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Pictorial kinetic methods in the theory of metals and semiconductors
Dingle, R . B . 1956
Physica XXlI 671-680
P I C T O R I A L K I N E T I C M E T H O D S IN T H E T H E O R Y OF METALS AND SEMICONDUCTORS b y R. B. D I N G L E Department of Physics, University of Western Australia, Nedlands, W. Australia
Synopsis Two methods have been used in the literature to calculate the electron current in transport problems, the strict Boltzmann method involving the solution of a partial differential equation, and the pictorial kinetic method based on calculations of the average drift velocity in the direction of the applied field. It is shown in the present paper that as customarily formulated this pictorial kinetic method gives erroneous results unless the collision time is independent of the electron speed Iv[. The fault is traced to an invalid superposition of the additional drift velocity Avz, caused (say) by an external electric field E z, upon the distribution function f0 in the absence of fields. It is shown that the error may be rectified by replacing, in this context, the ordinary equilibrium distribution function/0 by a "transport distribution function" It = - - vz. alol~vz.
Finally, a method is propounded for finding relations valid for semiconductors by averaging over the transport distribution function the corresponding relations for metals. 1. P i c t o r i a l k i n e t i c method. T h e strict B o l t z m a n n m e t h o d for solving t r a n s p o r t p r o b l e m s in the t h e o r y of m e t a l s a n d s e m i c o n d u c t o r s has somet i m e s been replaced b y the a r g u m e n t t h a t since the applied field can a l t e r the d i s t r i b u t i o n function b u t slightly, the c u r r e n t m a y be f o u n d b y a s s u m i n g (i) the n u m b e r of electrons per unit v e l o c i t y r a n g e to be as given b y t h e d i s t r i b u t i o n function w i t h o u t field,/0 (v), a n d (ii) the m e a n drift v e l o c i t y A v caused b y the applied field to be s u p e r p o s e d on the r a n d o m l y directed t h e r m a l velocities v. (This drift v e l o c i t y will in general d e p e n d on the collision t i m e ,). According to these ideas, the c u r r e n t d e n s i t y j, in the z-direction would be i , = (2m3e/ha) r i o Av~ d3v
(1)
in an isotropic m e d i u m , where d3v : dvx dv~ dye, e : electronic c h a r g e (negative), a n d the f a c t o r 2m3/h 3 arises f r o m the 2/h 3 electron s t a t e s per unit v o l u m e of p h a s e space. Before s u b m i t t i n g eq. (1) to a critical e x a m i n a t i o n , it is first n e c e s s a r y to i n v e s t i g a t e carefully the m e a n i n g a n d m e t h o d of e v a l u a t i o n of the t w o q u a n t i t i e s , : collision t i m e a n d A v : drift velocity. --
671
--
672
R.B. DINGLE
C o 11 i s i o n t i m e. If it is assumed t h a t the probability of an electron of given speed Ivl suffering a collision between the times t and t + dt is dt/~, where in general ~ = v ([vl), and P(t) is written for the probability of an electron travelling a time t without collision, then the proposition t h a t Probability of travelling a time t + dt without collision --[Probability of travelling a time t without collision] × [1 -- Probability of colliding in time dt] m a y be symbolised P ( t + d t ) = P ( t ) [1 - - d t / z ] , whence [P(t + dt) -- P(t)]/dt = ~P/Ot = -- Ply. Since the probability of travelling for zero time (t = 0) w i t h o u t collision must be unity, the appropriate solution is
P(t) = e -t/'.
(2)
The m e a n time to elapse before the next collision is
f~o te-'/T (dt/z) = 1:. Decay of current contribution on removal o f ext e r n a 1 f i e 1 d. Making the c u s t o m a r y assumption t h a t on collision an electron loses all its drift velocity, the contribution i([v[, t) w. Av ([vl, t) from electrons of speed Ivl to the current density must, on removal of the external field, decay with time proportionately to the n u m b e r of electrons which have not yet suffered collision. Thus, b y (2),
i(Ivl, t) oc exp [-- tit (Ivl)]
(3)
This deduction will later be required in establishing t h a t the q u a n t i t y
• (Ivl), which appears b o t h in the pictorial kinetic m e t h o d and in the Boltzm a n n equation, has precisely the same meaning in the two in spite of being introduced very differently. Method of evaluating drift velocity. If a , ( t ' ) i s the acceleration in the z-direction of an electron at time t', the increase in drift velocity between t' and t' + dt' will be a~(t') dt'. By (2), the probability of an electron not suffering a collision - i.e. the probability of retaining its drift velocity - at least up till the time t (t > t') is e-(t-v)/L so t h a t
Av~([vl, t) = f!oo a~(t') e -c~-t')lT dt' = f~o a~(t
-
-
T) e -r/~ d T ,
(4)
where T ~-- t -- t' is the interval of time which wilt elapse before an electron reaches the point at which the current is to be evaluated. The drift velocity is t h u s equal to the Laplace image in the T-Z-plane of the electron acceleration expressed in time-units measured backwards into the past from the time at which the current is to be evaluated *). *) In general,
Av(t) =
f~o
a(t --
T) e-- T/r dT.
673
PICTORIAL KINETIC METHODS
Examples of the evaluation o f Av. (a) In a uniform steady electric field Ez, the acceleration az = eE,/m does not depend on time, and furthermore the drift velocity cannot depend on the direction of the electron p a t h on account of the spatial u n i f o r m i t y of E,. Thus b y (4) eE, AV z --
m
t* oo
J
e-T/': d T = __er Ez. 0
(b) For a uniform alternating = (eEl~m) ei~(t-T~, so t h a t Av, = (eE,/m) f ~
e -T/"
(5)
m
electric
e''(t-T) d T =
field E , e~t,
az (t-
[ev/m(1 + iwv)] E , e'~t.
T)= (6)
(c) For an alternating electric field E,(x) e~°'~ which varies appreciably with the d e p t h x (e.g. within one mean free path) - the anomalous skin effect a, (t -- T) = (eE,/m) ei°(t-T) as in (b), b u t since E , = E,(x) it is now convenient to express each time interval T = t - t' in terms of the corresponding space interval x - x' travelled b y the electron. Since T = [ ( x ' - - x ) / v ~ l , d T = dx'/lv~[, a n d (4) gives Av: -- mlv,[
E:(x') exp
dx'. T
Vx
(7)
J
The limits of integration for x' which have been inserted in (7) take account of drift velocities incurred b y electrons travelling to the level x b o t h from all levels 0 < x' < x above x, and from all levels x' > x below x. It is assumed t h a t electrons reflected back into the metal after collision at the internal surface (located at x = 0) do not retain their previously acquired drift velocities. 2. Comparison with results obtained /rom the Boltzmann equation. Strict solutions of transport problems are obtained from the B o l t z m a n n equation. W h e n a collision time exists this m a y be written in the symbolic form
8/ ~/ 8--t-+v'-~ -+a"
~/ a~--
/ --/o _ ~
,,'t/ 3 '
(8)
where t----time, r = position vector, v = velocity, a = acceleration, ]0(r, v) = distribution function without fields, a n d / ( r , v, a) = distribution function in the presence of the fields responsible for the electron acceleration a. On this approach, the current density 1"~in the z-direction is, for an isotropic medium *), j~ = (2m3~/h3) f v~ / d3v = (2m3~/h 8) f v~ A~ d%.
(9)
*) I n general, j = ( 2 m S e / h S } f v A I dSv. Physica XXII
43
674
R.B.
DINGLE
Decay of current contribution on removal of e x t e r n a 1 f i e 1 d. The collision t i m e , appearing in (8) m a y be defined as a certain time of restoration of the distribution function ] to the value [0 when the disturbance A / = [ -- [o is uniform in space and no external forces are acting. For when these conditions appertain, (8) reduces to i.e. A / o c e -t/r, where in general ~ = ~(Ivl).
(10)
The contribution j(lvl, t) oc A[(Iv], t) from electrons of speed Iv] to the current density must therefore decay with time proportionately to e - t l * l l v l ) o n removal of the external fields. Since this is precisely the same result as contained in eq. (3), the collision time m u s t have identical meanings in the B o l t z m a n n m e t h o d and the pictorial kinetic method. Examples of the evaluation o f A[. (a) A uniform s t e a d y electric field E , causes an electron acceleration a, = eEl~m, and the operators a/at and a/ar are null on account of the u n i f o r m i t y in time a n d space. Hence b y (8) At =
-
• a, atlav, ~_ -
• a, alolav, =
-
( * * l m ) (atolav,) E , .
(b) For a uniform alternating electric field E, e~ , reduces to
(io~ + l/z)
A/~--
(11)
a/at =_ iw, so t h a t (8)
- - a , alo/aV,,
yielding AI =
-
[~,lm(1 +
io.,-~)] (alol~v,)
(12)
E , ~'"'.
(c) For an alternating electric field E , e~ v a r y i n g appreciably in the x-direction within one m e a n free p a t h - the anomalous skin effect - it is found t h a t 1) ee~"~ ~[o AI--
mlv~ I av,
F
E.(x') exp
o
{
1 + im~ ] ~
x' -- x
v~
}
dx',
(13)
provided t h a t electrons lose their drift velocity on reflection at the internal metal surface located at x ---- 0. Comparison between results of pictorial-kinetic and Boltzmann methods. Comparing (ll), (12) and (13) with (5), (6) and (7), it is seen t h a t AI -
-
(alolav,) A v , ,
(14)
at least for the instances considered *). Reference to (1) a n d (9) t h e n reveals *) If zJv had more than the one component /Ivz assumed in the text, it would obviously be necessary to add the contributory differentials: At ~ -- {(~/o/Ov.) zlv~ + (~/./~v~) /tv~ + (~/o/Ov~) live} = -- (~/o/Ov) . ~Iv.
PICTORIAL K I N E T I C METHODS
675
that if the two methods are to lead to the same values for the current density j,, the factor v , /I! =_ -
v,
'°~iv,
appearing in the integrand of (9) can only differ b y a perfect differential from the factor [o dv, appearing in the integrand of (I). Unfortunately, this is not in general the case, for
f v, ff/o/aV,) Av , dv, = [v, /iv, t0]M:-+: -- f /o Av, dv,-- f lo V, (~/iv dav,) dr,..(15) The first term on the right hand side of (15) is certainly zero, but the third term will only vanish if either of the following conditions is satisfied: (c¢) ~,4v,/av, = 0; i.e. if /Iv, does not contain v,. In the examples considered, this condition will be fulfilled if ~ is independent of the electron speed Iv]. Or (fl) O/Iv,/av, is an even function of v,, for ]0 is certainly even in v,; i.e. if /Iv, is an odd function of v,. This, however, would be physically quite absurd, since it would imply that electrons drift either with the field or against it depending on the direction of their initial thermal velocities! Moreover, even if it does not itself vanish, this third term in (15) cannot give zero contribution on integration over vx or v~ unless it is odd in vx or v~. Since ]0 is even in v~ and v~, this would entail t h a t / I v , be odd in vx or v~, which is again physically absurd for the same reason as in (fl) above. Thus it is to be concluded that the customary ]ormulation o] the pictorial
kinetic method is in agreement with the strict Boltzmann theory only when the collision time can be considered as independent o] velocity. For metals with degenerate electron systems, the strict Boltzmann approach shows that only the value Z(VFer,nl) at the Fermi surface is required, since the factor O]o/~V appearing in /I] is negligible except for v ~___vr~ni. Correct results will then be obtained from the customary pictorial kinetic method provided v is taken to be a constant equal to v(vFe~,~i). For semiconductors, however, the customary pictorial kinetic method gives erroneous results, since for them it is essential to take explicit account of the known variation of v with Iv].
3. Cause o] the discrepancy. Particular cases of the discrepancy revealed in the previous section have been noted in previously published work 2) 3), and it has been suggested that it m a y be attributed to neglect of the facts that during the electron acceleration the value of [0 will change 2) from /0(iv[) to /0(Iv + Av, I), and correspondingly the value of T change 3) from ,(Ivl) to ,(I v + Av,[), where iv I and Iv + Av,I are respectively the initial and final electron speeds. However, with Av, already appearing as a factor
676
R.B. DINGLE
in the integrand of eq. (1), these particular shortcomings in the pictorial kinetic theory can only explain discrepancies of order (Avz) ~, i.e. quadratic in Ez, whereas the discrepancy under discussion is already present in the first order. The major fallacy in the customary kinetic approach lies deeper, and can be traced to the invalidity of the superposition implied in the assumptions (i) and (ii) introduced at the beginning of § 1. In order to show that it is is in general not permissible to superpose on the distribution/o the additional drift velocity caused by the external field, it is to be noted that in general such a superposition might: (a) Violate those requirements of quantum theory which originally determined the very form of the statistical distribution function/0. To take an example, suppose that for some hypothetical system the number of accessible states decreases rapidly with increase in electron speed, and that practically all accessible states are occupied in the absence of an external field. The assumption of direct superposition would then altogether fail to take account of the essential fact that electrons could not pass into the inaccessible states even after the expected general increase in electron speed caused by a large electric field. (b) Fail to represent a steady state of the system. For even if the superposition did not involve actual violation of the requirements of quantum theory, as in (a), it would still in general only represent the transient state of affairs immediately succeeding the imposition of the external field. These objections show that the customary formulation of the kinetic approach cannot be universally valid. In order to determine the particular conditions for which the method, as it stands, would (by chance) lead to correct results, it is only necessary to observe that both objections (a) and (b) become nugatory when the distribution function / only differs from/o by either a uniform translational motion or a uniform rotational motion of the entire electron system, since it is known that such motions do not upset the statistical equilibrium 4). Since a value of the drift velocity Av, independent of v, would be equivalent to a uniform translational motion of the electron system with velocity Av~ in the z-direction, the customary kinetic approach will thus give correct results if Av, does not contain v,. This conclusion is in agreement with (~) of § 2. As noted there, such a condition will normally be fulfilled if the collision time is independent of electron velocity.
4. Previous attempts to evolve a correct pictorial kinetic method. I. It has been suggested ~) that since the contribution to the current is proportional to the change in the number of electrons (oc/0) of given speed consequent upon the presence of the field E,, and to the velocity v,, the product/0. Av, appearing in the integrand of eq. (1) should simply be replaced by the
PICTORIAL KINETIC METHODS
677
product
v~ A/o = v~(~/o/~V~) Av~. This "explanation" is unfortunately subject to grave objections: (i) In the strict derivation of the statistical distribution function/0 it is assumed that either (a) the momentum of the system as a whole, P say, is completely unknown, in which case the statistical average momentum P vanishes since all directions of P are equally likely, or (b) the momentum of the system as a whole is known to be identically zero, i.e. P ---- 0. It is therefore logica!ly unsound in any pictorial kinetic theory to apply the distribution function /0 directly to a system of electrons known to be contributing a current, since then it would be known that _P ve 0 in contradiction to both possibilities (a) and (b). The net current therefore cannot logically be determined b y simply imagining a change A/0 to occur in the distribution function/0 as a result of the imposition of an external field *). (ii) The implication that explicit account must be taken of the change in /0 occurring while the velocity of an electron is increasing due to the presence of the external field, on an average from v to v -6 Avz, gives rise to a number of serious conceptional difficulties. For instance, (~/o/SV,) Av, is presumably intended as an approximation to/0(v -6 Av,) --/0(v), of which only the first t e r m / o ( v -6 Av,) contributes to the current. But why should the weighting factor be taken as that appropriate to the final velocity v -6 Av, ? What happens to the drift momentum of electrons which, according to the variation in/o from start to finish, do have to be considered at the commencement of their path, but not at the end ? (The effect of collisions has already been taken into account in the calculation of Avz). On what grounds is it supposed that all those electron states which must be considered according to the final distribution function/0(v -6 Avz) taken, have all 'survived' while the argument of [o changed from v to v -6 Av, ? For if they did not 'survive' throughout, the acquired average drift velocity cannot have attained the full calculated value Av,. (iii) The suggested replacement of/o.Av, in eq. (1) b y v,.A/o = v,(a/o/~V,). •Av~ leads to a current of the wrong sign, as m a y be seen b y comparing equations (1), (9) and (14). As will become clear later, in § 5, the reason for this incorrect sign is that in reality the current is not determined b y the change in/o(V) caused b y the drift velocity Av,(v) as suggested in 2) ; instead, it is *) I t should be r e m a r k e d here t h a t it is implied in ~) t h a t allowance is to be made for the fact t h a t / o (Ivl) ~ ]o (Iv -t- zJvz[), b u t the m o d u l u s sign then appears to have been i n a d v e r t e n t l y dropped. If it had been retained, it would have been found t h a t on such a basis v, A/o = v, (8/o/a Iv, I) Av,, Av, always being of uniform sign for electrons of all velocities; since ]o is even in vz, this would h a v e led to the conclusion t h a t the current is always zero!]
678
R . B . DINGLE
determined by the velocity --Av,(v) which would have to be superposed on the electrons in order to permit the use of the stationary equilibrium distribution function/o(v). II. A mode of approach has been proposed 3) which involves calculations of the following quantities: (a) Change in probability of collision in the presence of an external field. (b) Average displacement in the direction of the applied field of an electron with given initial velocity. (c) Rate at which electrons are scattered into a differential volume of momentum space. (d) Statistical average of the sum of all electron velocities, and hence of the current. This, however, can hardly be classed as a true pictorial kinetic method, since in step (c) the collision t i m e , appears as a time of restoration of the distribution function / to the value/0, a concept typical of the Boltzmann approach. [3 appears also in the pictorial kinetic sense in steps (a) and (b)]. It is the aim of the present paper to show that the fallacy in the customary formulation of the kinetic approach m a y be removed merely by replacing the "equilibrium distribution function"/o in (1) by a "transport distribution function"/t; the quantity ~ will thus appear only in the true pictorial kinetic sense involved in evaluating the drift velocity Av, in accordance with eq. (4). III. General formulae have been derived 4) by finding / by integrating up the number of electrons scattered into a trajectory at previous points along it -- proportional to/o (E -- AE), where AE is the energy remaining to be acquired from the applied fields before the electrons reach their destination -- weighted by their probability of reaching this destination. But in this derivation, the concept of 'drift velocity', perhaps the most interesting contribution of the customary pictorial kinetic approach to the full understanding of the 'mechanistics' of the problem, has been abandoned -- though of course it could be reintroduced artificially.
5. Re/ormulation o/ the pictorial kinetic method. In reformulating the pictorial kinetic method, it is necessary to start from the relation (9), viz. i, : (2m3e/h3)f v,. A/dZv,
(16)
since it has already been demonstrated in § 3 that eq. (1) is subject to the superposition fallacy. The remaining problem is then to determine A/from the quantity Av, calculated on the kinetic approach. If the electrons in the statistical system' exhibited no drift velocity (Av, --= 0), the distribution function would be/o(E), where E is the electron energy. The required distribution function / : / 0 + A/applicable when the electrons do exhibit a drift velocity Avz, relative to a stationary observer, m a y be determined from the consideration that the distribution function
PICTORIAL
KINETIC
METHODS
679
will still be ]o according to an observer moving with velocity Av e, since such an observer would be unaware of the drift velocity. In the general case in which A v e is a function of Ivl, this observer is to be imagined as moving with the appropriate drift velocity Av,(Ivl) when observing electrons of thermal speed tvl. Thus
/(e) =/0(e'),
(17}
where 8' is the apparent energy (Hamiltonian) of an electron according to the observer moving with velocity A v , , whilst ~ is the energy of the same electron according to the stationary observer. Now it is known'that n) 6) e'
e
=
p.Av,
--
(18)
where p = momentum of particle or excitation, and Av = drift velocity. Hence by (17)
l(e)
=
lo(e
-
p, ~v,),
(19)
so that ~1 = 1 - - 1o ~-- -
P , A,,, ~to(e)l~e.
(20)
Assuming that the electrons behave as though free, with an effective mass m , p , = m y , , £ = ½mY 2, and v2 = v~ + v~ + v~, whence At =
- - v , Ave O/o/v av =
- - (~lo/~V,) A v , .
(21)
Inserting this expression for A / i n t o (16), it is seen that i,
(2raze/ha) f h Ave day,
=
(22)
where the "transport distribution function" /t is defined by *) It
=
--
v,
~/olbv,
(23)
and A v e is calculated by pictorial kinetic methods according to eq. (4). The result of our reformulation thus differs from that of the customary formulation only by the replacement of the stationary equilibrium distribution function [o in the current-relation (1) by the "transport distribution function"/t, equal to - - v z ~[o/OV, when the field and current are in a direction parallel to the z-axis. This reformulation of the pictorial kinetic method is in complete accord with the strict Boltzmann theory, the replacement of [0 by /t having removed the superposition fallacy inherent in the customary formulation. *) If z l v h a d m o r e t h a n o n e c o m p o n e n t - - see f o o t n o t e s to (9) a n d (14) - - it w o u l d be n e c e s s a r y t o d e f i n e It as t h e o p e r a t o r [ t = - - v (aldOv). ( N o t e t h a t w h i l e c o n t r i b u t i o n s to t h e d i f f e r e n t i a l q u a n t i t y / I ] c a n be a d d e d t o g e t h e r , n o s u c h sinple l a w of a d d i t i o n a p p l i e s t o t h e n o n - d i f f e r e n t i a l q u a n t i t y I t ) .
680
PICTORIAL KINETIC METHODS
6. Averaging over the distribution/unction. Since b y (22) the contribution j(v) from electrons of Velocity v to the current density is proportional t o / t and not to [0 (as often erroneously assumed), it follows that in any averaging over the velocity or energy distribution of quantities involved in the theory of transport phenomena, the appropriate distribution function is ]t and not /o- For instance, the relation o = ns%/m for the d.c. conductivity of a metal should be generalised to read = ( m 2 / m ) f , / , d3v/f/t d3v
(24)
for a semiconductor. Again, the Drude-Kronig optical absorptivity A = = (miens2) ½z -i for a metal in the near infra-red should be generalised to read A = (m/~n~2)½f ~-1/~ d 3 v / f / , d3v (25) for a semiconductor. Further examples of this convenient technique will be found in a forthcoming paper entitled "The anomalous skin effect and the optical absorptivity of semiconductors, I". Received 2-2.56.
REFERENCES I) R e u t e r , G . E . H . and S o n d h e i m e r , E . H . , P r o c . roy. Soc. A I 9 5 ( 1 9 4 8 ) 336. 2) C h a m b e r s , R . G . , P r o c . roy. Soc. A 2 0 2 ( 1 9 5 0 ) 380. 3) S h o c k 1 e y, W., Electrons and Holes in Semiconductors (1950), va n Nostrand, N.Y., p. 293. See also the detailed discussion of drift velocities in H u x l e y, L. G. H., Proc. phys. Soc. B 64
(1951) 844. 4) C h a m b e r s , R. G., Proc. phys. Soc. A 6 5 (1952) 458. 5) L a n d a u, L. D. and L i f s h i t z, E., Statistical Physics (1938), Oxford U ni ve rs i t y Press. 6) D i n g l e , R. B., Advances in Physics I (1952) 138.
Physica XXlI 671-680
P I C T O R I A L K I N E T I C M E T H O D S IN T H E T H E O R Y OF METALS AND SEMICONDUCTORS b y R. B. D I N G L E Department of Physics, University of Western Australia, Nedlands, W. Australia
Synopsis Two methods have been used in the literature to calculate the electron current in transport problems, the strict Boltzmann method involving the solution of a partial differential equation, and the pictorial kinetic method based on calculations of the average drift velocity in the direction of the applied field. It is shown in the present paper that as customarily formulated this pictorial kinetic method gives erroneous results unless the collision time is independent of the electron speed Iv[. The fault is traced to an invalid superposition of the additional drift velocity Avz, caused (say) by an external electric field E z, upon the distribution function f0 in the absence of fields. It is shown that the error may be rectified by replacing, in this context, the ordinary equilibrium distribution function/0 by a "transport distribution function" It = - - vz. alol~vz.
Finally, a method is propounded for finding relations valid for semiconductors by averaging over the transport distribution function the corresponding relations for metals. 1. P i c t o r i a l k i n e t i c method. T h e strict B o l t z m a n n m e t h o d for solving t r a n s p o r t p r o b l e m s in the t h e o r y of m e t a l s a n d s e m i c o n d u c t o r s has somet i m e s been replaced b y the a r g u m e n t t h a t since the applied field can a l t e r the d i s t r i b u t i o n function b u t slightly, the c u r r e n t m a y be f o u n d b y a s s u m i n g (i) the n u m b e r of electrons per unit v e l o c i t y r a n g e to be as given b y t h e d i s t r i b u t i o n function w i t h o u t field,/0 (v), a n d (ii) the m e a n drift v e l o c i t y A v caused b y the applied field to be s u p e r p o s e d on the r a n d o m l y directed t h e r m a l velocities v. (This drift v e l o c i t y will in general d e p e n d on the collision t i m e ,). According to these ideas, the c u r r e n t d e n s i t y j, in the z-direction would be i , = (2m3e/ha) r i o Av~ d3v
(1)
in an isotropic m e d i u m , where d3v : dvx dv~ dye, e : electronic c h a r g e (negative), a n d the f a c t o r 2m3/h 3 arises f r o m the 2/h 3 electron s t a t e s per unit v o l u m e of p h a s e space. Before s u b m i t t i n g eq. (1) to a critical e x a m i n a t i o n , it is first n e c e s s a r y to i n v e s t i g a t e carefully the m e a n i n g a n d m e t h o d of e v a l u a t i o n of the t w o q u a n t i t i e s , : collision t i m e a n d A v : drift velocity. --
671
--
672
R.B. DINGLE
C o 11 i s i o n t i m e. If it is assumed t h a t the probability of an electron of given speed Ivl suffering a collision between the times t and t + dt is dt/~, where in general ~ = v ([vl), and P(t) is written for the probability of an electron travelling a time t without collision, then the proposition t h a t Probability of travelling a time t + dt without collision --[Probability of travelling a time t without collision] × [1 -- Probability of colliding in time dt] m a y be symbolised P ( t + d t ) = P ( t ) [1 - - d t / z ] , whence [P(t + dt) -- P(t)]/dt = ~P/Ot = -- Ply. Since the probability of travelling for zero time (t = 0) w i t h o u t collision must be unity, the appropriate solution is
P(t) = e -t/'.
(2)
The m e a n time to elapse before the next collision is
f~o te-'/T (dt/z) = 1:. Decay of current contribution on removal o f ext e r n a 1 f i e 1 d. Making the c u s t o m a r y assumption t h a t on collision an electron loses all its drift velocity, the contribution i([v[, t) w. Av ([vl, t) from electrons of speed Ivl to the current density must, on removal of the external field, decay with time proportionately to the n u m b e r of electrons which have not yet suffered collision. Thus, b y (2),
i(Ivl, t) oc exp [-- tit (Ivl)]
(3)
This deduction will later be required in establishing t h a t the q u a n t i t y
• (Ivl), which appears b o t h in the pictorial kinetic m e t h o d and in the Boltzm a n n equation, has precisely the same meaning in the two in spite of being introduced very differently. Method of evaluating drift velocity. If a , ( t ' ) i s the acceleration in the z-direction of an electron at time t', the increase in drift velocity between t' and t' + dt' will be a~(t') dt'. By (2), the probability of an electron not suffering a collision - i.e. the probability of retaining its drift velocity - at least up till the time t (t > t') is e-(t-v)/L so t h a t
Av~([vl, t) = f!oo a~(t') e -c~-t')lT dt' = f~o a~(t
-
-
T) e -r/~ d T ,
(4)
where T ~-- t -- t' is the interval of time which wilt elapse before an electron reaches the point at which the current is to be evaluated. The drift velocity is t h u s equal to the Laplace image in the T-Z-plane of the electron acceleration expressed in time-units measured backwards into the past from the time at which the current is to be evaluated *). *) In general,
Av(t) =
f~o
a(t --
T) e-- T/r dT.
673
PICTORIAL KINETIC METHODS
Examples of the evaluation o f Av. (a) In a uniform steady electric field Ez, the acceleration az = eE,/m does not depend on time, and furthermore the drift velocity cannot depend on the direction of the electron p a t h on account of the spatial u n i f o r m i t y of E,. Thus b y (4) eE, AV z --
m
t* oo
J
e-T/': d T = __er Ez. 0
(b) For a uniform alternating = (eEl~m) ei~(t-T~, so t h a t Av, = (eE,/m) f ~
e -T/"
(5)
m
electric
e''(t-T) d T =
field E , e~t,
az (t-
[ev/m(1 + iwv)] E , e'~t.
T)= (6)
(c) For an alternating electric field E,(x) e~°'~ which varies appreciably with the d e p t h x (e.g. within one mean free path) - the anomalous skin effect a, (t -- T) = (eE,/m) ei°(t-T) as in (b), b u t since E , = E,(x) it is now convenient to express each time interval T = t - t' in terms of the corresponding space interval x - x' travelled b y the electron. Since T = [ ( x ' - - x ) / v ~ l , d T = dx'/lv~[, a n d (4) gives Av: -- mlv,[
E:(x') exp
dx'. T
Vx
(7)
J
The limits of integration for x' which have been inserted in (7) take account of drift velocities incurred b y electrons travelling to the level x b o t h from all levels 0 < x' < x above x, and from all levels x' > x below x. It is assumed t h a t electrons reflected back into the metal after collision at the internal surface (located at x = 0) do not retain their previously acquired drift velocities. 2. Comparison with results obtained /rom the Boltzmann equation. Strict solutions of transport problems are obtained from the B o l t z m a n n equation. W h e n a collision time exists this m a y be written in the symbolic form
8/ ~/ 8--t-+v'-~ -+a"
~/ a~--
/ --/o _ ~
,,'t/ 3 '
(8)
where t----time, r = position vector, v = velocity, a = acceleration, ]0(r, v) = distribution function without fields, a n d / ( r , v, a) = distribution function in the presence of the fields responsible for the electron acceleration a. On this approach, the current density 1"~in the z-direction is, for an isotropic medium *), j~ = (2m3~/h3) f v~ / d3v = (2m3~/h 8) f v~ A~ d%.
(9)
*) I n general, j = ( 2 m S e / h S } f v A I dSv. Physica XXII
43
674
R.B.
DINGLE
Decay of current contribution on removal of e x t e r n a 1 f i e 1 d. The collision t i m e , appearing in (8) m a y be defined as a certain time of restoration of the distribution function ] to the value [0 when the disturbance A / = [ -- [o is uniform in space and no external forces are acting. For when these conditions appertain, (8) reduces to i.e. A / o c e -t/r, where in general ~ = ~(Ivl).
(10)
The contribution j(lvl, t) oc A[(Iv], t) from electrons of speed Iv] to the current density must therefore decay with time proportionately to e - t l * l l v l ) o n removal of the external fields. Since this is precisely the same result as contained in eq. (3), the collision time m u s t have identical meanings in the B o l t z m a n n m e t h o d and the pictorial kinetic method. Examples of the evaluation o f A[. (a) A uniform s t e a d y electric field E , causes an electron acceleration a, = eEl~m, and the operators a/at and a/ar are null on account of the u n i f o r m i t y in time a n d space. Hence b y (8) At =
-
• a, atlav, ~_ -
• a, alolav, =
-
( * * l m ) (atolav,) E , .
(b) For a uniform alternating electric field E, e~ , reduces to
(io~ + l/z)
A/~--
(11)
a/at =_ iw, so t h a t (8)
- - a , alo/aV,,
yielding AI =
-
[~,lm(1 +
io.,-~)] (alol~v,)
(12)
E , ~'"'.
(c) For an alternating electric field E , e~ v a r y i n g appreciably in the x-direction within one m e a n free p a t h - the anomalous skin effect - it is found t h a t 1) ee~"~ ~[o AI--
mlv~ I av,
F
E.(x') exp
o
{
1 + im~ ] ~
x' -- x
v~
}
dx',
(13)
provided t h a t electrons lose their drift velocity on reflection at the internal metal surface located at x ---- 0. Comparison between results of pictorial-kinetic and Boltzmann methods. Comparing (ll), (12) and (13) with (5), (6) and (7), it is seen t h a t AI -
-
(alolav,) A v , ,
(14)
at least for the instances considered *). Reference to (1) a n d (9) t h e n reveals *) If zJv had more than the one component /Ivz assumed in the text, it would obviously be necessary to add the contributory differentials: At ~ -- {(~/o/Ov.) zlv~ + (~/./~v~) /tv~ + (~/o/Ov~) live} = -- (~/o/Ov) . ~Iv.
PICTORIAL K I N E T I C METHODS
675
that if the two methods are to lead to the same values for the current density j,, the factor v , /I! =_ -
v,
'°~iv,
appearing in the integrand of (9) can only differ b y a perfect differential from the factor [o dv, appearing in the integrand of (I). Unfortunately, this is not in general the case, for
f v, ff/o/aV,) Av , dv, = [v, /iv, t0]M:-+: -- f /o Av, dv,-- f lo V, (~/iv dav,) dr,..(15) The first term on the right hand side of (15) is certainly zero, but the third term will only vanish if either of the following conditions is satisfied: (c¢) ~,4v,/av, = 0; i.e. if /Iv, does not contain v,. In the examples considered, this condition will be fulfilled if ~ is independent of the electron speed Iv]. Or (fl) O/Iv,/av, is an even function of v,, for ]0 is certainly even in v,; i.e. if /Iv, is an odd function of v,. This, however, would be physically quite absurd, since it would imply that electrons drift either with the field or against it depending on the direction of their initial thermal velocities! Moreover, even if it does not itself vanish, this third term in (15) cannot give zero contribution on integration over vx or v~ unless it is odd in vx or v~. Since ]0 is even in v~ and v~, this would entail t h a t / I v , be odd in vx or v~, which is again physically absurd for the same reason as in (fl) above. Thus it is to be concluded that the customary ]ormulation o] the pictorial
kinetic method is in agreement with the strict Boltzmann theory only when the collision time can be considered as independent o] velocity. For metals with degenerate electron systems, the strict Boltzmann approach shows that only the value Z(VFer,nl) at the Fermi surface is required, since the factor O]o/~V appearing in /I] is negligible except for v ~___vr~ni. Correct results will then be obtained from the customary pictorial kinetic method provided v is taken to be a constant equal to v(vFe~,~i). For semiconductors, however, the customary pictorial kinetic method gives erroneous results, since for them it is essential to take explicit account of the known variation of v with Iv].
3. Cause o] the discrepancy. Particular cases of the discrepancy revealed in the previous section have been noted in previously published work 2) 3), and it has been suggested that it m a y be attributed to neglect of the facts that during the electron acceleration the value of [0 will change 2) from /0(iv[) to /0(Iv + Av, I), and correspondingly the value of T change 3) from ,(Ivl) to ,(I v + Av,[), where iv I and Iv + Av,I are respectively the initial and final electron speeds. However, with Av, already appearing as a factor
676
R.B. DINGLE
in the integrand of eq. (1), these particular shortcomings in the pictorial kinetic theory can only explain discrepancies of order (Avz) ~, i.e. quadratic in Ez, whereas the discrepancy under discussion is already present in the first order. The major fallacy in the customary kinetic approach lies deeper, and can be traced to the invalidity of the superposition implied in the assumptions (i) and (ii) introduced at the beginning of § 1. In order to show that it is is in general not permissible to superpose on the distribution/o the additional drift velocity caused by the external field, it is to be noted that in general such a superposition might: (a) Violate those requirements of quantum theory which originally determined the very form of the statistical distribution function/0. To take an example, suppose that for some hypothetical system the number of accessible states decreases rapidly with increase in electron speed, and that practically all accessible states are occupied in the absence of an external field. The assumption of direct superposition would then altogether fail to take account of the essential fact that electrons could not pass into the inaccessible states even after the expected general increase in electron speed caused by a large electric field. (b) Fail to represent a steady state of the system. For even if the superposition did not involve actual violation of the requirements of quantum theory, as in (a), it would still in general only represent the transient state of affairs immediately succeeding the imposition of the external field. These objections show that the customary formulation of the kinetic approach cannot be universally valid. In order to determine the particular conditions for which the method, as it stands, would (by chance) lead to correct results, it is only necessary to observe that both objections (a) and (b) become nugatory when the distribution function / only differs from/o by either a uniform translational motion or a uniform rotational motion of the entire electron system, since it is known that such motions do not upset the statistical equilibrium 4). Since a value of the drift velocity Av, independent of v, would be equivalent to a uniform translational motion of the electron system with velocity Av~ in the z-direction, the customary kinetic approach will thus give correct results if Av, does not contain v,. This conclusion is in agreement with (~) of § 2. As noted there, such a condition will normally be fulfilled if the collision time is independent of electron velocity.
4. Previous attempts to evolve a correct pictorial kinetic method. I. It has been suggested ~) that since the contribution to the current is proportional to the change in the number of electrons (oc/0) of given speed consequent upon the presence of the field E,, and to the velocity v,, the product/0. Av, appearing in the integrand of eq. (1) should simply be replaced by the
PICTORIAL KINETIC METHODS
677
product
v~ A/o = v~(~/o/~V~) Av~. This "explanation" is unfortunately subject to grave objections: (i) In the strict derivation of the statistical distribution function/0 it is assumed that either (a) the momentum of the system as a whole, P say, is completely unknown, in which case the statistical average momentum P vanishes since all directions of P are equally likely, or (b) the momentum of the system as a whole is known to be identically zero, i.e. P ---- 0. It is therefore logica!ly unsound in any pictorial kinetic theory to apply the distribution function /0 directly to a system of electrons known to be contributing a current, since then it would be known that _P ve 0 in contradiction to both possibilities (a) and (b). The net current therefore cannot logically be determined b y simply imagining a change A/0 to occur in the distribution function/0 as a result of the imposition of an external field *). (ii) The implication that explicit account must be taken of the change in /0 occurring while the velocity of an electron is increasing due to the presence of the external field, on an average from v to v -6 Avz, gives rise to a number of serious conceptional difficulties. For instance, (~/o/SV,) Av, is presumably intended as an approximation to/0(v -6 Av,) --/0(v), of which only the first t e r m / o ( v -6 Av,) contributes to the current. But why should the weighting factor be taken as that appropriate to the final velocity v -6 Av, ? What happens to the drift momentum of electrons which, according to the variation in/o from start to finish, do have to be considered at the commencement of their path, but not at the end ? (The effect of collisions has already been taken into account in the calculation of Avz). On what grounds is it supposed that all those electron states which must be considered according to the final distribution function/0(v -6 Avz) taken, have all 'survived' while the argument of [o changed from v to v -6 Av, ? For if they did not 'survive' throughout, the acquired average drift velocity cannot have attained the full calculated value Av,. (iii) The suggested replacement of/o.Av, in eq. (1) b y v,.A/o = v,(a/o/~V,). •Av~ leads to a current of the wrong sign, as m a y be seen b y comparing equations (1), (9) and (14). As will become clear later, in § 5, the reason for this incorrect sign is that in reality the current is not determined b y the change in/o(V) caused b y the drift velocity Av,(v) as suggested in 2) ; instead, it is *) I t should be r e m a r k e d here t h a t it is implied in ~) t h a t allowance is to be made for the fact t h a t / o (Ivl) ~ ]o (Iv -t- zJvz[), b u t the m o d u l u s sign then appears to have been i n a d v e r t e n t l y dropped. If it had been retained, it would have been found t h a t on such a basis v, A/o = v, (8/o/a Iv, I) Av,, Av, always being of uniform sign for electrons of all velocities; since ]o is even in vz, this would h a v e led to the conclusion t h a t the current is always zero!]
678
R . B . DINGLE
determined by the velocity --Av,(v) which would have to be superposed on the electrons in order to permit the use of the stationary equilibrium distribution function/o(v). II. A mode of approach has been proposed 3) which involves calculations of the following quantities: (a) Change in probability of collision in the presence of an external field. (b) Average displacement in the direction of the applied field of an electron with given initial velocity. (c) Rate at which electrons are scattered into a differential volume of momentum space. (d) Statistical average of the sum of all electron velocities, and hence of the current. This, however, can hardly be classed as a true pictorial kinetic method, since in step (c) the collision t i m e , appears as a time of restoration of the distribution function / to the value/0, a concept typical of the Boltzmann approach. [3 appears also in the pictorial kinetic sense in steps (a) and (b)]. It is the aim of the present paper to show that the fallacy in the customary formulation of the kinetic approach m a y be removed merely by replacing the "equilibrium distribution function"/o in (1) by a "transport distribution function"/t; the quantity ~ will thus appear only in the true pictorial kinetic sense involved in evaluating the drift velocity Av, in accordance with eq. (4). III. General formulae have been derived 4) by finding / by integrating up the number of electrons scattered into a trajectory at previous points along it -- proportional to/o (E -- AE), where AE is the energy remaining to be acquired from the applied fields before the electrons reach their destination -- weighted by their probability of reaching this destination. But in this derivation, the concept of 'drift velocity', perhaps the most interesting contribution of the customary pictorial kinetic approach to the full understanding of the 'mechanistics' of the problem, has been abandoned -- though of course it could be reintroduced artificially.
5. Re/ormulation o/ the pictorial kinetic method. In reformulating the pictorial kinetic method, it is necessary to start from the relation (9), viz. i, : (2m3e/h3)f v,. A/dZv,
(16)
since it has already been demonstrated in § 3 that eq. (1) is subject to the superposition fallacy. The remaining problem is then to determine A/from the quantity Av, calculated on the kinetic approach. If the electrons in the statistical system' exhibited no drift velocity (Av, --= 0), the distribution function would be/o(E), where E is the electron energy. The required distribution function / : / 0 + A/applicable when the electrons do exhibit a drift velocity Avz, relative to a stationary observer, m a y be determined from the consideration that the distribution function
PICTORIAL
KINETIC
METHODS
679
will still be ]o according to an observer moving with velocity Av e, since such an observer would be unaware of the drift velocity. In the general case in which A v e is a function of Ivl, this observer is to be imagined as moving with the appropriate drift velocity Av,(Ivl) when observing electrons of thermal speed tvl. Thus
/(e) =/0(e'),
(17}
where 8' is the apparent energy (Hamiltonian) of an electron according to the observer moving with velocity A v , , whilst ~ is the energy of the same electron according to the stationary observer. Now it is known'that n) 6) e'
e
=
p.Av,
--
(18)
where p = momentum of particle or excitation, and Av = drift velocity. Hence by (17)
l(e)
=
lo(e
-
p, ~v,),
(19)
so that ~1 = 1 - - 1o ~-- -
P , A,,, ~to(e)l~e.
(20)
Assuming that the electrons behave as though free, with an effective mass m , p , = m y , , £ = ½mY 2, and v2 = v~ + v~ + v~, whence At =
- - v , Ave O/o/v av =
- - (~lo/~V,) A v , .
(21)
Inserting this expression for A / i n t o (16), it is seen that i,
(2raze/ha) f h Ave day,
=
(22)
where the "transport distribution function" /t is defined by *) It
=
--
v,
~/olbv,
(23)
and A v e is calculated by pictorial kinetic methods according to eq. (4). The result of our reformulation thus differs from that of the customary formulation only by the replacement of the stationary equilibrium distribution function [o in the current-relation (1) by the "transport distribution function"/t, equal to - - v z ~[o/OV, when the field and current are in a direction parallel to the z-axis. This reformulation of the pictorial kinetic method is in complete accord with the strict Boltzmann theory, the replacement of [0 by /t having removed the superposition fallacy inherent in the customary formulation. *) If z l v h a d m o r e t h a n o n e c o m p o n e n t - - see f o o t n o t e s to (9) a n d (14) - - it w o u l d be n e c e s s a r y t o d e f i n e It as t h e o p e r a t o r [ t = - - v (aldOv). ( N o t e t h a t w h i l e c o n t r i b u t i o n s to t h e d i f f e r e n t i a l q u a n t i t y / I ] c a n be a d d e d t o g e t h e r , n o s u c h sinple l a w of a d d i t i o n a p p l i e s t o t h e n o n - d i f f e r e n t i a l q u a n t i t y I t ) .
680
PICTORIAL KINETIC METHODS
6. Averaging over the distribution/unction. Since b y (22) the contribution j(v) from electrons of Velocity v to the current density is proportional t o / t and not to [0 (as often erroneously assumed), it follows that in any averaging over the velocity or energy distribution of quantities involved in the theory of transport phenomena, the appropriate distribution function is ]t and not /o- For instance, the relation o = ns%/m for the d.c. conductivity of a metal should be generalised to read = ( m 2 / m ) f , / , d3v/f/t d3v
(24)
for a semiconductor. Again, the Drude-Kronig optical absorptivity A = = (miens2) ½z -i for a metal in the near infra-red should be generalised to read A = (m/~n~2)½f ~-1/~ d 3 v / f / , d3v (25) for a semiconductor. Further examples of this convenient technique will be found in a forthcoming paper entitled "The anomalous skin effect and the optical absorptivity of semiconductors, I". Received 2-2.56.
REFERENCES I) R e u t e r , G . E . H . and S o n d h e i m e r , E . H . , P r o c . roy. Soc. A I 9 5 ( 1 9 4 8 ) 336. 2) C h a m b e r s , R . G . , P r o c . roy. Soc. A 2 0 2 ( 1 9 5 0 ) 380. 3) S h o c k 1 e y, W., Electrons and Holes in Semiconductors (1950), va n Nostrand, N.Y., p. 293. See also the detailed discussion of drift velocities in H u x l e y, L. G. H., Proc. phys. Soc. B 64
(1951) 844. 4) C h a m b e r s , R. G., Proc. phys. Soc. A 6 5 (1952) 458. 5) L a n d a u, L. D. and L i f s h i t z, E., Statistical Physics (1938), Oxford U ni ve rs i t y Press. 6) D i n g l e , R. B., Advances in Physics I (1952) 138.