SM: Statistical Mechanics
relation between n– and n A (describing the acceptor activation), does not allow an analytical solution.
However, as was discussed above, in the most practical cases n A >> n i, we may use the approximate relations n– n A and n 0 at virtually any values of ’ within the locally shifted bandgap [V – e( x), C
– e( x)], so that the substitution of these relations, and the second of Eqs. (58), with the mentioned replacements, into Eq. (69) yields
e '
n
'
e
V
V
V
en exp
en
en
.
(6.71)
V
A
A
exp
exp
1
T
n
T
T
A
The x-independent electrochemical potential (a.k.a. Fermi level) ’ in this relation should be equal to the value of the chemical potential ( x ) in the semiconductor’s bulk, given by the last of Eqs. (67), which turns the expression in the parentheses into 1. With these substitutions, Eq. (68) becomes d 2
en
A
e
exp
1 ,
for
.
(6.72)
2
e
V
x ' e
C
x
dx
0
T
This nonlinear differential equation may be solved analytically, but in order to avoid a distraction by this (rather bulky) solution, let me first consider the case when the electrostatic potential is sufficiently small – either because the external field is small, or because we focus on the distances sufficiently far from the surface – see Fig. 8 again. In this case, in the Taylor expansion of the exponent in Eq. (72), with respect to small , we may keep only two leading terms, turning it into a linear equation:
1/ 2
2
2
2
d
e n
d
T
A
,
i.e.
,
where
0
,
(6.73)
2
2
2
D
2
dx
T
dx
e n
0
D
A
with the well-known exponential solution, satisfying also the boundary condition 0 at x :
x
C exp
,
e
at T .
(6.74)
D
The constant D given by the last of Eqs. (73) is called the Debye screening length. It may be rather substantial; for example, at T K = 300K, even for the relatively high doping, n A 1018cm-3 typical for modern silicon ( 12) integrated circuits, it is close to 4 nm – still much larger than the crystal lattice constant a ~ 0.3 nm, so that the above analysis is indeed quantitatively valid. Note also that D
does not depend on the charge’s sign; hence it should be no large surprise that repeating our analysis for an n-doped semiconductor, we may find out that Eqs. (73)-(74) are valid for that case as well, with the only replacement n A n D.
If the applied field E is weak, Eq. (74) is valid in the whole sample, and the constant C in it may be readily calculated using the boundary condition (70), giving
1/ 2
T
C
.
(6.75)
x0
E
0
D
E
2
e n A
This formula allows us to express the condition of validity of the linear approximation leading to Eq.
(74), e << T, in terms of the applied field:
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1/ 2
T
Tn
E E
,
with
A
E
;
(6.76)
max
max
e
D
0
in the above example, Emax ~ 60 kV/cm. On the lab scale, such field is not low at all (it is twice higher than the threshold of electric breakdown in the air at ambient conditions), but may be sustained by many solid-state materials that are much less prone to the breakdown.48 This is why we should be interested in what happens if the applied field is higher than this value.
The semi-quantitative answer is relatively simple if the field is directed out of the p-doped semiconductor (in our nomenclature, E < 0 – see Fig. 8b). As the valence band bends up by a few T, the local hole concentration p( x), and hence the charge density ( x), grow exponentially – see Eq. (71).
Hence the effective local length of the nonlinear field’s penetration, ef( x) -1/2( x), shrinks exponentially. A detailed analysis of this effect using Eq. (72) does not make much sense, because as soon as ef(0) decreases to ~ a, the macroscopic Poisson equation (68) is no longer valid quantitatively.
For typical semiconductors, this happens at the field that raises the edge V – e(0) of the bent valence band at the sample’s surface above the Fermi level ’. In this case, the valence-band electrons near the surface form a degenerate Fermi gas, with an “open” Fermi surface – essentially a metal, which a very small (atomic-size) Thomas-Fermi screening length:49
1/ 2
0 ~
.
(6.77)
ef
0
TF
2
e g
3 F
The effects taking place at the opposite polarity of the field, E > 0, are much more interesting –
and more useful for applications. Indeed, in this case, the band bending down leads to an exponential decrease of ( x) as soon as the valence band edge V – e( x) drops down by just a few T below its unperturbed value V. If the applied field is large enough, E > Emax (as it is in the situation shown in Fig.
8c), it forms, on the left of such point x 0 the so-called depletion layer, of a certain width w. Within this layer, not only the electron density n, but the hole density p as well, are negligible, so that the only substantial contribution to the charge density is given by the fully ionized acceptors: – en– – en A, and Eq. (72) becomes very simple:
2
d
en
A
const,
for x w x x .
(6.78)
2
0
0
dx
0
Let us use this equation to calculate the largest possible width w of the depletion layer, and the critical value, Ec, of the applied field necessary for this. (By definition, at E = Ec, the left boundary of the layer, where V – e( x) = C, i.e. e( x) = V – A , just touches the semiconductor surface: x 0 – w = 0, i.e. x 0 = w. (Figure 8c shows the case when E is slightly larger than Ec.) For this, Eq. (78) has to be solved with the following boundary conditions:
d
d
0 ,
0 E
w
w .
(6.79)
c ,
,0
0
e
dx
dx
48 Even some amorphous thin-film insulators, such as properly grown silicon and aluminum oxides, can withstand fields up to ~10 MV/cm.
49 As a reminder, the derivation of this formula was the task of Problem 3.14.
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Note that the first of these conditions is strictly valid only if T << , i.e. at the assumption we have made from the very beginning, while the last two conditions are asymptotically correct only if D << w – the assumption we should not forget to check after the solution.
After all the undergraduate experience with projective motion problems, the reader certainly knows by heart that the solution of Eq. (78) is a quadratic parabola, so that let me immediately write its final form satisfying the boundary conditions (79):
1/ 2
en
w x 2
2
2
x
A
,
w
0
with
, at E
.
(6.80)
2
e 2 n
c
eε w
0
A
0
Comparing the result for w with Eq. (73), we see that if our basic condition T << is fulfilled, then D
<< w, confirming the qualitative validity of the whole solution (80). For the same particular parameters as in the example before ( n A 1018cm-3, 10), and 1 eV, Eqs. (80) give w 40 nm and Ec 600
kV/cm – still a practicable field. (As Fig. 8c shows, to create it, we need a gate voltage only slightly larger than / e, i.e. close to 1 V for typical semiconductors.)
Figure 8c also shows that if the applied field exceeds this critical value, near the surface of the semiconductor the conduction band edge drops below the Fermi level. This is the so-called inversion layer, in which electrons with energies below ’ form a highly conductive degenerate Fermi gas.
However, typical rates of electron tunneling from the bulk through the depletion layer are very low, so that after the inversion layer has been created (say, by the gate voltage application), it may be only populated from another source – hence the hatched blue points in Fig. 8c. This is exactly the fact used in the workhorse device of semiconductor integrated circuits – the field-effect transistor (FET) – see Fig.
9.50
gate
(a)
gate
(b)
insulator
source
drain
insulators
"fin"
n
n
p
p
Fig. 6.9. Two main species of the n-FET: (a) the bulk FET, and (b) the FinFET. While on panel (a), the current flow from the source to the drain is parallel to the plane of the drawing, on panel (b) it is normal to the plane, with the n-doped source and drain contacting the thin “fin” from two sides off this plane.
In the “bulk” variety of this structure (Fig. 9a), a gate electrode overlaps a gap between two similar highly- n-doped regions near the surface, called source and drain, formed by n-doping inside a p-
doped semiconductor. It is more or less obvious (and will be shown in a moment) that in the absence of gate voltage, the electrons cannot pass through the p-doped region, so that virtually no current flows between the source and the drain, even if a modest voltage is applied between these electrodes.
However, if the gate voltage is positive and large enough to induce the electric field E > Ec at the surface of the p-doped semiconductor, it creates the inversion layer as shown in Fig. 8c, and the electron current 50 This device was invented (by Julius E. Lilienfeld) in 1930, but demonstrated only in the mid-1950s.
Chapter 6
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SM: Statistical Mechanics
between the source and drain electrodes may readily flow through this surface channel. (Very unfortunately, in this course I would not have time/space for a detailed analysis of transport properties of this keystone electron device, and have to refer the reader to special literature.51) Fig. 9a makes it obvious that another major (and virtually unavoidable) structure of semiconductor integrated circuits is the famous p-n junction – an interface between p- and n-doped regions. Let us analyze its simple model, in which the interface is in the plane x = 0, and the doping profiles n D( x) and n A( x) are step-like, making an abrupt jump at the interface:
n const, at x ,
0
x
A
,
0
at ,
0
n x
n x
(6.81)
A
D
,
0
at x ,
0
n const, at x 0.
D
(This model is very reasonable for modern integrated circuits, where the doping in performed by implantation, using high-energy ion beams.)
To start with, let us assume that no voltage is applied between the p- and n-regions, so that the system may be in thermodynamic equilibrium. In the equilibrium, the Fermi level ’ should be flat through the structure, and at x – and x +, where 0, the level structure has to approach the positions shown, respectively, on panels (a) and (b) of Fig. 7. In addition, the distribution of the electric potential ( x), shifting the level structure vertically by – e( x), has to be continuous to avoid unphysical infinite electric fields. With that, we inevitably arrive at the band-edge diagram that is (schematically) shown in Fig. 10.
n - doping
e
C
x
w
e
n
'
const
Fig. 6.10. The band-edge diagram of a
p-n
junction
in
thermodynamic
wp
e
V
x
equilibrium ( T = const, ’ = const). The
p - doping
notation is the same as in Figs. 7 and 8.
x 0
The diagram shows that the contact of differently doped semiconductors gives rise to a built-in electric potential difference , equal to the difference of their values of in the absence of the contact
– see Eqs. (65) and (67):
n n
e e e
C V
T ln
,
(6.82)
n
p
n n
D A
which is usually just slightly smaller than the bandgap.52 (Qualitatively, this is the same contact potential difference that was discussed, for the case of metals, in Sec. 3 – see Fig. 5.) The arising 51 The classical monograph in this field is S. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley 1981. (The 3rd edition, circa 2006, co-authored with K. Ng, is more tilted toward technical details.) I can also recommend a detailed textbook by R. Pierret, Semiconductor Device Fundamentals, 2nd ed., Addison Wesley, 1996.
Chapter 6
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SM: Statistical Mechanics
internal electrostatic field E = – d/ dx induces, in both semiconductors, depletion layers similar to that induced by an external field (Fig. 8c). Their widths wp and wn may also be calculated similarly, by solving the following boundary problem of electrostatics, mostly similar to that given by Eqs. (78)-(79): 2
d
e
n ,
for w x ,
0
A
p
(6.83)
2
dx
n
x
w
0
D,
0
for
,
n
d
d
d
d
w
, (6.84)
n
wp ,
wn wp 0, 0 0,
0 0
dx
dx
dx
dx
also exact only in the limit << , n i << n D, n A. Its (easy) solution gives the result similar to Eq. (80):
en w
x
w
x
A
p
2 /2 ,
for
,
0
const
0
p
(6.85)
en w
x
x
w
D
n
2 / 2 ,
0
for
,
0
n
with expressions for wp and wn giving the following formula for the full depletion layer width: 1/ 2
2
n n
1
1
1
0
A D
w w w
,
with n
,
i.e.
.
(6.86)
p
n
ef
en
n n
n
n
n
ef
A
D
ef
A
D
This expression is similar to that given by Eq. (80), so that for typical highly doped semiconductors ( n ef ~1018cm-3) it gives for w a similar estimate of a few tens nm.53 Returning to Fig. 9a, we see that this scale imposes an essential limit on the reduction of bulk FETs (whose scaling down is at the heart of the well-known Moore’s law),54 explaining why such high doping is necessary. In the early 2010s, the problems with implementing even higher doping, plus issues with dissipated power management, have motivated the transition of advanced silicon integrated circuit technology from the bulk FETs to the FinFET (also called “double-gate”, or “tri-gate”, or “wrap-around-gate”) variety of these devices, schematically shown in Fig. 9b, despite their essentially 3D structure and hence a more complex fabrication technology. In the FinFETs, the role of p-n junctions is reduced, but these structures remain an important feature of semiconductor integrated circuits.
Now let us have a look at the p-n junction in equilibrium from the point of view of Eq. (52). In the simple model we are considering now (in particular, at T << ), this equation is applicable separately to the electron and hole subsystems, because in this model the gases of these charge carriers are classical in all parts of the system, and the generation-recombination processes55 coupling these subsystems have relatively small rates – see below. Hence, for the electron subsystem, we may rewrite Eq. (52) as 52 Frequently, Eq. (82) is also rewritten in the form e = T ln( n
2
D n A/ n i ). In the view of the second of Eqs. (60),
this equality is formally correct but may be misleading because the intrinsic carrier density n i is an exponential function of temperature and is physically irrelevant for this particular problem.
53 Note that such w is again much larger than D – the fact that justifies the first two boundary conditions (84).
54 Another important limit is quantum-mechanical tunneling through the gate insulator, whose thickness has to be scaled down in parallel with lateral dimensions of a FET, including its channel length.
55 In the semiconductor physics lingo, the “carrier generation” event is the thermal excitation of an electron from the valence band to the conduction band, leaving a hole behind, while the reciprocal event of filling such a hole by a conduction-band electron is called the “carrier recombination”.
Chapter 6
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Essential Graduate Physics
SM: Statistical Mechanics
n
j n q
,
(6.87)
m E D
n
n
x
where q = – e. Let us discuss how each term of the right-hand of this equality depends on the system’s parameters. Because of the n-doping at x > 0, there are many more electrons in this part of the system.
According to the Boltzmann distribution (58), some number of them,
e
n exp
,
(6.88)
T
have energies above the conduction band edge in the p-doped part (see Fig. 11a) and try to diffuse into this part through the depletion layer; this diffusion flow of electrons from the n-side to the p-side of the structure (in Fig. 11, from the right to the left) is described by the second term on the right-hand side of Eq. (87). On the other hand, the intrinsic electric field E = –/ x inside the depletion layer, directed as Fig. 11a shows, exerting on the electrons the force F = q E – e E pushing them in the opposite direction (from p to n), is described by the first, “drift” term on the right-hand side of Eq. (87).56
(a)
(b)
p
n
p
n
F
E
e
j n
e V
e
j e
'
x
E
V
e
'
const
w
w
w
w
p
n
p
n
Fig. 6.11. Electrons in the conduction band of a p-n junction at: (a) V = 0, and (b) V > 0.
For clarity, other charges (of the holes and all ionized dopant atoms) are not shown.
The explicit calculation of these two flows57 shows, unsurprisingly, that in the equilibrium, they are exactly equal and opposite, so that jn = 0, and such analysis does not give us any new information.
However, the picture of two electron counter-flows, given by Eq. (87), enables us to predict the functional dependence of jn on a modest external voltage V, with V < , applied to the junction.
Indeed, since the doped semiconductor regions outside the depletion layer are much more conductive 56 Note that if an external photon with energy > generates an electron-hole pair somewhere inside the depletion layer, this electric field immediately drives its electron component to the right, and the hole component to the left, thus generating a pulse of electric current through the junction. This is the physical basis of the whole vast technological field of photovoltaics, currently strongly driven by the demand for renewable electric power.
Due to the progress of this technology, the cost of solar power systems has dropped from ~$300 per watt in the mid-1950s to the current ~$1 per watt, and its global generation has increased to almost 1015 watt-hours per year –
though this is still below 2% of the whole generated electric power.
57 I will not try to reproduce this calculation (which may be found in any of the semiconductor physics books mentioned above), because getting all its scaling factors right requires using some model of the recombination process, and in this course, there is just no time for their quantitative discussion. However, see Eq. (93) below.
Chapter 6
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SM: Statistical Mechanics
than it, virtually all applied voltage (i.e. the difference of values of the electrochemical potential ’) drops across this layer, changing the total band edge shift – see Fig. 11b:58
e
e μ'
e q V e V .
(6.89)
This change results in an exponential change of the number of electrons able to diffuse into the p-side of the junction – cf. Eq. (88):
e V
n V
n
(6.90)
0 exp
,
T
and hence in a proportional change of the diffusion flow jn of electrons from the n-side to the p-side of the system, i.e. of the oppositely directed density of the electron current je = – ejn – see Fig. 11b.
On the other hand, the drift counter-flow of electrons is not altered too much by the applied voltage: though it does change the electrostatic field E = – inside the depletion layer, and also the depletion layer width,59 these changes are incremental, not exponential. As the result, the net density of the current carried by electrons may be approximately expressed as
e V
j
(6.91a)
e V j
j
je
diffusion
0
drift
exp
const.
T
As was discussed above, at V = 0, the net current has to vanish, so that the constant in Eq. (91a) has to equal je(0), and we may rewrite this equality as
e V
j
(6.91b)
e V je 0 exp
1 .
T
Now repeating this analysis for the current jh of the holes (the exercise highly recommended to the reader), we get a similar expression, with the same sign before e V,60 though with a different scaling factor, jh(0) instead of je(0). As a result, the total electric current density obeys the famous Shockley law
e V
jV j V j V j
j
j
j
,
(6.92)
e
h 0 exp
1 , with 0
e 0
h 0
T
describing the main p-n junction’s property as an electric diode – a two-terminal device passing the current more “readily” in one direction (from the p- to the n-terminal) than in the opposite one.61
58 In our model, the positive sign of V ’/ q – ’/ e corresponds to the additional electric field, – ’/ q
’/ e, directed in the positive direction of the x-axis (in Fig. 11, from the left to the right), i.e. to the positive terminal of the voltage source connected to the p-doped semiconductor – which is the common convention.
59 This change, schematically shown in Fig. 11b, may be readily calculated by making the replacement (89) in the first of Eqs. (86).
60 This sign invariance may look strange, due to the opposite (positive) electric charge of the holes. However, this difference in the charge sign is compensated by the opposite direction of the hole diffusion – see Fig. 10. (Note also that the actual charge carriers in the valence band are still electrons, and the positive charge of holes is just a convenient representation of the specific dispersion law in this energy band, with a negative effective mass – see Fig. 6, the second line of Eq. (53), and a more detailed discussion of this issue in QM Sec. 2.8.) Chapter 6
Page 27 of 38
SM: Statistical Mechanics
Besides numerous practical applications in electrical and electronic engineering, such diodes have very interesting statistical properties, in particular performing very non-trivial transformations of the spectra of deterministic and random signals. Very unfortunately, I would not have time for their discussion and have to refer the interested reader to the special literature.62
Still, before proceeding to our next (and last!) topic, let me give for the reader reference, without proof, the expression for the scaling factor j(0) in Eq. (92), which follows from a simple, but broadly used model of the recombination process:
D
D
j0
2
en
e
h
.
(6.93)
i
l n
l n
e A
h D
Here le and lh are the characteristic lengths of diffusion of electrons and holes before their recombination, which may be expressed by Eq. (5.113), le = (2 De e)1/2 and lh = (2 Dh h)1/2, with e and h being the characteristic times of recombination of the so-called minority carriers – of electrons in the p-
doped part, and of holes in the n-doped part of the structure. Since the recombination is an inelastic process, its times are typically rather long – of the order of 10-7s, i.e. much longer than the typical times of elastic scattering of the same carriers, that define their diffusion coefficients – see Eq. (51).
6.5. Heat transfer and thermoelectric effects
Now let us return to our analysis of kinetic effects using the Boltzmann-RTA equation, and extend it even further, to the effects of a non-zero (albeit small) temperature gradient. Again, since for any of the statistics (20), the average occupancy N() is a function of just one combination of all its arguments, ( – )/ T, its partial derivatives obey not only Eq. (37), but also the following relation:
N
N
N
.
(6.94)
T
T 2
T
As a result, Eq. (38) is generalized as
w 0
w
,
(6.95)
0
T
T
giving the following generalization of Eq. (39):
w
~
0
w
v '
T .
(6.96)
T
Now, calculating current density as in Sec. 3, we get the result that is traditionally represented as
'
j
S T
,
(6.97)
q
61 Some metal-semiconductor junctions, called Schottky diodes, have similar rectifying properties (and may be better fitted for high-power applications than silicon p-n junctions), but their properties are more complex because of the rather involved chemistry and physics of interfaces between different materials.
62 See, e.g., the monograph by R. Stratonovich cited in Sec. 4.2.
Chapter 6
Page 28 of 38
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where the constant S, called the Seebeck coefficient 63 (or the “thermoelectric power”, or just
“thermopower”) is given by the following relation:
Seebeck
gq
4
1/ 2
3
( ) N
coefficient
S
.
(6.98)
3
8
m
d
2
3
T
0
Working out this integral for the most important case of a degenerate Fermi gas, with T << F, we have to be careful because the center of the sharp peak of the last factor under the integral coincides with the zero point of the previous factor, ( – )/ T. This uncertainty may be resolved using the Sommerfeld expansion formula (3.59). Indeed, for a smooth function f() obeying Eq. (3.60), so that f(0)
= 0, we may use Eq. (3.61) to rewrite Eq. (3.59) as
N
2 2 2
T d f
f
( )
d f ()
.
(6.99)
2
6
d
0
In particular, for working out the integral (98), we may take f() (8 m3)1/2( – )/ T. (For this function, the condition f(0) = 0 is evidently satisfied.) Then f() = 0, d 2 f/ d2= = 3(8 m)1/2/ T 3(8 mF)1/2/ T, and Eq. (98) yields
gq
2
4 T 2
3 8 m F 1/2
S
.
(6.100)
23 3
6
T
Comparing the result with Eqs. (3.54) and (32), for the constant S we get a simple expression independent of :64
2
T
c
V
S
,
for T ,
(6.101)
F
2 q
q
F
where cV CV/ N is the heat capacity of the gas per unit particle, in this case given by Eq. (3.70).
In order to understand the physical meaning of the Seebeck coefficient, it is sufficient to consider a conductor carrying no current. For this case, Eq. (97) yields
Seebeck
effect
'
/ q ST 0.
(6.102)
So, at these conditions, a temperature gradient creates a proportional gradient of the electrochemical potential ’, and hence the effective electric field E defined by Eq. (42). This is the Seebeck effect.
Figure 12 shows the standard way of its measurement, using an ordinary (electrodynamic) voltmeter that measures the difference of ’/ e at its terminals, and a pair of junctions (in this context, called the thermocouple) of two materials with different coefficients S.
63 Named after Thomas Johann Seebeck who experimentally discovered, in 1822, the effect described by the second term in Eq. (97) – and hence by Eq. (103).
64 Again, such independence hints that Eq. (101) has a broader validity than in our simple model of an isotropic gas. This is indeed the case: this result turns out to be valid for any form of the Fermi surface, and for any dispersion law (p). Note, however, that all calculations of this section are valid for the simplest RTA model in that is an energy-independent parameter; for real metals, a more accurate description of experimental results may be obtained by tweaking this model to take this dependence into account – see, e.g., Chapter 13 in the monograph by N. Ashcroft and N. D. Mermin, cited in Sec. 3.5.
Chapter 6
Page 29 of 38
SM: Statistical Mechanics
T"
S 1
S 2
"
A
V ( T' T" )( S
)
1
S 2
B
A
'
A
Fig. 6.12. The Seebeck effect in a thermocouple.
T'
Integrating Eq. (102) around the loop from point A to point B, and neglecting the temperature drop across the voltmeter, we get the following simple expression for the thermally-induced difference of the electrochemical potential, usually called either the thermoelectric power or the “thermo e.m.f.”: μ'
μ'
1 B
B
A "
A '
B
B
A
V
'
dr S T dr S T dr S
T dr T d
r
1
2
q
q
q
A
A
A '
A
A"
(6.103)
S T" T' S T' T" S S T' T"
1
2
(
) (
) .
1
2
(Note that according to Eq. (103), any attempt to measure such voltage across any two points of a uniform conductor would give results depending on the voltmeter wire materials, due to an unintentional gradient of temperature in them.)
Using thermocouples is a very popular, inexpensive method of temperature measurement –
especially in the few-hundred-C range where gas- and fluid-based thermometers are not too practicable, if a 1C-scale accuracy is sufficient. The temperature responsivity ( S 1 – S 2) of a typical popular thermocouple, chromel-constantan,65 is about 70 V/C. To understand why the typical values of S are so small, let us discuss the Seebeck effect’s physics. Superficially, it is very simple: particles, heated by an external source, diffuse from it toward the colder parts of the conductor, carrying electrical current with them if they are electrically charged. However, this naïve argument neglects the fact that at j = 0, there is no total flow of particles. For a more accurate interpretation, note that inside the integral (98), the Seebeck effect is described by the factor ( – )/ T, which changes its sign at the Fermi surface, i.e. at the same energy where the term [- N()/], describing the availability of quantum states for transport (due to their intermediate occupancy 0 < N() < 1), reaches its peak. The only reason why that integral does not vanish completely, and hence S 0, is the growth of the first factor under the integral (which describes the density of available quantum states on the energy scale) with , so the hotter particles (with > ) are more numerous and hence carry more heat than the colder ones.
The Seebeck effect is not the only result of a temperature gradient; the same diffusion of particles also causes the less subtle effect of heat flow from the region of higher T to that with lower T, i.e. the effect of thermal conductivity, well known from our everyday practice. The density of this flow (i.e. that of thermal energy) may be calculated similarly to that of the electric current – see Eq. (26), with the natural replacement of the electric charge q of each particle with its thermal energy ( – ): 65 Both these materials are alloys, i.e. solid solutions: chromel is 10% chromium in 90% nickel, while constantan is 45% nickel and 55% copper.
Chapter 6
Page 30 of 38
SM: Statistical Mechanics
j v
.
(6.104)
h
wd 3 p
(Indeed, we may look at this expression is as at the difference between the total energy flow density, j =
v wd 3 p, and the product of the average energy needed to add a particle to the system () by the particle flow density, j n = v wd 3 p j/ q.)66 Again, at equilibrium ( w = w 0) the heat flow vanishes, so that w in Eq.
(104) may be replaced with its perturbation w
~ , which already has been calculated – see Eq. (96). The
substitution of that expression into Eq. (104), and its transformation exactly similar to the one performed above for the electric current j, yields
'
j Π
,
(6.105)
h
T
q
with the coefficients and given, in our approximation, by the following formulas:
Peltier
gq
4
1/ 2
N
coefficient
8
,
(6.106)
3
m 3
d
2
3
0
Thermal
g
4
1/ 2 (
)
3
2 N
conductivity
8
.
(6.107)
3
m
d
2
3
T
0
Besides the missing factor T in the denominator, the integral in Eq. (106) is the same as the one in Eq. (98), so that the constant (called the Peltier coefficient67), is simply and fundamentally related to the Seebeck coefficient:
vs.
S
ST .
(6.108)
The simplicity of this relation (first discovered experimentally in 1854 by W. Thompson, a.k.a. Lord Kelvin) is not occasional. This is one of the so-called Onsager reciprocal relations between kinetic coefficients (suggested by L. Onsager in 1931), which are model-independent, i.e. valid within very general assumptions. Unfortunately, I have no time/space left for a discussion of this interesting topic (closely related to the fluctuation-dissipation theorem discussed in Sec. 5.5), and have to refer the interested reader to its detailed discussions available in the literature.68
On the other hand, the integral in Eq. (107) is different, but may be readily calculated, for the most important case of a degenerate Fermi gas, using the Sommerfeld expansion in the form (99), with f() (8 m3)1/2( – )2/ T, for which f() = 0 and d 2 f/ d2
3
= = 2(8 m3)1/2/ T 2(8 mF )1/2/ T, so that
g
4 2 2
m
8
(
2
3)1/2
F
2 n T
.
(6.109)
2
T
3
3 6
T
3 m
66 An alternative explanation of the factor ( – ) in Eq. (104) is that according to Eqs. (1.37) and (1.56), for a uniform system of N particles this factor is just ( E – G)/ N ( TS – PV)/ N. The full differential of the numerator is TdS + SdT –PdV – VdP, so that in the absence of the mechanical work d W = – PdV, and changes of temperature and pressure, it is just TdS dQ – see Eq. (1.19).
67 Named after Jean Charles Athanase Peltier who experimentally discovered, in 1834, the effect expressed by the first term in Eq. (105) – and hence by Eq. (112).
68 See, for example, Sec. 15.7 in R. Pathria and P. Beale, Statistical Mechanics, 3rd ed., Elsevier, 2011. Note, however, that the range of validity of the Onsager relations is still debated – see, e.g., K.-T. Chen and P. Lee, Phys. Rev. B 79, 18 (2009).
Chapter 6
Page 31 of 38
SM: Statistical Mechanics
Comparing the result with Eq. (32), we get the so-called Wiedemann-Franz law 69
2
T
.
(6.110) Wiedemann-
2
3 q
Franz law
This relation between the electric conductivity and the thermal conductivity is more general than our formal derivation might imply. Indeed, it may be shown that the Wiedemann-Franz law is also valid for an arbitrary anisotropy (i.e. an arbitrary Fermi surface shape) and, moreover, well beyond the relaxation-time approximation. (For example, it is also valid for the scattering integral (12) with an arbitrary angular dependence of rate , provided that the scattering is elastic.) Experiments show that the law is well obeyed by most metals, but only at relatively low temperatures, when the thermal conductance due to electrons is well above the one due to lattice vibrations, i.e. phonons – see Sec. 2.6.
Moreover, for a non-degenerate gas, Eq. (107) should be treated with the utmost care, in the context of the definition (105) of this coefficient . (Let me leave this issue for the reader’s analysis.) Now let us discuss the effects described by Eq. (105), starting from the less obvious, first term on its right-hand side. It describes the so-called Peltier effect, which may be measured in the loop geometry similar to that shown in Fig. 12, but now driven by an external voltage source – see Fig. 13.
( ) I
1
2
I
2
I jA
I
1
I
2
Fig. 6.13. The Peltier effect at T = const.
( ) I
1
2
The voltage drives a certain dc current I = jA (where A is the area of conductor’s cross-section), necessarily the same in the whole loop. However, according to Eq. (105), if materials 1 and 2 are different, the power P = j h A of the associated heat flow is different in two parts of the loop.70 Indeed, if the whole system is kept at the same temperature ( T = 0), the integration of that relation over the cross-sections of each part yields
69 It was named after Gustav Wiedemann and Rudolph Franz who noticed the constancy of ratio / for various materials, at the same temperature, as early as 1853. The direct proportionality of the ratio to the absolute temperature was noticed by Ludwig Lorenz in 1872. Due to his contribution, the Wiedemann-Franz law is frequently represented, in the SI temperature units, as / = LT K, where the constant L (2/3) k B/ e 2, called the Lorenz number, is close to 2.4510-8WK-2. Theoretically, Eq. (110) was derived in 1928 by A. Sommerfeld.
70 Let me emphasize that here we are discussing the heat transferred through a conductor, not the Joule heat generated in it by the current. (The latter effect is quadratic, rather than linear, in current, and hence is much smaller at I 0.)
Chapter 6
Page 32 of 38
SM: Statistical Mechanics
'
A
P
A j I I ,
(6.111)
,
1 2
,
1 2
,
1 2
,
1 2 q
,
1 2
,
1 2
,
1 2
,
1 2
,
1 2
,
1 2
,12
where, at the second step, Eq. (41) for the electric current density has been used. This equality means that to sustain a constant temperature, the following power difference,
Peltier
effect
P
,
(6.112)
1
I
2
has to be extracted from one junction of the two materials (in Fig. 13, shown on the top), and inserted into the counterpart junction.
If a constant temperature is not maintained, the former junction is heated (in excess of the bulk, Joule heating), while the latter one is cooled, thus implementing a thermoelectric heat pump/refrigerator.
Such Peltier refrigerators, which require neither moving parts nor fluids, are very convenient for modest (by a few tens C) cooling of relatively small components of various systems – from sensitive radiation detectors on mobile platforms (including spacecraft), all the way to cold drinks in vending machines. It is straightforward to use the above formulas to show that the practical efficiency of active materials used in such thermoelectric refrigerators may be characterized by the following dimensionless figure-of-merit,
2
ZT
S
T .
(6.113)
For the best thermoelectric materials found so far, the values of ZT at room temperature are in the range from 2 to 3, providing the COPcooling, defined by Eq. (1.69), of the order of 0.5 – a few times lower than that of traditional, mechanical refrigerators. The search for composite materials (including those with nanoparticles) with higher values of ZT is one of very active fields of applied solid-state physics.71
Finally, let us discuss the second term of Eq. (105), in the absence of ’ (and hence of the electric current) giving
Fourier
law
j
T,
(6.114)
h
This equality should be familiar to the reader because it describes the very common effect of thermal conductivity. Indeed, this linear relation is much more general than the particular expression (107) for : for sufficiently small temperature gradients it is valid for virtually any medium – for example, for insulators. (The left column in Table 6.1 gives typical values of for most common and/or representative materials.) Due to its universality and importance, Eq. (114) has deserved its own name –
the Fourier law.72
Acting absolutely similarly to the derivation of other continuity equations, such as Eqs. (5.117) for the classical probability, and Eq. (49) for the electric charge,73 let us consider the conservation of the aggregate variable corresponding to jh – the internal energy E within a time-independent volume V.
According to the basic Eq. (1.18), in the absence of media’s expansion ( dV = 0 and hence d W = 0), the 71 See, e.g., D. Rowe (ed.), Thermoelectrics Handbook: Macro to Nano, CRC Press, 2005.
72 It was suggested (in 1822) by the same universal scientific genius J.-B. J. Fourier who has not only developed such a key mathematical tool as the Fourier series but also discovered what is now called the greenhouse effect!
73 They are all similar to continuity equations for other quantities – e.g., the mass (see CM Sec. 8.3) and the quantum-mechanical probability (see QM Secs. 1.4 and 9.6).
Chapter 6
Page 33 of 38
SM: Statistical Mechanics
energy change74 has only the thermal component, so its only cause may be the heat flow through its boundary surface S:
dE
j d 2r .
(6.115)
dt
h
S
In the simplest case of thermally-independent heat capacity CV, we may integrate Eq. (1.22) over temperature to write75
E C T c Td 3 r ,
(6.116)
V
V
V
where cV is the volumic specific heat, i.e. the heat capacity per unit volume (see the right column in Table 6.1).
Table 6.1. Approximate values of two major thermal coefficients of some materials at 20C.
Material
(Wm-1K-1)
cV (JK-1m-3)
Air(a),(b)
0.026
1.2103
Teflon ([C
2F4] n)
0.25
0.6106
Water(b)
0.60
4.2106
Amorphous silicon dioxide
1.1-1.4
1.5106
Undoped silicon
150
1.6106
Aluminum(c)
235
2.4106
Copper(c)
400
3.4106
Diamond
2,200
1.8106
(a) At ambient pressure.
(b) In fluids (gases and liquids), heat flow may be much enhanced by temperature-gradient-induced turbulent circulation – convection, which is highly dependent on the system’s geometry. The given values correspond to conditions preventing the convection.
(c) In the context of the Wiedemann-Franz law (valid for metals only!), the values of for Al and Cu correspond to the Lorenz numbers, respectively, 2.2210-8 WK-2 and 2.3910-8 WK-2, in a pretty i
mpressive comparison with the universal theoretical value of 2.4510-8WK-2 given by Eq. (110).
Now applying to the right-hand side of Eq. (115) the divergence theorem,76 and taking into account that for a time-independent volume the full and partial derivatives over time are equivalent, we get
T
3
c
j d r
,
(6.117)
V
h
0
t
V
74 According to Eq. (1.25), in the case of negligible thermal expansion, it does not matter whether we speak about the internal energy E or the enthalpy H.
75 If the dependence of cV on temperature may be ignored only within a limited temperature interval, Eqs. (116) and (118) may be still used within that interval, for temperature deviations from some reference value.
76 I hope the reader knows it by heart by now, but if not – see, e.g., MA Eq. (12.2).
Chapter 6
Page 34 of 38
SM: Statistical Mechanics
This equality should hold for any time-independent volume V, which is possible only if the function under the integral equals zero at any point. Using Eq. (114), we get the following partial differential equation, called the heat conduction equation (or, rather inappropriately, the “heat equation”): Heat
conduction
r
r
,
(6.118)
V T
c
Τ 0
equation
t
where the spatial arguments of the coefficients cV and are spelled out to emphasize that this equation is valid even for nonuniform media. (Note, however, that Eq. (114) and hence Eq. (118) are valid only if the medium is isotropic.)
In a uniform medium, the thermal conductivity may be taken out from the external spatial differentiation, and the heat conduction equation becomes mathematically similar to the diffusion equation (5.116), and also to the drift-diffusion equation (50) in the absence of drift ( U = 0): T
2
D T,
with D
.
(6.119)
T
T
t
cV
This means, in particular, that the solutions of these equations, discussed earlier in this course (such as Eqs. (5.112)-(5.113) for the evolution of the delta-functional initial perturbation) are valid for Eq. (119) as well, with the only replacement D DT. This is why I will leave a few other examples of the solution of this equation for the reader’s exercise.
Let me finish this chapter (and this course as a whole) by emphasizing again that due to time/space restrictions I was able to barely scratch the surface of physical kinetics.77
6.6. Exercise problems
6.1. Use the Boltzmann equation in the relaxation-time approximation to derive the Drude formula for the complex ac conductivity (), and give a physical interpretation of the result’s trend at high frequencies.
6.2. Apply the variable separation method78 to Eq. (50) to calculate the time evolution of the particle density distribution in an unlimited uniform medium, in the absence of external forces, provided that at t = 0 the particles are released from their uniform distribution in a plane layer of thickness 2 a:
n ,
for a x a,
n 0
,
0
otherwise.
77 A much more detailed coverage of this important part of physics may be found, for example, in the textbook by L. Pitaevskii and E. Lifshitz, Physical Kinetics, Butterworth-Heinemann, 1981. A deeper discussion of the Boltzmann equation is given, e.g., in the monograph by S. Harris, An Introduction to the Theory of the Boltzmann Equation, Dover 2011. For a discussion of applied aspects of kinetics see, e.g., T. Bergman et al., Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, 2011.
78 A detailed introduction to this method (repeatedly used in this series) may be found, for example, in EM Sec.
2.5.
Chapter 6
Page 35 of 38
Essential Graduate Physics
SM: Statistical Mechanics
6.3. Solve the previous problem using an appropriate Green’s function for the 1D version of the diffusion equation, and discuss the relative convenience of the results.
6.4.* Calculate the electric conductance of a narrow, uniform conducting link between two bulk conductors, in the low-voltage and low-temperature limit, neglecting the electron interaction and scattering inside the link.
6.5. Calculate the effective capacitance (per unit area) of a broad plane sheet of a degenerate 2D
electron gas, separated by distance d from a metallic ground plane.
6.6. Give a quantitative description of the dopant atom ionization, which would be consistent with the conduction and valence band occupation statistics, using the same simple model of an n-doped semiconductor as in Sec. 4 (see Fig. 7a), and taking into account that the ground state of the dopant atom is typically doubly degenerate, due to two possible spin orientations of the bound electron. Use the results to verify Eq. (65), within the displayed limits of its validity.
6.7. Generalize the solution of the previous problem to the case when C
the n-doping of a semiconductor by n D donor atoms per unit volume is
complemented with its simultaneous p-doping by n
D
A acceptor atoms per unit
volume, whose energy A – V of activation, i.e. of accepting an additional
electron and hence becoming a negative ion, is much lower than the bandgap A
– see the figure on the right.
V
6.8. A nearly-ideal classical gas of N particles with mass m, was in thermal equilibrium at temperature T, in a closed container of volume V. At some moment, an orifice of a very small area A is open in one of the container’s walls, allowing the particles to escape into the surrounding vacuum.79 In the limit of very low density n N/ V, use simple kinetic arguments to calculate the r.m.s. velocity of the escaped particles during the time period when the total number of such particles is still much smaller than N. Formulate the limits of validity of your results in terms of V, A, and the mean free path l.
Hint: Here and below, the term “nearly-ideal” means that l is so large that particle collisions do not affect the basic statistical properties of the gas.
6.9. For the system analyzed in the previous problem, calculate the rate of particle flow through the orifice – the so-called effusion rate. Discuss the limits of validity of your result.
6.10. Use simple kinetic arguments to estimate:
(i) the diffusion coefficient D,
(ii) the thermal conductivity , and
(iii) the shear viscosity ,
of a nearly-ideal classical gas with mean free path l. Compare the result for D with that calculated in Sec. 3 from the Boltzmann-RTA equation.
79 In chemistry-related fields, this process is frequently called effusion.
Chapter 6
Page 36 of 38
SM: Statistical Mechanics
Hint: In fluid dynamics, the shear viscosity (frequently called simply "viscosity") may be defined as the coefficient in the following relation:80
d F
v
j'
j'
,
dA
r
j
j
where d F j' is the j' th Cartesian component of the tangential force between two parts of a fluid, separated by an imaginary interface normal to some direction n j (with j j' , and hence n j n j' ), exerted over an elementary area dAj of this surface, and v(r) is the velocity of the fluid at the interface.
6.11. Use simple kinetic arguments to relate the mean free path l in a nearly-ideal classical gas, with the full cross-section of mutual scattering of its particles.81 Use the result to evaluate the thermal conductivity and the viscosity coefficient estimates made in the previous problem, for the molecular nitrogen, with the molecular mass m 4.710-26 kg and the effective (“van der Waals”) diameter d ef
4.510-10m, at ambient conditions, and compare them with experimental results.
6.12. Use the Boltzmann-RTA equation to calculate the thermal conductivity of a nearly-ideal classical gas, measured in conditions when the applied thermal gradient does not create a net particle flow. Compare the result with that following from the simple kinetic arguments (Problem 6.10), and discuss their relationship.
6.13. Use the heat conduction equation (6.119) to calculate the time evolution of temperature in the center of a uniform solid sphere of radius R, initially heated to a uniformly distributed temperature T ini, and at t = 0 placed into a heat bath that keeps its surface at temperature T 0.
6.14. Suggest a reasonable definition of the entropy production rate (per unit volume), and calculate this rate for stationary thermal conduction, assuming that it obeys the Fourier law, in a material with negligible thermal expansion. Give a physical interpretation of the result. Does the stationary temperature distribution in a sample correspond to the minimum of the total entropy production in it?
6.15.82 Use the Boltzmann-RTA equation to calculate the shear viscosity of a nearly-ideal gas.
Spell out the result in the classical limit, and compare it with the estimate made in the solution of Problem 10.
80 See, e.g., CM Eq. (8.56). Please note the difference between the shear viscosity coefficient considered in this problem and the drag coefficient whose calculation was the task of Problem 3.2. Despite the similar (traditional) notation, and belonging to the same realm (kinematic friction), these coefficients have different definitions and even different dimensionalities.
81 I am sorry for using the same letter for the cross-section as for the electric Ohmic conductivity. (Both notations are very traditional.) Let me hope this would not lead to confusion, because the conductivity is not discussed in this problem.
82 This problem does not follow Problem 12 only for historic reasons.
Chapter 6
Page 37 of 38
SM: Statistical Mechanics
This page is
intentionally left
blank
Chapter 6
Page 38 of 38
Essential Graduate Physics
Lecture Notes and Problems
Open online access at
https://sites.google.com/site/likharevegp/
and
http://commons.library.stonybrook.edu/egp/
Appendix MA
Selected Mathematical Formulas
that are used in this series, but not always remembered by students
(and some instructors :-)
Last corrections: 2021/08/20
1. Constants
– Euclidean circle’s length-to-diameter ratio:
141
.
3
592
...;
653
1/ 2
77
.
1
.
(1.1)
– Natural logarithm base:
1 n
e lim
;
(1.2a)
n 1
2.718 281 828...
n
from that value, the logarithm base conversion factors are as follows ( > 0):
ln
log
10
1
ln10 2.
,
303
0.434 .
(1.2b)
log
ln
ln10
10
– The Euler (or “Euler-Mascheroni”) constant:
1 1
1
lim
n
e
.
(1.3)
n 1
... ln
90
0.57715664
;
781
.
1
2 3
n
2. Combinatorics, sums, and series
(i) Combinatorics
– The number of different permutations, i.e. ordered sequences of k elements selected from a set of n distinct elements ( n k), is
n
!
n
P n ( n )
1 ... ( n k )
1
;
(2.1a)
k
( n k)!
© K. Likharev
MA: Math Appendix
in particular, the number of different permutations of all elements of the set ( n = k) is k P k ( k )
1 ... 2 1 k!
(2.1b)
k
– The number of different combinations, i.e. unordered sequences of k elements from a set of n
k distinct elements, is equal to the binomial coefficient
n
n
n
Pk
!
n
C
.
(2.2)
k
k k
P
k!( n k)!
k
In an alternative, very popular “ball/box language”, nCk is the number of different ways to put in a box, in an arbitrary order, k balls selected from n distinct balls.
– A generalization of the binomial coefficient notion is the multinomial coefficient, n
n
!
l
C
,
with n
k k
k
k ,
, ,...
(2.3)
j
1 2
l
k ! k !... k !
1
2
l
j 1
which, in the standard mathematical language, is a number of different permutations in a multiset of l distinct element types from an n-element set which contains kj ( j = 1, 2,… l) elements of each type. In the less formal “ball/box language”, the coefficient (2.3) is the number of different ways to distribute n distinct balls between l distinct boxes, each time keeping the number ( kj) of balls in the j th box fixed, but ignoring their order inside the box. The binomial coefficient nCk, given by Eq. (2.2), is a particular case of the multinomial coefficient (2.3) for l = 2 – counting the explicit box for the first one, and the remaining space for the second box, so that if k 1 k, then k 2 = n – k.
– One more important combinatorial quantity is the number M ( k) n
of different ways to place n
indistinguishable balls into k distinct boxes. It may be readily calculated from Eq. (2.2) as the number of different ways to select ( k – 1) partitions between the boxes in an imagined linear row of ( k – 1 + n)
“objects” (balls in the boxes and partitions between them):
k
n
k
k
n
( )
1
1 !
M
C
.
(2.4)
n
k 1
k 1! ! n
(ii) Sums and series
– Arithmetic progression:
n
n( r nr)
r 2 r ... nr kr
;
(2.5a)
k 1
2
in particular, at r = 1 it is reduced to the sum of n first natural numbers: n
n( n )
1
1 2 ... n k
.
(2.5b)
k 1
2
– Sums of squares and cubes of n first natural numbers:
n
n n
n
2
2
2
2
(
)(
1 2
)
1
1 2 ... n k
;
(2.6a)
k 1
6
n
2
n n
3
3
3
3
(
)
1 2
1 2 ... n k
.
(2.6b)
k 1
4
Selected
Mathematical
Formulas Page
2
of
16
MA: Math Appendix
– The Riemann zeta function:
1
1
1
( s) 1
...
;
(2.7a)
s
s
2
3
s
k 1 k
the particular values frequently met in applications are
3
2
5
4
2.
,
612
2
, 1.
,
341 3 1.
,
202
(4)
, 5 1.037. (2.7b)
2
6
2
90
– Finite geometric progression (for real 1):
n 1
n
2
n1
k
1
1 ...
;
(2.8a)
k0
1
in particular, if 2 < 1, the progression has a finite limit at n (called the geometric series): n 1
k
k
1
lim
(2.8b)
n
.
k
k
0
0
1
– Binomial sum (also called the “binomial theorem”):
n
1 a n
n
k
C a ,
(2.9)
k
k 0
where nCk are the binomial coefficients given by Eq. (2.2).
– The Stirling formula:
1
1
1
lim
ln n
n
n
n
;
(2.10)
n
! (ln )1 ln(2 )
...
2
12 n 360 3
n
for most applications in physics, the first term1 is sufficient.
– The Taylor (or “Taylor-Maclaurin”) series: for any infinitely differentiable function f (): 2
~
df
~
1 d f
~
1 k
d f
~
2
lim
k
~
x
f ( ) f ( )
( )
( )
...
0
( )
;
(2.11a)
2
d
!
2
k
d
k 0 k! d
note that for many functions this series converges only within a limited, sometimes small range of deviations ~ . For a function of several arguments, f(1,2,…, N), the first terms of the Taylor series are
~
~
N
f
~
1 N
2
f ~ ~
lim ~
f ( , ,...) f
(2.11b)
k
0
1
1
2
2
,1 2
,...
,1 2
,...
...
k
k'
k
k k
k k
1
!
2 , ' 1 k
k'
– The Euler-Maclaurin formula, valid for any infinitely differentiable function f(): n
n
df
df
f ( k) f ( d 1
)
f ( n) f (0) 1 1
( n)
(0)
k
d
d
1
2
6
!
2
0
1 Actually, this leading term was conjectured by A. de Moivre in1733, before J. Stirling’s proof of the series.
Selected
Mathematical
Formulas Page
3
of
16
MA: Math Appendix
1 1
3
3
d f
d f
1 1
5
5
d f
d f
( n)
(0)
( n)
(0) ...;
(2.12a)
30
!
4
3
3
d
d
42
!
6
5
5
d
d
the coefficients participating in this formula are the so-called Bernoulli numbers:2
1
1
1
1
1
B , B , B ,
0
B
, B ,
0
B
, B ,
0
