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Mass correction of heavy particles in metals
Physica 123B (1984) 175--182 North-Holland, Amsterdam
MASS C O R R E C T I O N OF HEAVY P A R T I C L E S IN M E T A L S Jun K O N D O Elec.trotechnical Laboratory, Tsukuba Research Center, Sakuramura, Niiharigun, Ibaraki-ken, Japan Received 7 July 1983 Mass correction of heavy particles interacting with degenerate fermions has been calculated. Particle-hole pair excitation near the fermi level makes an important contribution to the mass correction. The unperturbed energy spectrum of the heavy particle is taken to he either of Bloch type or of free-particle type. It is only in the former case that a logarithmic mass correction occurs. At high temperatures, all the most divergent logarithmic terms are summed. The result is that the band width is reduced by a factor proportional to a power of the temperature. At low temperatures, only the second-order correction term has been calculated. The result is applied to the 3d band of nickel screened by 4s electrons. A rather large reduction of the band width is expected.
1. Introduction Screening by a degenerate electron gas has been a subject of interest for more than 30 years. There are two types of screening by the degenerate electron gas. One is due to the plasma oscillation of the electrons, which is of high frequency and screens the long range part of the Coulomb interaction. The other is of low frequency, due to the excitation of electron-hole pairs near the fermi surface, and has a tendency to an infrared divergence. The latter effect has its origin in A n d e r s o n ' s orthogonality catastrophe [1] and is related, for example, to the problems of dilute magnetic alloys and soft X-ray absorption and emission spectra in metals. Contrary to plasma screening, this effect is not peculiar to a charged system. We have previously considered two subjects of the latter type. A heavy particle moving in a degenerate liquid 3He has a vertex correction which is proportional to a power of the temperature [2]. On the other hand, a particle hopping between two sites in a metal acquires an effective hopping integral which is also proportional to a power of the temperature [3]. This is essentially a mass correction of the particle. In this paper we calculate a one-particle Green's function of a heavy particle interacting with degenerate fermions. The particle may have a tight-binding energy spectrum, or a free-particle spectrum. (By a heavy particle we mean that the band width is much smaller than the fermi energy in the former case or that the particle mass is much larger than the fermion mass in the latter case.) It turns out that it is in the former case that a logarithmic mass correction occurs. At high temperatures, the mass correction involves a logarithm of the ratio of k T to the fermi energy. It is easy to collect all the most divergent logarithmic terms. The result is quite consistent with the result obtained for a particle jumping between two sites: The band width is reduced by a factor proportional to a power of the temperature. At lower temperatures, the logarithm becomes that of the ratio of the band width of the particle to the fermi energy. It is rather hard to calculate higher-order logarithmic terms in this case, and we have calculated only the second-order mass correction, including non-logarithmic terms, too. The result is applied to the band mass of 3d electrons in nickel, which are screened by 4s electrons. It is pointed out that a rather large reduction of the overall band width is expected. It is also noted that a non-logarithmic correction is not small in this case. 0378-4363/84/$03.00 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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J. Kondo / Mass correctionof heavy particles in metals
2. Mass correction at h i g h e r t e m p e r a t u r e s We treat the following system: H = Z ekaLau + Z E,b~b, + Vo Z a~,saub~-k,+kb,, ks
q
(1)
kk'sq
where at, and bq represent the fermi particle and the heavy particle, respectively, s denotes the spin. We take eL = heke/2m, but the form of Eq will be specified later. We calculate Green's function of the heavy particle:
The lowest-order contribution to E is represented by fig. 1: ~¢z)= -2V~0 Z klA1 Ekl
fa,(1-fQ --
EA1Jr
Eq+Al_kl
(3) - - O)
The factor of 2 comes from the spin of the fermions. We use the following notations for later convenience:
g=2~,
(4)
F, = f,,(1 - f,,),
(5)
al = eL,- ca,,
(6)
rl = k l - A1.
(7)
Expanding the denominator of (3), we obtain Za)
F1 a~ = - g Z aa + E , _ , _ ca ~- - g Z + g Z ( E , - , , - ca)-~l ,
(8)
where it is implied that al contains an infinitesimal imaginary part - i8. Since the main contribution to the sum of (8) comes from those regions where [all - kT, the above expansion is valid for [ca - E,_,I[ <~ kT. This condition is satisfied when Ica - E,] ~ k T and [E, - E,_,I] a kT. The first condition implies that the obtained result holds in the long-time regime. The last condition is also expressed as Eq=~ "~ k T,
(9)
since the recoil momentum rl is of the order of kF.
4E
Fig. 1. The second-orderself-energyof a heavy particle. The thick line represents the heavy particle, and the thin line the fermions,
4D
Fig. 2. The fourth-order most divergent self-energy. The dotted line implies that the denominator which corresponds to the position of that line is expanded. The most divergent contribution is obtained when the dotted line is placed at the center of the figures, as shown here.
J. Kondo / Mass correction of heavy particles in metals
177
The most divergent 4th-order terms are represented by fig. 2 [2]:
ElF2 ~(4D)= _g2 ~" (al + Eq-rl - to)(al + a2 + Eq_~_~: - to)(a2 + Eq-, 2- to)' F1F2
(4E)= __g2~" (al + Eq_,.,- to)2(al + a2 + Eq-,-,-r2- to)' where a2, F2 and #'2 are defined in a similar way to a~, F1 and r~. We expand the denominators of these expressions. This is equivalent to introducing a vertical line in fig. 2. The most divergent contribution occurs when it is introduced at the center of the figure [2]. Retaining only those terms, we have
F~F2
E (4D)=_g2 ~_. ala2(al+ a2) tFlEE
~, t4E)= _g2 ~2 a2( a' + a2) +
g2 -Z
F1F2
(Eq-.,-,'z- to) ata2(--~~l+a2)2,
g2 ~. (Eq_,,_,.: - to) al(aF1F2 ' + a2):"
Adding the two terms, we find
~t4m + ~t4E)
=
_g2 X" F~F2 + 1 g2 . FxF2 z_, ala2 2 Z (E,_,,_n- to)a~ia~.
We can treat the general terms by using the method of [2]. We find the most divergent 2nth order terms as E(2n)
=
-
(n ~n- 1)! •
. . --Ft" "F,,
.
. g~ .~ (Eq_,, ..... ,, - to) F~. . . F,,
a 2...a~-aa,, Vn!
a]
.a~"
(10)
Here, four forms of Eq will be considered:
(i) Eq = 2A (cos q~b + cos qyb + cos q,b) ,
s.c.
(ii) E q = 4 A k c o s
f.c.c.
2 cos 2 +cos 2 cos 2 + c ° s 2 cos 2 ) '
(iii) E q = 8 A c ° s ~ - c ° s
2-q~c°sq~b2 '
b.c.c.
(iv) Eq = h2q2/2M, where b is the lattice constant of the fundamental simple cubic lattice. A is the transfer integral between nearest neighbour sites and is assumed to be <0. Since ek is spherically symmetric as a function of k, the value of the second term of (10) does not change if Eq-r~ ..... r. is replaced by the following expressions:
O) Oi) (iii) (iv)
E, exp{i(rt + . . . + r.)xb}, E, exp~i[(r, + . . - + r,)x + (r, + . . . Eq e x p ~ i [ ( r l + - ' ' + r,)x + ( r l + ' - " E, + h2(rl + . . . + r.)2/2M.
+ r.),lb}, + r,), + ( r l + " " " + r.),lb},
Furthermore, in cases (ii) and (iii), we make a transformation of the coordinate axes, so that the (110) axis in case (ii) or the (111) axis in case (iii) becomes the x' axis. This does not change ek, because of spherical symmetry but changes the above expressions to (ii) (iii)
Eq e x p [ i ( r , + . . . + r,)x,b/g/2], Eq e x p [ i ( r l + . - . + r.)x,X/3b/2l.
178
J. Kondo / Mass correction of heavy particles in metals
Note that b/X/2 in case (ii) or X/-3b/2 in case (iii) is the nearest neighbour atomic distance. Since the result of the summation does n o t ' d e p e n d on whether the reference axis is x or x', we can always (in cases (i)-(iii)) use Eq exp[i(rl + .-- + r,)xa], where a is the nearest neighbour atomic distance. Thus, in cases (i)-(iii), (10) consists of products of three kinds of sums:
i F _ g ~ F al x = g ~" / A ( 1 - / k ) = i2~rV~op2kT, ek - e~ - i6
(11)
F1 ,fx(1 - fL) = 2 V~op2 log(D/k T) L-:-- g ~_~-~= g ~ (eL- ea - i 8 ) 2
(12)
M = - g ~ e i q : F t = 2a g ~" ei(kx-XDa (eLf'(1 f Le~ ) -_ - i8) 2 kFa -~-~-d~ log(D/kT),
,~ • ,9 2 sin2
=z v~p
(13)
where D is an energy of order eF, p is the density of states of the fermions. W e finally find t o E- q
- ~E
(2n) =
to e L - Eq e M + i F e L ,
n=l
so we have
G(q, to) =
e-L
to - Eq e M-L + iF
(cases (i) - (iii))
(14)
e -L is equal to Z ( T ) of [2]. e M-L is written as (kT/D) r, where K = 2V~0p2(1
sin ~ 2 kFa~ }.
(15)
Thus, in cases (i)-(iii), the band width is reduced by the same factor as occurred in the system of [3]. In case (iv), Eq_,~ ..... ,, of (10) can be replaced by Eq, because the rest gives rise only to a q-independent term. Then we obtain the result of [2]:
G(q, to)=
e-L
t o - Eq + iF
(case (iv))
No logarithmic mass correction occurs. The vertex correction, on the other hand, always exists, and is expressed as in [2] for the four cases.
3. M a s s c o r r e c t i o n
at lower temperatures
The perturbation expansion of the last section which is valid under (9) was essentially in terms of
V~olog(kT/D). For k T ~ Eq=kF, we would expect that the same kind of expansion holds with log(kT/D) replaced by log(lA [/D) or log(m/M). That such is the case was demonstrated for the system of [3]. It is very difficult in the present system to collect all the higher order logarithmic terms. So, in this section we consider the second-order mass correction at T = 0 exactly with terms other than the logarithmic one included, and see the qualitative features of the result. The spectrum of the heavy particle with the second-order mass correction is expressed by
J. Kondo / Mass correctionof heavyparticlesin metals
179
1 #,q = E q - 2V~o
~
(16)
+ E,_ o - E, "
IklkF6 k+O- F'k
We are interested in the q-dependence of the second term. The summation is performed over k and Q with the shown restriction. First, we fix Q, take it as the z axis and perform the integral over k first. The latter integral is over Ikt, 0 and ~b. Instead of integrating over 0, it is convenient to integrate over Ik + QI -- k' (fig. 3). Then we have
;Eq=Eq
2V~0Vx? I f
(--~ ~-~
kdk
/k'
1
(17)
dk'(h2k2/2m)_(h2k,2/2m)+Eq_o_Eq,
where V is the volume of the system and the region of integration, which depends on Q, is shown in fig. 4. The double integral in (17) is easily performed and is expressed as
2mk 2 t4 [ E "-° -~F) h ~ " ' \ eF '
"
H(x, y) contains x °, x loglxl, x, x 2, etc. x is small when A/e~ or m/M is small. We are not interested in x ° terms, because it is independent of q. So, we retain only the x loglxl and x terms:
n(x, y) =
1 1"~x[loglxl- 2 l°g(4y2 - y')] x
(~y
1 y+2~ - ~ log Y _ 2]
0< y< 2
(18)
2
We first consider the tight-binding energy spectrum, cases (i)-(iii) of the last section. We set
Eq =-2AFq, Fq representing the cosine factor. After carrying out Q summation as far as possible, we obtain
F_.q=Eq +2V20p2Eq[(1
sin2kFa'~ ~-~h-~ ] log 2A-L~e+ F O ( k F a ) ] - 2V~0p22Aff~,,
(19)
where a is the nearest neighbour atomic distance and 2a 2a sin 2ot~ / sin x dx sin2a a2 f l - c O s x dx, 1( G(a)= 1 2~ ] x x 0
(20)
o
Fig. 3. The k integral in (16) is performed over the shaded region. Instead of k, O and 4~,we use k, Ik + QI and 4~as the integration variables.
kF
k-
Fig. 4. The double integral in (17) is performed over the double-hatched region.
180
J. Kondo / Mass correction of heavy particles in metals /~q = ~
1
f
1 (F,-Q - Fq) loglF,-Q - Fql. d3O --~
(21)
IQ]~2kF AS expected, (19) contains the logarithmic term as well as non-logarithmic terms (G and Fq). G ( a ) is shown in fig. 5. Fq depends sensitively on the value of kFa. Fig. 6 shows Fq=0 as a function of kva. In general, the value of -~q=0 represents the overall magnitude of Fq for q ¥ 0. Three cases can be distinguished (figs. 7 and 8): ( A ) kFa <~0.4: Fq decreases sharply as q increases. For small kFa, the G and the Fq terms in (19) are linear in kFa, whereas the log[A I/2ev term is quadratic, and so is smaller than the others. Then, the mass correction consists of two terms: one is proportional to Eq, the other to Fq. They have quite different qdependences. Fq decreases linearly for small q. (For very small q it is quadratic.) On the other hand, Eq increases quadratically. One must have a minimum (or minima) outside of the origin. (B) 0.4 <~kFa <~2.5: Fq oscillates as q increases. (C) 2.5 <<.kva: Fq increases with q. Not only the value of ,~q=0 but also the overall features of Fq are nearly independent of kFa. Furthermore, it is to be noted that the overall features of Fq in this case are fairly similar to the original band structure Fq as one sees from figs. 7 and 8. Roughly speaking, we may set Fq = -1.3Fq for b.c.c, and Fq ~- - 1.8Fq for f.c.c. Since G(kFa) as well as sin 2 kFa/k~a 2 is negligible in this case, we can write E,q = Eq[1 + 2V~0p2(log([A I/2eF)+ 1.8)] for f.c.c. (For b.c.c., 1.8 should be replaced by 1.3.) This means that the band structure with mass correction due to s-electrons screening is nearly identical to the unperturbed one, except that the overall band width is reduced (assuming [A I/eF ~ 1). Finally let us consider case (iv). It is easy in this case to perform Q integration in (17) with (18) inserted for the double integral. We find
Eq=Eq{l-34 V~op2[1- -1~q'~2 5\kF/ log k~F]} " We do not find a mass correction involving, for example, log(m/M), but find a small singular one involving q2 log q.
5
kro
oC
S.C.
Fq=o
G (a)
-0.7
.¢.C
-14 Fig. 5. G(a) in (20) as a f u n c t i o n of a .
Fig. 6. Fq at q = 0 as a f u n c t i o n of kva.
J. Kondo I Mass correction of heavy particles in metals
0.05
181
0.2
, ..........~
0
~(~0o)
~
/
//7
~
11o) -12
E
00
000)
kFo =0.707
k F o = 0.0 7 0 7
kFO
= 3.535
Fig. 7. Fq for three different values of kFa (f.c.c.).
0.05
02
0
I , l~/I ,(]
E
(111)
0 ~0~.(110)
-0.2
11)~
-0.4
(1001
E
00 kF o =
0.0866
-6 kF a =
0.866
kF o =
2.60
Fig. 8. Fq for three different values of kra (b.c.c.).
4. Discussion In section 2 we have taken into account only the most divergent terms. Consequently, K in (15) involves only the V~0p2 term. If K is defined in such a way that the reduction factor of the band width is expressed as (kT/D) r, K should actually involve higher-order terms. Finding them would be a difficult task. One would consider that V~0p2 should be replaced by (8/¢r)2, etc. Concerning this point, Yamada, Sakurai and Takeshige's work [4] is interesting. This work follows Yamada and Yosida's elaborate calculation [5] of the overlap integral between two Slater determinants, one corresponding to the impurity potential at the origin, the other to that at a distance a from the origin. Their result states that the overlap integral is expressed as N -r', where N is the number of the fermions, and K ' is a complicated function of 8 and kFa. For example, for 8 = ~r/2 and kFa -> 1, one has K ' = 1/2 (including spins). There are some reasons to believe that K and K ' are identical. In fact, the lowest terms are identical. As an application of the result of section 3, we consider the screening of the 3d band of nickel by 4s electrons. Here, we are certainly in the region k T ~ IA[. Since each nickel atom has 0.6 s-electrons, eF = 5 eV and kFa = 2.9, which belongs to case (C). When the overall band width 121/tl is taken to be
182
J. Kondo / Mass correction of heavy particles in metals
3 eV, we have 1og(lAl/2eF)=-3.7, and so/~q = Eq[1 + 2V~0p2(-1.9)]. This is a rather large reduction of the band width. The answer depends on the magnitude of V0p. If the charge of the 3d electrons is screened solely by 4s electrons, the phase shift must be 7r/2, so V0p is not small. If we replace 2V~op2 by K' = 1/2, the reduction is too large. This seems to imply that the screening by other 3d electrons is also important. In conclusion, we have calculated the mass correction of heavy particles interacting with a degenerate fermi system. The energy spectrum of the heavy particle may be either of Bloch type or of free-particle type. It is only in the former case that a logarithmic mass correction occurs. At high temperatures, all the most divergent logarithmic terms are summed. The result is that the band width of the heavy particle is reduced by a factor proportional to a power of the temperature. At low temperatures, only the second-order term has been calculated. The result depends sensitively on the value of kFa. The 3d band of nickel has been considered and a fairly large reduction of the band width is expected.
References [1] [2] [3] [4] [5]
P.W. Anderson, Phys. Rev. Len. 18 (1967) 1049. J. Kondo and T. Soda, J. Low Temp. Phys. 50 (1983) 21. J. Kondo, Physica 84B (1976) 40. K. Yamada, A. Sakurai and M. Takeshige, preprint. K. Yamada and K. Yosida, Prog. Theor. Phys. 68 (1982) 1504; 59 (1978) 1061; 60 (1978) 353; 62 (1979) 363.
MASS C O R R E C T I O N OF HEAVY P A R T I C L E S IN M E T A L S Jun K O N D O Elec.trotechnical Laboratory, Tsukuba Research Center, Sakuramura, Niiharigun, Ibaraki-ken, Japan Received 7 July 1983 Mass correction of heavy particles interacting with degenerate fermions has been calculated. Particle-hole pair excitation near the fermi level makes an important contribution to the mass correction. The unperturbed energy spectrum of the heavy particle is taken to he either of Bloch type or of free-particle type. It is only in the former case that a logarithmic mass correction occurs. At high temperatures, all the most divergent logarithmic terms are summed. The result is that the band width is reduced by a factor proportional to a power of the temperature. At low temperatures, only the second-order correction term has been calculated. The result is applied to the 3d band of nickel screened by 4s electrons. A rather large reduction of the band width is expected.
1. Introduction Screening by a degenerate electron gas has been a subject of interest for more than 30 years. There are two types of screening by the degenerate electron gas. One is due to the plasma oscillation of the electrons, which is of high frequency and screens the long range part of the Coulomb interaction. The other is of low frequency, due to the excitation of electron-hole pairs near the fermi surface, and has a tendency to an infrared divergence. The latter effect has its origin in A n d e r s o n ' s orthogonality catastrophe [1] and is related, for example, to the problems of dilute magnetic alloys and soft X-ray absorption and emission spectra in metals. Contrary to plasma screening, this effect is not peculiar to a charged system. We have previously considered two subjects of the latter type. A heavy particle moving in a degenerate liquid 3He has a vertex correction which is proportional to a power of the temperature [2]. On the other hand, a particle hopping between two sites in a metal acquires an effective hopping integral which is also proportional to a power of the temperature [3]. This is essentially a mass correction of the particle. In this paper we calculate a one-particle Green's function of a heavy particle interacting with degenerate fermions. The particle may have a tight-binding energy spectrum, or a free-particle spectrum. (By a heavy particle we mean that the band width is much smaller than the fermi energy in the former case or that the particle mass is much larger than the fermion mass in the latter case.) It turns out that it is in the former case that a logarithmic mass correction occurs. At high temperatures, the mass correction involves a logarithm of the ratio of k T to the fermi energy. It is easy to collect all the most divergent logarithmic terms. The result is quite consistent with the result obtained for a particle jumping between two sites: The band width is reduced by a factor proportional to a power of the temperature. At lower temperatures, the logarithm becomes that of the ratio of the band width of the particle to the fermi energy. It is rather hard to calculate higher-order logarithmic terms in this case, and we have calculated only the second-order mass correction, including non-logarithmic terms, too. The result is applied to the band mass of 3d electrons in nickel, which are screened by 4s electrons. It is pointed out that a rather large reduction of the overall band width is expected. It is also noted that a non-logarithmic correction is not small in this case. 0378-4363/84/$03.00 O Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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J. Kondo / Mass correctionof heavy particles in metals
2. Mass correction at h i g h e r t e m p e r a t u r e s We treat the following system: H = Z ekaLau + Z E,b~b, + Vo Z a~,saub~-k,+kb,, ks
q
(1)
kk'sq
where at, and bq represent the fermi particle and the heavy particle, respectively, s denotes the spin. We take eL = heke/2m, but the form of Eq will be specified later. We calculate Green's function of the heavy particle:
The lowest-order contribution to E is represented by fig. 1: ~¢z)= -2V~0 Z klA1 Ekl
fa,(1-fQ --
EA1Jr
Eq+Al_kl
(3) - - O)
The factor of 2 comes from the spin of the fermions. We use the following notations for later convenience:
g=2~,
(4)
F, = f,,(1 - f,,),
(5)
al = eL,- ca,,
(6)
rl = k l - A1.
(7)
Expanding the denominator of (3), we obtain Za)
F1 a~ = - g Z aa + E , _ , _ ca ~- - g Z + g Z ( E , - , , - ca)-~l ,
(8)
where it is implied that al contains an infinitesimal imaginary part - i8. Since the main contribution to the sum of (8) comes from those regions where [all - kT, the above expansion is valid for [ca - E,_,I[ <~ kT. This condition is satisfied when Ica - E,] ~ k T and [E, - E,_,I] a kT. The first condition implies that the obtained result holds in the long-time regime. The last condition is also expressed as Eq=~ "~ k T,
(9)
since the recoil momentum rl is of the order of kF.
4E
Fig. 1. The second-orderself-energyof a heavy particle. The thick line represents the heavy particle, and the thin line the fermions,
4D
Fig. 2. The fourth-order most divergent self-energy. The dotted line implies that the denominator which corresponds to the position of that line is expanded. The most divergent contribution is obtained when the dotted line is placed at the center of the figures, as shown here.
J. Kondo / Mass correction of heavy particles in metals
177
The most divergent 4th-order terms are represented by fig. 2 [2]:
ElF2 ~(4D)= _g2 ~" (al + Eq-rl - to)(al + a2 + Eq_~_~: - to)(a2 + Eq-, 2- to)' F1F2
(4E)= __g2~" (al + Eq_,.,- to)2(al + a2 + Eq-,-,-r2- to)' where a2, F2 and #'2 are defined in a similar way to a~, F1 and r~. We expand the denominators of these expressions. This is equivalent to introducing a vertical line in fig. 2. The most divergent contribution occurs when it is introduced at the center of the figure [2]. Retaining only those terms, we have
F~F2
E (4D)=_g2 ~_. ala2(al+ a2) tFlEE
~, t4E)= _g2 ~2 a2( a' + a2) +
g2 -Z
F1F2
(Eq-.,-,'z- to) ata2(--~~l+a2)2,
g2 ~. (Eq_,,_,.: - to) al(aF1F2 ' + a2):"
Adding the two terms, we find
~t4m + ~t4E)
=
_g2 X" F~F2 + 1 g2 . FxF2 z_, ala2 2 Z (E,_,,_n- to)a~ia~.
We can treat the general terms by using the method of [2]. We find the most divergent 2nth order terms as E(2n)
=
-
(n ~n- 1)! •
. . --Ft" "F,,
.
. g~ .~ (Eq_,, ..... ,, - to) F~. . . F,,
a 2...a~-aa,, Vn!
a]
.a~"
(10)
Here, four forms of Eq will be considered:
(i) Eq = 2A (cos q~b + cos qyb + cos q,b) ,
s.c.
(ii) E q = 4 A k c o s
f.c.c.
2 cos 2 +cos 2 cos 2 + c ° s 2 cos 2 ) '
(iii) E q = 8 A c ° s ~ - c ° s
2-q~c°sq~b2 '
b.c.c.
(iv) Eq = h2q2/2M, where b is the lattice constant of the fundamental simple cubic lattice. A is the transfer integral between nearest neighbour sites and is assumed to be <0. Since ek is spherically symmetric as a function of k, the value of the second term of (10) does not change if Eq-r~ ..... r. is replaced by the following expressions:
O) Oi) (iii) (iv)
E, exp{i(rt + . . . + r.)xb}, E, exp~i[(r, + . . - + r,)x + (r, + . . . Eq e x p ~ i [ ( r l + - ' ' + r,)x + ( r l + ' - " E, + h2(rl + . . . + r.)2/2M.
+ r.),lb}, + r,), + ( r l + " " " + r.),lb},
Furthermore, in cases (ii) and (iii), we make a transformation of the coordinate axes, so that the (110) axis in case (ii) or the (111) axis in case (iii) becomes the x' axis. This does not change ek, because of spherical symmetry but changes the above expressions to (ii) (iii)
Eq e x p [ i ( r , + . . . + r,)x,b/g/2], Eq e x p [ i ( r l + . - . + r.)x,X/3b/2l.
178
J. Kondo / Mass correction of heavy particles in metals
Note that b/X/2 in case (ii) or X/-3b/2 in case (iii) is the nearest neighbour atomic distance. Since the result of the summation does n o t ' d e p e n d on whether the reference axis is x or x', we can always (in cases (i)-(iii)) use Eq exp[i(rl + .-- + r,)xa], where a is the nearest neighbour atomic distance. Thus, in cases (i)-(iii), (10) consists of products of three kinds of sums:
i F _ g ~ F al x = g ~" / A ( 1 - / k ) = i2~rV~op2kT, ek - e~ - i6
(11)
F1 ,fx(1 - fL) = 2 V~op2 log(D/k T) L-:-- g ~_~-~= g ~ (eL- ea - i 8 ) 2
(12)
M = - g ~ e i q : F t = 2a g ~" ei(kx-XDa (eLf'(1 f Le~ ) -_ - i8) 2 kFa -~-~-d~ log(D/kT),
,~ • ,9 2 sin2
=z v~p
(13)
where D is an energy of order eF, p is the density of states of the fermions. W e finally find t o E- q
- ~E
(2n) =
to e L - Eq e M + i F e L ,
n=l
so we have
G(q, to) =
e-L
to - Eq e M-L + iF
(cases (i) - (iii))
(14)
e -L is equal to Z ( T ) of [2]. e M-L is written as (kT/D) r, where K = 2V~0p2(1
sin ~ 2 kFa~ }.
(15)
Thus, in cases (i)-(iii), the band width is reduced by the same factor as occurred in the system of [3]. In case (iv), Eq_,~ ..... ,, of (10) can be replaced by Eq, because the rest gives rise only to a q-independent term. Then we obtain the result of [2]:
G(q, to)=
e-L
t o - Eq + iF
(case (iv))
No logarithmic mass correction occurs. The vertex correction, on the other hand, always exists, and is expressed as in [2] for the four cases.
3. M a s s c o r r e c t i o n
at lower temperatures
The perturbation expansion of the last section which is valid under (9) was essentially in terms of
V~olog(kT/D). For k T ~ Eq=kF, we would expect that the same kind of expansion holds with log(kT/D) replaced by log(lA [/D) or log(m/M). That such is the case was demonstrated for the system of [3]. It is very difficult in the present system to collect all the higher order logarithmic terms. So, in this section we consider the second-order mass correction at T = 0 exactly with terms other than the logarithmic one included, and see the qualitative features of the result. The spectrum of the heavy particle with the second-order mass correction is expressed by
J. Kondo / Mass correctionof heavyparticlesin metals
179
1 #,q = E q - 2V~o
~
(16)
+ E,_ o - E, "
Ikl
We are interested in the q-dependence of the second term. The summation is performed over k and Q with the shown restriction. First, we fix Q, take it as the z axis and perform the integral over k first. The latter integral is over Ikt, 0 and ~b. Instead of integrating over 0, it is convenient to integrate over Ik + QI -- k' (fig. 3). Then we have
;Eq=Eq
2V~0Vx? I f
(--~ ~-~
kdk
/k'
1
(17)
dk'(h2k2/2m)_(h2k,2/2m)+Eq_o_Eq,
where V is the volume of the system and the region of integration, which depends on Q, is shown in fig. 4. The double integral in (17) is easily performed and is expressed as
2mk 2 t4 [ E "-° -~F) h ~ " ' \ eF '
"
H(x, y) contains x °, x loglxl, x, x 2, etc. x is small when A/e~ or m/M is small. We are not interested in x ° terms, because it is independent of q. So, we retain only the x loglxl and x terms:
n(x, y) =
1 1"~x[loglxl- 2 l°g(4y2 - y')] x
(~y
1 y+2~ - ~ log Y _ 2]
0< y< 2
(18)
2
We first consider the tight-binding energy spectrum, cases (i)-(iii) of the last section. We set
Eq =-2AFq, Fq representing the cosine factor. After carrying out Q summation as far as possible, we obtain
F_.q=Eq +2V20p2Eq[(1
sin2kFa'~ ~-~h-~ ] log 2A-L~e+ F O ( k F a ) ] - 2V~0p22Aff~,,
(19)
where a is the nearest neighbour atomic distance and 2a 2a sin 2ot~ / sin x dx sin2a a2 f l - c O s x dx, 1( G(a)= 1 2~ ] x x 0
(20)
o
Fig. 3. The k integral in (16) is performed over the shaded region. Instead of k, O and 4~,we use k, Ik + QI and 4~as the integration variables.
kF
k-
Fig. 4. The double integral in (17) is performed over the double-hatched region.
180
J. Kondo / Mass correction of heavy particles in metals /~q = ~
1
f
1 (F,-Q - Fq) loglF,-Q - Fql. d3O --~
(21)
IQ]~2kF AS expected, (19) contains the logarithmic term as well as non-logarithmic terms (G and Fq). G ( a ) is shown in fig. 5. Fq depends sensitively on the value of kFa. Fig. 6 shows Fq=0 as a function of kva. In general, the value of -~q=0 represents the overall magnitude of Fq for q ¥ 0. Three cases can be distinguished (figs. 7 and 8): ( A ) kFa <~0.4: Fq decreases sharply as q increases. For small kFa, the G and the Fq terms in (19) are linear in kFa, whereas the log[A I/2ev term is quadratic, and so is smaller than the others. Then, the mass correction consists of two terms: one is proportional to Eq, the other to Fq. They have quite different qdependences. Fq decreases linearly for small q. (For very small q it is quadratic.) On the other hand, Eq increases quadratically. One must have a minimum (or minima) outside of the origin. (B) 0.4 <~kFa <~2.5: Fq oscillates as q increases. (C) 2.5 <<.kva: Fq increases with q. Not only the value of ,~q=0 but also the overall features of Fq are nearly independent of kFa. Furthermore, it is to be noted that the overall features of Fq in this case are fairly similar to the original band structure Fq as one sees from figs. 7 and 8. Roughly speaking, we may set Fq = -1.3Fq for b.c.c, and Fq ~- - 1.8Fq for f.c.c. Since G(kFa) as well as sin 2 kFa/k~a 2 is negligible in this case, we can write E,q = Eq[1 + 2V~0p2(log([A I/2eF)+ 1.8)] for f.c.c. (For b.c.c., 1.8 should be replaced by 1.3.) This means that the band structure with mass correction due to s-electrons screening is nearly identical to the unperturbed one, except that the overall band width is reduced (assuming [A I/eF ~ 1). Finally let us consider case (iv). It is easy in this case to perform Q integration in (17) with (18) inserted for the double integral. We find
Eq=Eq{l-34 V~op2[1- -1~q'~2 5\kF/ log k~F]} " We do not find a mass correction involving, for example, log(m/M), but find a small singular one involving q2 log q.
5
kro
oC
S.C.
Fq=o
G (a)
-0.7
.¢.C
-14 Fig. 5. G(a) in (20) as a f u n c t i o n of a .
Fig. 6. Fq at q = 0 as a f u n c t i o n of kva.
J. Kondo I Mass correction of heavy particles in metals
0.05
181
0.2
, ..........~
0
~(~0o)
~
/
//7
~
11o) -12
E
00
000)
kFo =0.707
k F o = 0.0 7 0 7
kFO
= 3.535
Fig. 7. Fq for three different values of kFa (f.c.c.).
0.05
02
0
I , l~/I ,(]
E
(111)
0 ~0~.(110)
-0.2
11)~
-0.4
(1001
E
00 kF o =
0.0866
-6 kF a =
0.866
kF o =
2.60
Fig. 8. Fq for three different values of kra (b.c.c.).
4. Discussion In section 2 we have taken into account only the most divergent terms. Consequently, K in (15) involves only the V~0p2 term. If K is defined in such a way that the reduction factor of the band width is expressed as (kT/D) r, K should actually involve higher-order terms. Finding them would be a difficult task. One would consider that V~0p2 should be replaced by (8/¢r)2, etc. Concerning this point, Yamada, Sakurai and Takeshige's work [4] is interesting. This work follows Yamada and Yosida's elaborate calculation [5] of the overlap integral between two Slater determinants, one corresponding to the impurity potential at the origin, the other to that at a distance a from the origin. Their result states that the overlap integral is expressed as N -r', where N is the number of the fermions, and K ' is a complicated function of 8 and kFa. For example, for 8 = ~r/2 and kFa -> 1, one has K ' = 1/2 (including spins). There are some reasons to believe that K and K ' are identical. In fact, the lowest terms are identical. As an application of the result of section 3, we consider the screening of the 3d band of nickel by 4s electrons. Here, we are certainly in the region k T ~ IA[. Since each nickel atom has 0.6 s-electrons, eF = 5 eV and kFa = 2.9, which belongs to case (C). When the overall band width 121/tl is taken to be
182
J. Kondo / Mass correction of heavy particles in metals
3 eV, we have 1og(lAl/2eF)=-3.7, and so/~q = Eq[1 + 2V~0p2(-1.9)]. This is a rather large reduction of the band width. The answer depends on the magnitude of V0p. If the charge of the 3d electrons is screened solely by 4s electrons, the phase shift must be 7r/2, so V0p is not small. If we replace 2V~op2 by K' = 1/2, the reduction is too large. This seems to imply that the screening by other 3d electrons is also important. In conclusion, we have calculated the mass correction of heavy particles interacting with a degenerate fermi system. The energy spectrum of the heavy particle may be either of Bloch type or of free-particle type. It is only in the former case that a logarithmic mass correction occurs. At high temperatures, all the most divergent logarithmic terms are summed. The result is that the band width of the heavy particle is reduced by a factor proportional to a power of the temperature. At low temperatures, only the second-order term has been calculated. The result depends sensitively on the value of kFa. The 3d band of nickel has been considered and a fairly large reduction of the band width is expected.
References [1] [2] [3] [4] [5]
P.W. Anderson, Phys. Rev. Len. 18 (1967) 1049. J. Kondo and T. Soda, J. Low Temp. Phys. 50 (1983) 21. J. Kondo, Physica 84B (1976) 40. K. Yamada, A. Sakurai and M. Takeshige, preprint. K. Yamada and K. Yosida, Prog. Theor. Phys. 68 (1982) 1504; 59 (1978) 1061; 60 (1978) 353; 62 (1979) 363.