – see, e.g., EM Chapter 5. Such devices are sometimes called galvanometers, honoring another pioneer of electricity, Luigi Galvani.
Chapter 6
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Hence the current density may be viewed as consisting of two independent parts: one due to particle drift induced by the “usual” electric field E = –, and another due to their diffusion – see Eq. (5.118) and its discussion. This is exactly the physics of the “mysterious” term in Eq. (42), though its simple form (48) is valid only in the classical limit.
Besides being very useful for applications, Eq. (48) also gives us a pleasant surprise. Namely, plugging it into the continuity equation for electric charge,29
( qn) j 0,
(6.49)
t
we get (after the division of all terms by q/ m) the so-called drift-diffusion equation:30
m n
Drift-
n U T2
n,
U
with
q .
(6.50) diffusion
t
equation
Comparing it with Eq. (5.122), we see that the drift-diffusion equation is identical to the Smoluchowski equation,31 provided that we parallel the ratio / m with the mobility m = 1/ of the Brownian particle.
Now using the Einstein relation (5.78), we see that the effective diffusion constant D of the classical gas of similar particles is
T
D
.
(6.51a)
m
This important relation is more frequently represented in either of two other forms. First, since the rare scattering events we are considering do not change the statistics of the gas in thermal equilibrium, we may still use the Maxwell-distribution result (3.9) for the average-square velocity v 2, to recast Eq. (51a) as
1
2
D
v .
(6.51b)
3
One more popular form of the same relation uses the notion of the mean free path l, which may be defined as the average distance passed by the particle between two sequential scattering events: 1
1/ 2
1/ 2
2
2
D l v
,
with l v
.
(6.51c)
3
In the forms (51b)-(51c), the result for D makes more physical sense, because it may be readily derived (admittedly, with some uncertainty of the numerical coefficient) from simple kinematic arguments – the task left for the reader’s exercise. Note that since the definition of in Eq. (17) is phenomenological, so is the above definition of l; this is why several definitions of this parameter, which may differ by a numerical factor of the order of 1, are possible.
Note also that using Eq. (51a), Eq. (48) may be rewritten as an expression for the particle flow density j n nj w = j/ q:
j nμ q
,
(6.52)
m E D n
n
29 If this relation is not evident, please revisit EM Sec. 4.1.
30 Sometimes this term is associated with Eq. (52). One may also run into the term “convection-diffusion equation” for Eq. (50) with the replacement (51a).
31 And hence, at negligible U, identical to the diffusion equation (5.116).
Chapter 6
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SM: Statistical Mechanics
with the first term on the right-hand side describing particles’ drift, while the second one, their diffusion.
I will discuss the application of this equation to the most important case of non-degenerate (“quasiclassical”) gases of electrons and holes in semiconductors, in the next section.
To complete this section, let me emphasize again that the mathematically similar drift-diffusion equation (50) and the Smoluchowski equation (5.122) describe different physical situations. Indeed, our (or rather Einstein and Smoluchowski’s :-) treatment of the Brownian motion in Chapter 5 was based on a strong hierarchy of the system, consisting of a large “Brownian particle” in an environment of many smaller particles – “molecules”. On the other hand, in this chapter we are considering a gas of similar particles. Nevertheless, the equations describing the dynamics of their probability distribution, are the same – at least within the framework of the Boltzmann transport equation with the relaxation-time approximation (17) of the scattering integral. The origin of this similarity is the fact that Eq. (12) is clearly applicable to a Brownian particle as well, with each “scattering” event being the particle’s hit by a random molecule of its environment. Since, due to the mass hierarchy, the particle momentum change at each such event is very small, the scattering integral has to be local, i.e. depend only on w at the same momentum p as the left-hand side of the Boltzmann equation, so that the relaxation time approximation (17) is absolutely natural – indeed, more natural than for our current case of similar particles.
6.4. Charge carriers in semiconductors
Now let me demonstrate the application of the concepts discussed in the last section to understanding the basic kinetic properties of semiconductors and a few key semiconductor structures –
which are the basis of most modern electronic and optoelectronic devices, and hence of all our IT
civilization. For that, I will need to take a detour to discuss their equilibrium properties first.
I will use an approximate but reasonable picture in which the energy of the electron subsystem in a solid may be partitioned into the sum of effective energies of independent electrons. Quantum mechanics says32 that in such periodic structures as crystals, the stationary state energy of a particle interacting with the atomic lattice follows one of periodic functions n(q) of the quasimomentum q, oscillating between two extreme values nmin and nmax. These allowed energy bands are separated by bandgaps, of widths n nmin – n-1max, with no allowed states inside them. Semiconductors and insulators (dielectrics) are defined as such crystals that in equilibrium at T = 0, all electron states in several energy bands (with the highest of them called the valence band) are completely filled, N(v) =
1, while those in the upper bands, starting from the lowest, conduction band, are completely empty,
N(c) = 0.33, 34 Since the electrons follow the Fermi-Dirac statistics (2.115), this means that at T 0, 32 See, e.g., QM Sec. 2.7 and 3.4, but the thorough knowledge of this material is not necessary for following discussions of this section. If the reader is not familiar with the notion of quasimomentum (alternatively called the
“crystal momentum”), its following semi-quantitative interpretation may be useful: q is the result of quantum averaging of the genuine electron momentum p over the crystal lattice period. In contrast to p, which is not conserved because of the electron’s interaction with the atomic lattice, q is an integral of motion – in the absence of other forces.
33 This mapping of electrical properties of crystals on their band structure was pioneered in 1931-32 by Alan H.
Wilson.
34 In insulators, the bandgap is so large (e.g., ~9 eV in SiO2) that the conduction band remains unpopulated in all practical situations, so that the following discussion is only relevant for semiconductors, with their moderate bandgaps – such as 1.14 eV in the most important case of silicon at room temperature.
Chapter 6
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the Fermi energy F (0) is located somewhere between the valence band’s maximum vmax (usually called simply V), and the conduction band’s minimum cmin (called C) – see Fig. 6.
q
c
C
V
Fig. 6.6. Calculating in an
q
v
intrinsic semiconductor.
q
0
Let us calculate the population of both branches n(q), and the chemical potential in equilibrium at T > 0. Since the functions n(q) are typically smooth, near the bandgap edges the dispersion laws c(q) and v(q) may be well approximated with quadratic parabolas. For our analysis, let us take the parabolas the simplest, isotropic form, with origins at the same quasimomentum, taking it for the reference point:35
2
q / 2 m , for ,
C
C
C
with .
(6.53)
2
C
V
q / 2 m , for ,
V
V
V
The positive constants m C and m V are usually called the effective masses of, respectively, electrons and holes. (In a typical semiconductor, m C is a few times smaller than the free electron mass m e, while m V is closer to m e.)
Due to the similarity between the top line of Eq. (53) and the dispersion law (3.3) of free particles, we may re-use Eq. (3.40), with the appropriate particle mass m, the degeneracy factor g, and the energy origin, to calculate the full spatial density of populated states (in semiconductor physics, called electrons in the narrow sense of the word):
3 / 2
N
g m
e
n
N g
d
N
d
,
(6.54)
3
C
C
~
C
~1/ 2 ~
V
2 2 3
0
C
where ~ – C 0. Similarly, the density p of “no-electron” excitations (called holes) in the valence band is the number of unfilled states in the band, and hence may be calculated as
V
3 / 2
N
g m
h
p
1 N g d
N
d ,
(6.55)
3
V
V
~
1
V
~1/2 ~
V
2 2 3
0
where in this case, ~ 0 is defined as (V – ). If the electrons and holes36 are in the thermal and chemical equilibrium, the functions N() in these two relations should follow the Fermi-Dirac 35 It is easy (and hence is left for the reader’s exercise) to verify that all equilibrium properties of charge carriers remain the same (with some effective values of m C and m V) if c(q) and v(q) are arbitrary quadratic forms of the Cartesian components of the quasimomentum. A mutual displacement of the branches c(q) and v(q) in the quasimomentum space is also unimportant for statistical and most transport properties of the semiconductors, though it is very important for their optical properties – which I will not have time to discuss in any detail.
36 The collective name for them in semiconductor physics is charge carriers – or just “carriers”.
Chapter 6
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SM: Statistical Mechanics
distribution (2.115) with the same temperature T and the same chemical potential . Moreover, in our current case of an undoped ( intrinsic) semiconductor, these densities have to be equal, n p n ,
(6.56)
i
because if this electroneutrality condition was violated, the volume would acquire a non-zero electric charge density = e( p – n), which would result, in a bulk sample, in an extremely high electric field energy. From this condition, we get a system of two equations,
3 / 2
~1/ 2 ~
3 / 2
~1/ 2 ~
g m
d
g m
d
n
C
C
,
(6.57)
i
2
3
V
V
2
~
exp / T
2
3
1
2
~
exp / T1
0
C
0
V
whose solution gives both the requested charge carrier density n i and the Fermi level .
For an arbitrary ratio / T, this solution may be found only numerically, but in most practical cases, this ratio is very large. (Again, for Si at room temperature, 1.14 eV, while T 0.025 eV.) In this case, we may use the same classical approximation as in Eq. (3.45), to reduce Eqs. (54) and (55) to simple expressions
C
V
n n exp
,
(6.58)
C
,
p n exp
V
,
T
for
T
T
where the temperature-dependent parameters
3 / 2
3 / 2
g m T
g m T
C
C
V
V
n
and
n
(6.59)
C
3
V
3
2
2
may be interpreted as the effective numbers of states (per unit volume) available for occupation in, respectively, the conduction and valence bands, in thermal equilibrium. For usual semiconductors (with g C ~ g V ~ 1, and m C ~ m V ~ m e), at room temperature, these numbers are of the order of 31025m-3
31019cm-3. (Note that all results based on Eqs. (58) are only valid if both n and p are much lower than, respectively, n C and n V.)
With the substitution of Eqs. (58), the system of equations (56) allows a straightforward solution:
V
C
T
g V
3
m
V
1/ 2
ln
ln
,
n
.
(6.60)
i
n n
C V
exp
2
2 g
2
m
C
C
T
2
Since in all practical materials the logarithms in the first of these expressions are never much larger than 1,37 it shows that the Fermi level in intrinsic semiconductors never deviates substantially from the so-called midgap value (V +C)/2 – see the (schematic) Fig. 6. In the result for n i, the last (exponential) factor is very small, so that the equilibrium number of charge carriers is much lower than that of the atoms – for the most important case of silicon at room temperature, n i ~ 1010cm-3. The exponential temperature dependence of n i (and hence of the electric conductivity n i) of intrinsic semiconductors is the basis of several applications, for example simple germanium resistance thermometers, efficient in the whole range from ~0.5K to ~100K. Another useful application of the same fact is the extraction of 37 Note that in the case of simple electron spin degeneracy ( g V = g C = 2), the first logarithm vanishes altogether.
However, in many semiconductors, the degeneracy is factored by the number of similar energy bands (e.g., six similar conduction bands in silicon), and the factor ln( g V/ g C) may slightly affect quantitative results.
Chapter 6
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Essential Graduate Physics
SM: Statistical Mechanics
the bandgap of a semiconductor from the experimental measurement of the temperature dependence of
n i – frequently, in just two well-separated temperature points.
However, most applications require a much higher concentration of carriers. It may be increased quite dramatically by planting into a semiconductor a relatively small number of slightly different atoms – either donors (e.g., phosphorus atoms for Si) or acceptors (e.g., boron atoms for Si). Let us analyze the first opportunity, called n-doping, using the same simple energy band model (53). If the donor atom is only slightly different from those in the crystal lattice, it may be easily ionized – giving an additional electron to the conduction band, and hence becoming a positive ion. This means that the effective ground state energy D of the additional electrons is just slightly below the conduction band edge C – see Fig. 7a.38
(a)
(b)
C
Fig. 6.7. The Fermi levels in
D
(a) n-doped and (b) p-doped
semiconductors. Hatching shows
the ranges of unlocalized state
A
V
energies.
Reviewing the arguments that have led us to Eqs. (58), we see that at relatively low doping, when the strong inequalities n << n C and p << n V still hold, these relations are not affected by the doping, so that the concentrations of electrons and holes given by these equalities still obey a universal (doping-independent) relation following from Eqs. (58) and (60):39
2
np n .
(6.61)
i
However, for a doped semiconductor, the electroneutrality condition looks differently from Eq. (56), because the total density of positive charges in a unit volume is not p, but rather ( p + n+), where n+ is the density of positively-ionized (“activated”) donor atoms, so that the electroneutrality condition becomes n p n .
(6.62)
If virtually all dopants are activated, as it is in most practical cases,40 then we may take n+ = n D, where n D is the total concentration of donor atoms, i.e. their number per unit volume, and Eq. (62) becomes n p n .
(6.63)
D
Plugging in the expression p = n 2i/ n, following from Eq. (61), we get a simple quadratic equation for n, with the following physically acceptable (positive) solution:
38 Note that in comparison with Fig. 6, here the (for most purposes, redundant) information on the q-dependence of the energies is collapsed, leaving the horizontal axis of such a band-edge diagram free for showing their possible spatial dependences – see Figs. 8, 10, and 11 below.
39 Very similar relations may be met in the theory of chemical reactions (where it is called the law of mass action), and other disciplines – including such exotic examples as the theoretical ecology.
40 Let me leave it for the reader’s exercise to prove that this assumption is always valid unless the doping density n D becomes comparable to n C, and as a result, the Fermi energy moves into a ~ T-wide vicinity of D.
Chapter 6
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SM: Statistical Mechanics
1/ 2
2
n
n
D
D
2
n
n
.
(6.64)
i
2
4
This result shows that the doping affects n (and hence =
2
C – T ln( n C/ n) and p = n i / n) only if the dopant concentration n D is comparable with, or higher than the intrinsic carrier density n i given by Eq.
(60). For most applications, n D is made much higher than n i; in this case Eq. (64) yields 2
2
n
n
n
i
i
C
n n n ,
p
,
n
T ln
.
(6.65)
D
i
p
C
n
n
n
D
D
Because of the reasons to be discussed very soon, modern electron devices require doping densities above 1018cm-3, so that the logarithm in Eq. (65) is not much larger than 1. This means that the Fermi level rises from the midgap to a position only slightly below the conduction band edge C – see Fig. 7a.
The opposite case of purely p-doping, with n A acceptor atoms per unit volume, and a small activation (negative ionization) energy A – V << ,41 may be considered absolutely similarly, using the electroneutrality condition in the form
n n p ,
(6.66)
where n– is the number of activated (and hence negatively charged) acceptors. For the relatively high concentration ( n i << n A << n V), virtually all acceptors are activated, so that n– n A, Eq. (66) may be approximated as n + n A = p, and the analysis gives the results dual to Eq. (65): 2
2
n
n
n
i
i
V
p n n ,
n
p,
T ln
,
(6.67)
A
i
n
V
p
n
n
A
A
so that in this case, the Fermi level is just slightly above the valence band edge (Fig. 7b), and the number of holes far exceeds that of electrons – again, in the narrow sense of the word. Let me leave the analysis of the simultaneous n- and p-doping (which enables, in particular, so-called compensated semiconductors with the sign-variable difference n – p n D – n A) for the reader’s exercise.
Now let us consider how a sample of a doped semiconductor (say, a p-doped one) responds to a static external electrostatic field E applied normally to its surface.42 (In semiconductor integrated circuits, such field is usually created by the voltage applied to a special highly-conducting gate electrode separated from the semiconductor surface by a thin insulating layer.) Assuming that the field penetrates into the sample by a distance much larger than the crystal lattice period a (the assumption to be verified a posteriori), we may calculate the distribution of the electrostatic potential using the macroscopic version of the Poisson equation.43 Assuming that the semiconductor occupies the semi-space x > 0 and that E = n x E, the equation reduces to the following 1D form44
41 For the typical donors (P) and acceptors (B) in silicon, both ionization energies, (C – D) and (A – V), are close to 45 meV, i.e. are indeed much smaller than 1.14 eV.
42 A simplified version of this analysis was discussed in EM Sec. 2.1.
43 See, e.g., EM Sec. 3.4.
44 I am sorry for using, for the SI electric constant 0, the same Greek letter as for single-particle energies, but both notations are traditional, and the difference between these uses will be clear from the context.
Chapter 6
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Essential Graduate Physics
SM: Statistical Mechanics
2
d
x
.
(6.68)
2
dx
0
Here is the dielectric constant of the semiconductor matrix – excluding the dopants and charge carriers, which in this approach are treated as explicit (“stand-alone”) charges, with the volumic density
e p n n
.
(6.69)
(As a sanity check, Eqs. (68)-(69) show that if E – d/ dx = 0, then = 0, bringing us back to the electroneutrality condition (66), and hence the “flat” band-edge diagrams shown in Figs. 7b and 8a.) (a)
(b)
(c)
n ,
0 p 0
C
n 0
p n A
en 0
A
p n
p n
0
A
0
e
A
A
,
0
0
0
0
0
0
0
e 0
0
'
w
const
'
'
const
A
V
x x w
0
x 0
x 0
x ~
x x
D
x 0
0
Fig. 6.8. The band-edge diagrams of the electric field penetration into a uniform p-doped semiconductor: (a) E = 0, (b) E < 0, and (c) E > Ec > 0. Solid red points depict positive charges; solid blue points, negative charges; and hatched blue points, possible electrons in the inversion layer – all very schematically.
In order to get a closed system of equations for the case E 0, we should take into account that the electrostatic potential 0, penetrating into the sample with the field,45 adds the potential component q( x) = – e( x) to the energy of each electron, and hence shifts the whole local system of single-electron energy levels “vertically” by this amount – down for > 0, and up for < 0. As a result, the field penetration leads to what is called band bending – see the band-edge diagrams schematically shown in Figs. 8b,c for two possible polarities of the applied field, which affects the distribution ( x) via the boundary condition46
d
0 E
.
(6.70)
dx
Note that the electrochemical potential ’ (which, in accordance with the discussion in Sec. 3, replaces the chemical potential in presence of the electric field),47 has to stay constant through the system in equilibrium, keeping the electric current equal to zero – see Eq. (41). For arbitrary doping parameters, the system of equations (58) (with the replacements V V – e, and ’) and (68)-(70), plus the 45 It is common (though not necessary) to select the energy reference so that deep inside the semiconductor, = 0; in what follows I will use this convention.
46 Here E is the field just inside the semiconductor. The free-space field necessary to create it is times larger –
see, e.g., the same EM Sec. 3.4, in particular Eq. (3.56).
47 In semiconductor physics literature, the value of ’ is usually called the Fermi level, even in the absence of the degenerate Fermi sea typical for metals – cf. Sec. 3.3. In this section, I will follow this common terminology.
Chapter 6
Page 20 of 38
