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Diffusion of light interstitials in metals
Physica 125B (1984) 279-285 North-Holland, Amsterdam
D I F F U S I O N O F L I G H T I N T E R S T I T I A L S IN M E T A L S Jun K O N D O Electrotechnical Laboratory, Tsukuba Research Center, Ibaraki-ken, 305, Japan Received 30 April 1984 The muon tunneling among interstitial sites in metals is dressed by the cloud of conduction electrons and has a temperature-dependent tunneling matrix. Its hopping frequency is calculated by taking account of interaction both with conduction electrons and with lattice distortion. It is activation-type at high temperatures, whereas it is proportional to T 2r-1 at low temperatures. K is a coupling constant with electrons. The factor T 2K came from the orthogonality catastrophe, while T -1 from another infrared catastrophe. Due to this factor, the hopping frequency shows a minimum as a function of the temperature, in agreement with experiments.
1. Introduction
the electronic wave function to occur is [4]
In previous papers we have considered h e a v y particles m o v i n g in metals [1-4]. T h e particles are a s s u m e d both to have an interaction with c o n d u c t i o n electrons and to feel a periodic potential f r o m the atomic lattice, which has been a s s u m e d to be rigid. W e have distinguished two cases. W h e n the periodic potential is slowly varying, the screening charge of the c o n d u c t i o n electrons follows the particle a l m o s t adiabatically and there is no mass correction for the particle [4]. T h e r e is, however, a vertex correction in this case, so that the scattering probability of the particle by electron collision involves a factor which gives rise to a t e m p e r a t u r e - d e p e n d e n t mobility of the particle [2]. In the o t h e r case, the periodic potential m a y consist of a periodic array of d e e p potential wells. T h e particle will m o v e a r o u n d very fast in each of t h e m and j u m p to n e i g h b o u r i n g wells after a short time. W h e n the m o t i o n in a well is very fast, the electron cloud c a n n o t follow the particle, but is localized. By this, we m e a n that the electronic wave function does not involve the c o o r d i n a t e of the particle as a p a r a m e t e r , but is identical to what o n e would obtain w h e n the particle is fixed at the center of the well (with a r e d u c e d e l e c t r o n - p a r t i c l e interaction) [4]. T h e condition for the localization of
M k 2 b 2 ~ M In ~ - ,
(1)
w h e r e m / M is the mass ratio and b the radius of the particle w a v e function. T h e a b o v e description is for the case when the particle stays within a well. W h e n it j u m p s to a n e i g h b o u r i n g site, the electronic wave function must be r e a r r a n g e d so that it is c e n t e r e d a r o u n d the new site. Here, the o r t h o g o n a l i t y c a t a s t r o p h e [5] c o m e s in with the result that the transfer integral A for the j u m p acquires a factor ( T / D ) r at T > A or ( A / D ) r at T < A [1], w h e r e K is an e l e c t r o n - p a r t i c l e coupling, D the b a n d width (or the fermi energy) of the c o n d u c t i o n electrons. F o r c h a r g e d particles, K m a y not be small c o m p a r e d with unity, so that this is r a t h e r a large effect, if A is smaller than, say, 0.01 eV. Now, such an effect m a y be seen in diffusion processes of light interstitial a t o m s in metals. This was already predicted in [1]. T h e m u o n is especially suited to this, because the condition (1) seems to be satisfied for it. A complication arises f r o m the fact that it is also c o u p l e d to the lattice distortion, so that its diffusion process is d o m i n a t e d by thermally activated h o p p i n g process. A t low temperatures, however, w h e r e
0378-4363/84/$03.00 O Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
280
J. Kondo / Diffusion of light interstitials in metals
the activation ceases to predominate, we may expect that quantum tunneling becomes the main process. Recent experiments revealed that the diffusion constant of the positive muon in copper and aluminum shows a minimum as a function of the temperature at around 10 K [6-8]. We consider this to be due to a cross-over from the activation to the quantum tunneling. Thus, below the minimum, the particle tunnels to neighbouring sites, where we may expect the above mentioned effect to be seen. An important thing to r e m e m b e r is that, in this temperature region, the experiments show the diffusion constant to increase as the temperature is decreased. The purpose of this paper is to explain this fact in terms of the electron-particle interaction, as well as to see the part played by the ( T / D ) K effect in the temperature dependence of the diffusion constant. The first step is to decide whether the particle motion in the low-temperature region is coherent or hopping. Since the transfer integral increases with T as T K, the hopping seems to contradict the experiments. On the other hand, if the particle moves coherently and is scattered by conduction electrons, the scattering probability will increase with T. This agrees with the experiments. Recently, Yamada [9] made a discussion along this line of argument. However, from consideration of the magnitude of the parameters, the coherent motion seems to be impossible. One parameter is K, or more conveniently, K . D, the gain of the energy due to the electronparticle interaction. The other is A, the gain due to the coherent motion. As we mentioned, K is not much smaller than unity, whereas A / D may be 10-4 or less. (Later, we will see A / D ~ 10 7 for the muon in copper.) This indicates that the motion must be hopping. In the following we present a resolution of this contradiction. We will calculate the hopping probability by taking account of both the electron-particle and the particle-lattice interactions, and will explain the overall feature of temperature dependence of the diffusion constant. In the low temperature region, we will have an extra factor T -~ arising from a fermi surface effect, which resolves the contradiction.
2. Interaction with lattice
We consider the process where the particle hops from site R1 to R2. When it is at R~ (i = 1,2), the lattice system will be described by a Hamiltonian H ~ ), and the electron system by HI e). The total Hamiltonian is Hi = H ~ ) + H I ~. The probability W per unit time for the jump from R1 to R2 to occur is, to the lowest order of A, o~
W = A: ~ q~(t) at,
(2)
where
¢'(t)= (e i"'' e-~"2')1. ( ' " ) 1 denotes Tr(e oH,...)/Tr(e-OUl). Since the electron system and the lattice system are independent, q~(t) factorizes as q~(t)= qb~)(t). @~)(t).
(3)
Let us first discuss q~(P), which has been treated by many authors [10, 11]. We assume H ~i ) = ~toqbtqbq +- ~ (Aq/2)(bq + btq), q
(4)
q
w h e r e + a n d - c o r r e s p o n d to i = l and i = 2 , respectively. Let the maximum of ~oq be o~D. We find q~(P)(t) = e x p / - "~ (Aq/wq) 2 t
x [(1 - cos wqt)(1 + 2nq) + i sin ~0qt]/ -= exp{- F(P)(t)},
(5)
where the last equality defines F°')(t). (F(e)(t) will be defined similarly.) nq is the Planck distribution with wq. When wot ~ 1, we have FO')(t) = O2t 2 +'IEFc ,(P) t where G 2 = ~ (A~/2)(1 + 2nq),
(6)
J. K o n d o / Diffusion of light intersfitials in metals
F o r [t[~ % the sine and cosine t e r m s oscillate fast and tend to zero when integrated o v e r q: lim F~)(t) = ~ (Aq/wq)2(1 + 2nq) =- 2St. Itl--,~
(7)
Let us introduce a s p e c t r u m J(og):
J(og) =
~ , A ~ 6 ( w - ogq).
(8)
q
In o r d e r to simplify our model, we a s s u m e the following f o r m of J(og):
281
(10), so is m u c h larger than unity. Now, when one uses (6) at It[ = wB ', the real part of F °') b e c o m e s G2/o92, which is 2Sr/3 at T >> OgD and is m u c h larger than unity. T h u s qb°')(t) decreases practically to zero when It[ reaches ogB~. W e can use (6) in (5), which is then used in (2) and integrated to t = -+ oo. (Here, we assume qb(,)= 1.) This is H o l s t e i n ' s p r o c e d u r e to obtain the activation-type diffusion constant. Since the activation energy turns out to be Erc/4, (P) the range of t e m p e r a t u r e w h e r e the activation-type b e h a v i o r is o b s e r v e d is OgD"~ T < E F ~ 4 .
hw
J(w)=
0 ~< o9 ~< ogD,
0
OgD
At T ~ OgD, on the o t h e r hand, we find
For this form we find
Sr =
{,
iim O°')(t)=
2
~AOgD= S AOgDT
T = 0, T ~" OgD,
T = 0 ' T ~ OgD,
[1SO92
o,)
~EFc T EF~C) =
(15)
e -2s ,
Itl~
(10)
S r e p r e s e n t s a dimensionless particle-lattice interaction. W e also find
G2 =
(14)
(9)
(4/3)SOgD.
(11)
(12)
With (9), we also have
which m a y not be very small. T h e integral in (2), then, diverges. This implies that the q u a n t u m tunneling is now taking place, and here we must consider the interaction with the conduction electrons and see what effect the interaction has on the tunneling m o t i o n of the particle.
3. I n t e r a c t i o n w i t h e l e c t r o n s
W e assume that, when the particle is at Ri, the H a m i l t o n i a n of the electron system is expressed
toD
as
F~)(t) = A J o)[(1 - cos ogt)(2n(og)+ 1) ]4!e) __,
0
= E
"t e~a~a~,+ ~ Vi(k, k ')a~,ak,~, *
ka
(16)
kk'o"
+ i sin tot] do9
= 2S[Ra(T/Ogo, OgDt) + iR2(ogot)],
(13)
R~(x, y) = 1 + (2/y2)(1 - cos y - y sin y) 1 +
4 J so(1 - cos ~y)/(e e/x - 1) dsc ,
where V~(k, k') = V0 e i(k'-k)'R' •
W e define V by °,-
e)
0
R2(y) = (2/yZ)(sin y - y cos y ) . S m a y be e x p e c t e d to be larger than unity. At T >>OgD, Sr is m u c h larger than S as is seen f r o m
= ~ , [V2(k, k ' ) -
V,(k, k')]a~a,,~,
and e x p a n d O(o(t) as follows:
(17)
J. Kondo / Diffusion of light interstitials in metals
282
00
ff
@e)(t)= 1 + ~ ( - i ) "
t>tl>..-tn>O
x (V(t,)...
V(t.))l dtl..,
dt.,
(18)
(a)
(b)
where
Fig. 2. (V(tt)V(t2))l is represented by the sum of (a) and (b).
V ( t ) = eiH~)'Ve-i"~)'
where fk is the fermi distribution function. W h e n (20) is inserted in (19) and the integrations over tl and t2 are p e r f o r m e d , there occurs a term linear in t, which cancels the E~t term. In this way we obtain
For n = 1, we have
(V(I1))I~E~. This is r e p r e s e n t e d by a loop as in fig. 1. For n = 2, we have two terms:
(V(q)V(t2))l
= (E~)2+
x [1 - cos(et -
- fk,)
er)t + i sin(e~ - ek,)t].
Using the same p r o c e d u r e as in [I], we find D
F(e'(t) =
2K f f (e-e')-2f(e)[1-f(e')] -D
× [1 - cos(~
+ i sin(e -
-
~')t
e')t] de de',
(21)
where /
K
= 2~o0 2 [1 \
s i £ kFa'~ kZa 2 ] "
t1
p is the density of conducting states, D is the band width, and a is the distance between R1
dp(e)(t)=exp[-iE~)ct-f dt, f dt2G(t,-t2)]. 0
a n d RE.
(19) T o be consistent with our p r o c e d u r e of keeping only the most divergent terms, we calculate G correct to the order of V200:
G(t) -- 2 ~
k')- V~(k, k')[z
kk'
t2).
T h e first one is r e p r e s e n t e d by two loops (fig. 2(a)), and the second by a bubble (fig. 2(b)). T h e term with n > 2 is r e p r e s e n t e d by sum of m a n y diagrams in which there are m o r e than one loop a n d / o r m o r e than one bubble. F u r t h e r m o r e , there will be diagrams where m o r e than two vertices are c o n n e c t e d by a closed line (an example in fig. 3). W e neglect these terms. This a m o u n t s to keeping only the most strongly divergent terms. Thus, examples of the retained diagrams for n = 4 are shown in fig. 4(a), and those of a b a n d o n e d ones in fig. 4(b). A d d i n g all the most divergent terms, we find
0
2 ~ I V2(k,
x (~k - ~ , ) - 2 f ~ ( 1
G(tl-
t
Fte)(t) =
I V2(k, k') - V,(I, k')l 2
Now, it is convenient to change the integration variables in (21) to t o - e - e ' and e. T h e integration over e can be p e r f o r m e d : D
F(e)(t) =
2 K f to-l[(1 -- COS tot)(1 + 2n(to)) 0
kk'
X ei(e'-~')tfl(1 -- f l , ) ,
(20)
+ i sin tot] dto.
(22)
22 Fig. 1. A loop representing (V(t)h.
Fig. 3. A closed loop connecting more than two vertices.
J. Kondo / Diffusion of light interstitials in metals
AL©
integral. On the other hand, T -1 came from the second factor of (25). Because of this factor, the diffusion constant W a 2 decreases as the temperature goes up. (An argument that K does not exceed 1/2 was presented by Yamada et al. [12].)
AA9
(a) (bl Fig. 4. (a) Retained fourth-order diagrams. (b) Abandoned fourth-order diagrams.
Comparing this with (13), one sees that (22) is obtained for a phonon system with the spectrum J(o~) = 2Ko) and the cut-off of D. This is nothing but the Tomonaga phonon. The integration of o~ in (22) can also be done: FCO(t) = 2K ln{[(sinh 7rTt)/(~Tt)]
x X/1 + D2I 2} + 2i K tan -1 D t . For
Itl >
(23)
T-l, we have
(24)
or =
(2zrT/D) zK e-Z,,Krl'l.
(25)
The first factor is the square of the overlap integrals between two electronic wave functions and correspond to e -zsr in the previous section. Here, however, we have another factor, which makes 4~te)(t) tend to zero as I t l ~ . This comes from the fact that the sine and cosine terms in (21) cannot be neglected even when Itl is very large. This is due to the vanishing of the denominator at e = e', and may be called a fermi surface effect. When the integration over t in (2) is performed (assuming ~tp)= 1), we have, for small K, W=
A2COS zrK (27)'x~2K T2K_ 1 ~K \D/
4. Both interactions combined
Now, we discuss the diffusion constant when both interactions are present. We simply use (13) and (23) in (2) and integrate over t. In the general case, numerical integration is inevitable. Before doing so, let us discuss limiting cases. When (14) is satisfied, one sees that TrTG - l = X/~TT/E~F~C~ 1, and so (sinh ~rTG-I)/(rrTG -~) = 1. Now, from (6), we observe that the practical limit of t integration in (2) may be several times G -1. Within the limit, F(O(t) can be approximated by F(°(t) = 2K In ~/1 + D2t 2 + 2i K tan -1 D t .
FCO(t) = 2 K ln(D/2TrT)+ 27rKTlt I + 7r i K sgn(t),
I~(=)(t)l
283
(26)
A general analytic form for K is also easily obtained. The T 2r factor came from the overlap
(27)
This means that the T K factor disappears. The relevant time scale of the lattice system is shorter than T -1. Thus, it is not allowed to represent the effect of the interaction with electrons by a modified transfer integral Ae, = A ( T / D ) K and to follow Holstein's procedure with this modified transfer integral. In any case, (27) has a rather weak t dependence, and W becomes of activation-type. When T ~ O)D, the integration in (2) must be extended to several times T -1. When It] reaches wB1, ~(P) becomes - e -2s. The main contribution to the integral comes beyond ~oB1. Thus, it is allowed to represent the effect of the interaction with lattice by an effective transfer integral de~ = A e -s and follow the procedure of the previous section. In the general case we have made numerical integration using (13) and (23). The results will be expressed as
AZH( T-ff--,D,K,S ) W : --~ \O)DG tO
(28)
As an example, we show in fig. 5 the values of H
J. Kondo / Diffusion of light interstitials in metals
284
of singularity. chain. Let u, atom from its interested in
It is a one-dimensional harmonic be the displacement of the nth equilibrium position, then we are ((u,-u,,)2). At T = 0, we have ((u,,-u,,,)2)~lnln-n'l, while at finite temperatures, we have
10 2
s=0 10 ~
10 °
( ( u , - u,,) 2) : ( ~ m o w o ) - ' [ l n ( w o / T ) + T i n - n'l/Wo] . One can compare this with (24). The role played by the second term in dynamic response of a harmonic chain was pointed out by E m e r y and Axe [13]. It should also play an important role in MiSssbauer spectrum [14] and in optical spectrum [15] in a one-dimensional chain.
10-1 H
1021 10-3
/
Acknowledgements 1
0
-
4
~
,. ~. _.J
10"4
........ , | 10-3
.
.
.
.
.
.
d i i iiiiiii I I IP ,,,,I . . . . . . . . 10-2 10-~ 10° 101
The author expresses K. Y a m a d a for pointing for helpful discussion. thanks to Dr. S. A b e for
his thanks to Professor out to him ref. [8] and H e also expresses his useful discussion.
T (,,O D
Fig. 5. Normalized hopping frequency H (see (28)) vs. T/coD for K = 0.3 and D/coD = 100. The dotted part contains uncertainty arising from numerical error.
References [1] [2] [3] [4] [5] [6]
with K = 0.3 and D / o ) D = 100 together with the experimental points from ref. [8]. The straight lines at the low t e m p e r a t u r e region are represented by T 2K-I. The interval between them is governed by e -2s. As seen in the figure, the result of numerical calculation predicts a minimum. An agreement of absolute magnitudes is obtained for rOD = 625 K, D = 6.25 × 104 K, A = 9 . 6 K as well as for S = 5 and K = 0.3. These values are quite reasonable.
[7]
5. Discussion
[9]
What is the physics of the factor T-l? Its origin is rather singular, and seems to be difficult to be mentioned in a simple term. Here, we will mention another system which has the same kind
[8]
[10]
[11]
J. Kondo, Physica 84B (1976) 40. J. Kondo and T. Soda, J. Low Temp. Phys. 50 (1983) 2l. J. Kondo, Physica 123B (1984) 175. J. Kondo, Physica 124B (1984) 25. P.W. Anderson, Phys. Rev. Lett. 18 (1967) 1049. C.W. Clawson, K.M. Crowe, S.E. Kohn, S.S. Rosenblum, C.Y. Huang, J.L. Smith and J.H. Brewer, Physica 109-110B (1982) 2164. K.W. Kehr, D. Richter, J.-M. Welter, O. Hartmann, E. Karlsson, L.O. Norlin, T.O. Niinikoski and A. Yaouanc, Phys. Rev. 26B (1982) 567; J.-M. Welter, D. Richter, R. Hempelmann, O. Hartmann, E. Karlsson, L.O. Norlin, T.O. Niinikoski and D. Lenz, Z. Phys. 52B (1983) 303. R. Kadono, J. Imazato, K. Nishiyama, K. Nagamine, T. Yamazaki, D. Richter and J.-M. Welter, Hyperfine Interactions 17-19 (1984) 109. K. Yamada, at a meeting at Osaka University, Dec. 1983. T. Holstein, Ann. Phys. 8 (1959) 325 and 343; C.P. Flynn and A.M. Stoneham, Phys. Rev. B1 (1970) 3966; H. Teichler, Phys. Lett. 64A (1977) 78. H. Sumi, J. Phys. Soc. Japan 49 (1980) 1701.
J. Kondo / Diffusion of light interstitials in metals [12] K. Yamada, A. Sakurai and M. Takeshige, Prog. Theor. Phys. 70(1983) 73. [13] V.J. Emery and J.D. Axe, Phys. Rev. Lett. 40 (1978) 1507.
285
[14] J. Kondo, unpublished. [15] S. Abe, Thesis for Master's degree, University of Tokyo (1977).
D I F F U S I O N O F L I G H T I N T E R S T I T I A L S IN M E T A L S Jun K O N D O Electrotechnical Laboratory, Tsukuba Research Center, Ibaraki-ken, 305, Japan Received 30 April 1984 The muon tunneling among interstitial sites in metals is dressed by the cloud of conduction electrons and has a temperature-dependent tunneling matrix. Its hopping frequency is calculated by taking account of interaction both with conduction electrons and with lattice distortion. It is activation-type at high temperatures, whereas it is proportional to T 2r-1 at low temperatures. K is a coupling constant with electrons. The factor T 2K came from the orthogonality catastrophe, while T -1 from another infrared catastrophe. Due to this factor, the hopping frequency shows a minimum as a function of the temperature, in agreement with experiments.
1. Introduction
the electronic wave function to occur is [4]
In previous papers we have considered h e a v y particles m o v i n g in metals [1-4]. T h e particles are a s s u m e d both to have an interaction with c o n d u c t i o n electrons and to feel a periodic potential f r o m the atomic lattice, which has been a s s u m e d to be rigid. W e have distinguished two cases. W h e n the periodic potential is slowly varying, the screening charge of the c o n d u c t i o n electrons follows the particle a l m o s t adiabatically and there is no mass correction for the particle [4]. T h e r e is, however, a vertex correction in this case, so that the scattering probability of the particle by electron collision involves a factor which gives rise to a t e m p e r a t u r e - d e p e n d e n t mobility of the particle [2]. In the o t h e r case, the periodic potential m a y consist of a periodic array of d e e p potential wells. T h e particle will m o v e a r o u n d very fast in each of t h e m and j u m p to n e i g h b o u r i n g wells after a short time. W h e n the m o t i o n in a well is very fast, the electron cloud c a n n o t follow the particle, but is localized. By this, we m e a n that the electronic wave function does not involve the c o o r d i n a t e of the particle as a p a r a m e t e r , but is identical to what o n e would obtain w h e n the particle is fixed at the center of the well (with a r e d u c e d e l e c t r o n - p a r t i c l e interaction) [4]. T h e condition for the localization of
M k 2 b 2 ~ M In ~ - ,
(1)
w h e r e m / M is the mass ratio and b the radius of the particle w a v e function. T h e a b o v e description is for the case when the particle stays within a well. W h e n it j u m p s to a n e i g h b o u r i n g site, the electronic wave function must be r e a r r a n g e d so that it is c e n t e r e d a r o u n d the new site. Here, the o r t h o g o n a l i t y c a t a s t r o p h e [5] c o m e s in with the result that the transfer integral A for the j u m p acquires a factor ( T / D ) r at T > A or ( A / D ) r at T < A [1], w h e r e K is an e l e c t r o n - p a r t i c l e coupling, D the b a n d width (or the fermi energy) of the c o n d u c t i o n electrons. F o r c h a r g e d particles, K m a y not be small c o m p a r e d with unity, so that this is r a t h e r a large effect, if A is smaller than, say, 0.01 eV. Now, such an effect m a y be seen in diffusion processes of light interstitial a t o m s in metals. This was already predicted in [1]. T h e m u o n is especially suited to this, because the condition (1) seems to be satisfied for it. A complication arises f r o m the fact that it is also c o u p l e d to the lattice distortion, so that its diffusion process is d o m i n a t e d by thermally activated h o p p i n g process. A t low temperatures, however, w h e r e
0378-4363/84/$03.00 O Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)
280
J. Kondo / Diffusion of light interstitials in metals
the activation ceases to predominate, we may expect that quantum tunneling becomes the main process. Recent experiments revealed that the diffusion constant of the positive muon in copper and aluminum shows a minimum as a function of the temperature at around 10 K [6-8]. We consider this to be due to a cross-over from the activation to the quantum tunneling. Thus, below the minimum, the particle tunnels to neighbouring sites, where we may expect the above mentioned effect to be seen. An important thing to r e m e m b e r is that, in this temperature region, the experiments show the diffusion constant to increase as the temperature is decreased. The purpose of this paper is to explain this fact in terms of the electron-particle interaction, as well as to see the part played by the ( T / D ) K effect in the temperature dependence of the diffusion constant. The first step is to decide whether the particle motion in the low-temperature region is coherent or hopping. Since the transfer integral increases with T as T K, the hopping seems to contradict the experiments. On the other hand, if the particle moves coherently and is scattered by conduction electrons, the scattering probability will increase with T. This agrees with the experiments. Recently, Yamada [9] made a discussion along this line of argument. However, from consideration of the magnitude of the parameters, the coherent motion seems to be impossible. One parameter is K, or more conveniently, K . D, the gain of the energy due to the electronparticle interaction. The other is A, the gain due to the coherent motion. As we mentioned, K is not much smaller than unity, whereas A / D may be 10-4 or less. (Later, we will see A / D ~ 10 7 for the muon in copper.) This indicates that the motion must be hopping. In the following we present a resolution of this contradiction. We will calculate the hopping probability by taking account of both the electron-particle and the particle-lattice interactions, and will explain the overall feature of temperature dependence of the diffusion constant. In the low temperature region, we will have an extra factor T -~ arising from a fermi surface effect, which resolves the contradiction.
2. Interaction with lattice
We consider the process where the particle hops from site R1 to R2. When it is at R~ (i = 1,2), the lattice system will be described by a Hamiltonian H ~ ), and the electron system by HI e). The total Hamiltonian is Hi = H ~ ) + H I ~. The probability W per unit time for the jump from R1 to R2 to occur is, to the lowest order of A, o~
W = A: ~ q~(t) at,
(2)
where
¢'(t)= (e i"'' e-~"2')1. ( ' " ) 1 denotes Tr(e oH,...)/Tr(e-OUl). Since the electron system and the lattice system are independent, q~(t) factorizes as q~(t)= qb~)(t). @~)(t).
(3)
Let us first discuss q~(P), which has been treated by many authors [10, 11]. We assume H ~i ) = ~toqbtqbq +- ~ (Aq/2)(bq + btq), q
(4)
q
w h e r e + a n d - c o r r e s p o n d to i = l and i = 2 , respectively. Let the maximum of ~oq be o~D. We find q~(P)(t) = e x p / - "~ (Aq/wq) 2 t
x [(1 - cos wqt)(1 + 2nq) + i sin ~0qt]/ -= exp{- F(P)(t)},
(5)
where the last equality defines F°')(t). (F(e)(t) will be defined similarly.) nq is the Planck distribution with wq. When wot ~ 1, we have FO')(t) = O2t 2 +'IEFc ,(P) t where G 2 = ~ (A~/2)(1 + 2nq),
(6)
J. K o n d o / Diffusion of light intersfitials in metals
F o r [t[~ % the sine and cosine t e r m s oscillate fast and tend to zero when integrated o v e r q: lim F~)(t) = ~ (Aq/wq)2(1 + 2nq) =- 2St. Itl--,~
(7)
Let us introduce a s p e c t r u m J(og):
J(og) =
~ , A ~ 6 ( w - ogq).
(8)
q
In o r d e r to simplify our model, we a s s u m e the following f o r m of J(og):
281
(10), so is m u c h larger than unity. Now, when one uses (6) at It[ = wB ', the real part of F °') b e c o m e s G2/o92, which is 2Sr/3 at T >> OgD and is m u c h larger than unity. T h u s qb°')(t) decreases practically to zero when It[ reaches ogB~. W e can use (6) in (5), which is then used in (2) and integrated to t = -+ oo. (Here, we assume qb(,)= 1.) This is H o l s t e i n ' s p r o c e d u r e to obtain the activation-type diffusion constant. Since the activation energy turns out to be Erc/4, (P) the range of t e m p e r a t u r e w h e r e the activation-type b e h a v i o r is o b s e r v e d is OgD"~ T < E F ~ 4 .
hw
J(w)=
0 ~< o9 ~< ogD,
0
OgD
At T ~ OgD, on the o t h e r hand, we find
For this form we find
Sr =
{,
iim O°')(t)=
2
~AOgD= S AOgDT
T = 0, T ~" OgD,
T = 0 ' T ~ OgD,
[1SO92
o,)
~EFc T EF~C) =
(15)
e -2s ,
Itl~
(10)
S r e p r e s e n t s a dimensionless particle-lattice interaction. W e also find
G2 =
(14)
(9)
(4/3)SOgD.
(11)
(12)
With (9), we also have
which m a y not be very small. T h e integral in (2), then, diverges. This implies that the q u a n t u m tunneling is now taking place, and here we must consider the interaction with the conduction electrons and see what effect the interaction has on the tunneling m o t i o n of the particle.
3. I n t e r a c t i o n w i t h e l e c t r o n s
W e assume that, when the particle is at Ri, the H a m i l t o n i a n of the electron system is expressed
toD
as
F~)(t) = A J o)[(1 - cos ogt)(2n(og)+ 1) ]4!e) __,
0
= E
"t e~a~a~,+ ~ Vi(k, k ')a~,ak,~, *
ka
(16)
kk'o"
+ i sin tot] do9
= 2S[Ra(T/Ogo, OgDt) + iR2(ogot)],
(13)
R~(x, y) = 1 + (2/y2)(1 - cos y - y sin y) 1 +
4 J so(1 - cos ~y)/(e e/x - 1) dsc ,
where V~(k, k') = V0 e i(k'-k)'R' •
W e define V by °,-
e)
0
R2(y) = (2/yZ)(sin y - y cos y ) . S m a y be e x p e c t e d to be larger than unity. At T >>OgD, Sr is m u c h larger than S as is seen f r o m
= ~ , [V2(k, k ' ) -
V,(k, k')]a~a,,~,
and e x p a n d O(o(t) as follows:
(17)
J. Kondo / Diffusion of light interstitials in metals
282
00
ff
@e)(t)= 1 + ~ ( - i ) "
t>tl>..-tn>O
x (V(t,)...
V(t.))l dtl..,
dt.,
(18)
(a)
(b)
where
Fig. 2. (V(tt)V(t2))l is represented by the sum of (a) and (b).
V ( t ) = eiH~)'Ve-i"~)'
where fk is the fermi distribution function. W h e n (20) is inserted in (19) and the integrations over tl and t2 are p e r f o r m e d , there occurs a term linear in t, which cancels the E~t term. In this way we obtain
For n = 1, we have
(V(I1))I~E~. This is r e p r e s e n t e d by a loop as in fig. 1. For n = 2, we have two terms:
(V(q)V(t2))l
= (E~)2+
x [1 - cos(et -
- fk,)
er)t + i sin(e~ - ek,)t].
Using the same p r o c e d u r e as in [I], we find D
F(e'(t) =
2K f f (e-e')-2f(e)[1-f(e')] -D
× [1 - cos(~
+ i sin(e -
-
~')t
e')t] de de',
(21)
where /
K
= 2~o0 2 [1 \
s i £ kFa'~ kZa 2 ] "
t1
p is the density of conducting states, D is the band width, and a is the distance between R1
dp(e)(t)=exp[-iE~)ct-f dt, f dt2G(t,-t2)]. 0
a n d RE.
(19) T o be consistent with our p r o c e d u r e of keeping only the most divergent terms, we calculate G correct to the order of V200:
G(t) -- 2 ~
k')- V~(k, k')[z
kk'
t2).
T h e first one is r e p r e s e n t e d by two loops (fig. 2(a)), and the second by a bubble (fig. 2(b)). T h e term with n > 2 is r e p r e s e n t e d by sum of m a n y diagrams in which there are m o r e than one loop a n d / o r m o r e than one bubble. F u r t h e r m o r e , there will be diagrams where m o r e than two vertices are c o n n e c t e d by a closed line (an example in fig. 3). W e neglect these terms. This a m o u n t s to keeping only the most strongly divergent terms. Thus, examples of the retained diagrams for n = 4 are shown in fig. 4(a), and those of a b a n d o n e d ones in fig. 4(b). A d d i n g all the most divergent terms, we find
0
2 ~ I V2(k,
x (~k - ~ , ) - 2 f ~ ( 1
G(tl-
t
Fte)(t) =
I V2(k, k') - V,(I, k')l 2
Now, it is convenient to change the integration variables in (21) to t o - e - e ' and e. T h e integration over e can be p e r f o r m e d : D
F(e)(t) =
2 K f to-l[(1 -- COS tot)(1 + 2n(to)) 0
kk'
X ei(e'-~')tfl(1 -- f l , ) ,
(20)
+ i sin tot] dto.
(22)
22 Fig. 1. A loop representing (V(t)h.
Fig. 3. A closed loop connecting more than two vertices.
J. Kondo / Diffusion of light interstitials in metals
AL©
integral. On the other hand, T -1 came from the second factor of (25). Because of this factor, the diffusion constant W a 2 decreases as the temperature goes up. (An argument that K does not exceed 1/2 was presented by Yamada et al. [12].)
AA9
(a) (bl Fig. 4. (a) Retained fourth-order diagrams. (b) Abandoned fourth-order diagrams.
Comparing this with (13), one sees that (22) is obtained for a phonon system with the spectrum J(o~) = 2Ko) and the cut-off of D. This is nothing but the Tomonaga phonon. The integration of o~ in (22) can also be done: FCO(t) = 2K ln{[(sinh 7rTt)/(~Tt)]
x X/1 + D2I 2} + 2i K tan -1 D t . For
Itl >
(23)
T-l, we have
(24)
or =
(2zrT/D) zK e-Z,,Krl'l.
(25)
The first factor is the square of the overlap integrals between two electronic wave functions and correspond to e -zsr in the previous section. Here, however, we have another factor, which makes 4~te)(t) tend to zero as I t l ~ . This comes from the fact that the sine and cosine terms in (21) cannot be neglected even when Itl is very large. This is due to the vanishing of the denominator at e = e', and may be called a fermi surface effect. When the integration over t in (2) is performed (assuming ~tp)= 1), we have, for small K, W=
A2COS zrK (27)'x~2K T2K_ 1 ~K \D/
4. Both interactions combined
Now, we discuss the diffusion constant when both interactions are present. We simply use (13) and (23) in (2) and integrate over t. In the general case, numerical integration is inevitable. Before doing so, let us discuss limiting cases. When (14) is satisfied, one sees that TrTG - l = X/~TT/E~F~C~ 1, and so (sinh ~rTG-I)/(rrTG -~) = 1. Now, from (6), we observe that the practical limit of t integration in (2) may be several times G -1. Within the limit, F(O(t) can be approximated by F(°(t) = 2K In ~/1 + D2t 2 + 2i K tan -1 D t .
FCO(t) = 2 K ln(D/2TrT)+ 27rKTlt I + 7r i K sgn(t),
I~(=)(t)l
283
(26)
A general analytic form for K is also easily obtained. The T 2r factor came from the overlap
(27)
This means that the T K factor disappears. The relevant time scale of the lattice system is shorter than T -1. Thus, it is not allowed to represent the effect of the interaction with electrons by a modified transfer integral Ae, = A ( T / D ) K and to follow Holstein's procedure with this modified transfer integral. In any case, (27) has a rather weak t dependence, and W becomes of activation-type. When T ~ O)D, the integration in (2) must be extended to several times T -1. When It] reaches wB1, ~(P) becomes - e -2s. The main contribution to the integral comes beyond ~oB1. Thus, it is allowed to represent the effect of the interaction with lattice by an effective transfer integral de~ = A e -s and follow the procedure of the previous section. In the general case we have made numerical integration using (13) and (23). The results will be expressed as
AZH( T-ff--,D,K,S ) W : --~ \O)DG tO
(28)
As an example, we show in fig. 5 the values of H
J. Kondo / Diffusion of light interstitials in metals
284
of singularity. chain. Let u, atom from its interested in
It is a one-dimensional harmonic be the displacement of the nth equilibrium position, then we are ((u,-u,,)2). At T = 0, we have ((u,,-u,,,)2)~lnln-n'l, while at finite temperatures, we have
10 2
s=0 10 ~
10 °
( ( u , - u,,) 2) : ( ~ m o w o ) - ' [ l n ( w o / T ) + T i n - n'l/Wo] . One can compare this with (24). The role played by the second term in dynamic response of a harmonic chain was pointed out by E m e r y and Axe [13]. It should also play an important role in MiSssbauer spectrum [14] and in optical spectrum [15] in a one-dimensional chain.
10-1 H
1021 10-3
/
Acknowledgements 1
0
-
4
~
,. ~. _.J
10"4
........ , | 10-3
.
.
.
.
.
.
d i i iiiiiii I I IP ,,,,I . . . . . . . . 10-2 10-~ 10° 101
The author expresses K. Y a m a d a for pointing for helpful discussion. thanks to Dr. S. A b e for
his thanks to Professor out to him ref. [8] and H e also expresses his useful discussion.
T (,,O D
Fig. 5. Normalized hopping frequency H (see (28)) vs. T/coD for K = 0.3 and D/coD = 100. The dotted part contains uncertainty arising from numerical error.
References [1] [2] [3] [4] [5] [6]
with K = 0.3 and D / o ) D = 100 together with the experimental points from ref. [8]. The straight lines at the low t e m p e r a t u r e region are represented by T 2K-I. The interval between them is governed by e -2s. As seen in the figure, the result of numerical calculation predicts a minimum. An agreement of absolute magnitudes is obtained for rOD = 625 K, D = 6.25 × 104 K, A = 9 . 6 K as well as for S = 5 and K = 0.3. These values are quite reasonable.
[7]
5. Discussion
[9]
What is the physics of the factor T-l? Its origin is rather singular, and seems to be difficult to be mentioned in a simple term. Here, we will mention another system which has the same kind
[8]
[10]
[11]
J. Kondo, Physica 84B (1976) 40. J. Kondo and T. Soda, J. Low Temp. Phys. 50 (1983) 2l. J. Kondo, Physica 123B (1984) 175. J. Kondo, Physica 124B (1984) 25. P.W. Anderson, Phys. Rev. Lett. 18 (1967) 1049. C.W. Clawson, K.M. Crowe, S.E. Kohn, S.S. Rosenblum, C.Y. Huang, J.L. Smith and J.H. Brewer, Physica 109-110B (1982) 2164. K.W. Kehr, D. Richter, J.-M. Welter, O. Hartmann, E. Karlsson, L.O. Norlin, T.O. Niinikoski and A. Yaouanc, Phys. Rev. 26B (1982) 567; J.-M. Welter, D. Richter, R. Hempelmann, O. Hartmann, E. Karlsson, L.O. Norlin, T.O. Niinikoski and D. Lenz, Z. Phys. 52B (1983) 303. R. Kadono, J. Imazato, K. Nishiyama, K. Nagamine, T. Yamazaki, D. Richter and J.-M. Welter, Hyperfine Interactions 17-19 (1984) 109. K. Yamada, at a meeting at Osaka University, Dec. 1983. T. Holstein, Ann. Phys. 8 (1959) 325 and 343; C.P. Flynn and A.M. Stoneham, Phys. Rev. B1 (1970) 3966; H. Teichler, Phys. Lett. 64A (1977) 78. H. Sumi, J. Phys. Soc. Japan 49 (1980) 1701.
J. Kondo / Diffusion of light interstitials in metals [12] K. Yamada, A. Sakurai and M. Takeshige, Prog. Theor. Phys. 70(1983) 73. [13] V.J. Emery and J.D. Axe, Phys. Rev. Lett. 40 (1978) 1507.
285
[14] J. Kondo, unpublished. [15] S. Abe, Thesis for Master's degree, University of Tokyo (1977).