Thermal conductivity of rare earth superconductors 1. Main principles

Let us consider qualitatively some conclusions of the theory of thermal conductivity of standard superconductors which may be useful in analysing tc data of rare earth

T H E R M A L C O N D U C T I V I T Y O F R E C s t83

superconductors. This theory has been developed by Geilikman (1958), Geilikman and Kresin (1958, 1959, 1972) and Bardeen et al. (1959).

The most important properties of superconducting electrons are that they do not transport energy and do not interact with phonons. At T < Tc the number of heat transporting electrons decreases by an exponent determined by the energy gap

A(T)

at the Fermi surface and the electron thermal conductivity

~c~ ~ exp [ -

A(T)/T].

(50)

The relative contributions to the total thermal conductivity, ~c~ot, from ~c~ and ~c[ in the superconductivity region and their temperature dependences are determined, mainly, by

A(T)

and Tc (superconducting and normal states are marked, respectively, by the indices s and n).

Let us consider the case of a metal with t<~ ~ ~c~.

(a) The value Tc is small and electrons are mainly scattered by defects. The electron thermal conductivity in a normal state ~c n oc T. In the superconducting state at T < T~

the value ~c s decreases rapidly due to the decreasing number of normal electrons. At very low temperatures, when ~:~ is practically negligible, ~c[ shows up. ~[ is somewhat larger than ~c[ at the same temperature due to absence of phonon-electron scattering.

It is possible at very low temperature to reach the region ~c[ oc T 3, where boundary scattering dominates (see fig. 102).

(b) The value To is not so small and electrons are scattered by defects and lattice vibrations. At T < T~, ~c~ decreases at first (as in the previous case) and then can increase slightly due to a weakening of the electron phonon scattering. At very low temperatures ~c[ can show up on the background of the small ~ (fig. 103).

In the case ~c~ >> ~c~ at T < T~ the ~c[ becomes greater than the ~c~ due to a decrease of the phonon-electron scattering (fig. 104). A combination of variants of figs. 102 104 is possible.

"[¢

J/

~T3/ I T

P, T

Fig. 102. Schematic shape of the temperature dependence of l<[,

~c~ and K~ for a metal when electron-impurity interaction is the m a i n electron scattering mechanism. K L ~ ~co.

184 I.A. SMIRNOV and V.S. OSKOTSKI

f \

/ \

]'c

/ /

~ / / /

~ - T

Fig. 103. Schematic shape of the temperature dependences of 1c[,

~c2 and tc~ for a metal when electron impurity and electron-phonon interactions are the main electron scattering mechanisms. ~L ~ Kc.

Fig. 104. Schematic shape of the temperature dependences of ~ t I, T and t¢~o t for the case K L ~> 1%.

An external magnetic field H > Her (critical field) destroys s u p e r c o n d u c t i v i t y and transfers the s u p e r c o n d u c t o r to the n o r m a l state. F o r pure metals Her can be small.

I n s u p e r c o n d u c t o r s of the second kind at T < To in a mixed state (in a m a g n e t i c field between He1 and Ho2, where Hol is the lower and H~2 the u p p e r critical field) a new effect in the thermal c o n d u c t i v i t y is possible: p h o n o n a n d electron scattering by A b r i k o s o v vortex lines penetrating the s u p e r c o n d u c t o r (Red'ko a n d Chakalski

1987).

8.2. Thermal conductivity of heavy-fermion systems

Steglich et al. (1979) have observed in CeCu2Si 2 a new effect in solid state physics:

heavy-fermion superconductivity (HFS) with Tc = 0.56 K. In the following years the

T H E R M A L C O N D U C T I V I T Y O F RECs 185

HFS has also been observed in UBe~3 (To = 0.854 K) (Ott et al. 1983) and UPt3 (To = 0.520 K) (Stewart et al. 1984). For the sake of completeness we will consider data on the thermal conductivity of all three heavy-fermion superconductors.

HFS has a number of distinguishing features. Let us note some of them:

(1) In these systems triplet superconductivity is possible with a parallel orientation of the electron spins in a pair with total spin 1 (superconductivity in HFS-systems is connected with Cooper pairs, formed by fermions with a very large effective mass (m*/mo ~ 100-1000, m0 is the free-electron mass)).

(2) A non-phonon mechanism of superconductivity is possible (Gudak 1985).

(3) The heavy-fermion superconductors of the second kind (such as CeCu2Si2, UBe~3 and UPt3) have a ratio Ho(O)/Tc(O) significantly higher than the classical superconductors. According to the theory H¢(O)/Tc(O) does not depend on the type of superconductor. So, at T c = 1 K the magnetic field Ho(0) which destroys superconductivity (paramagnetic limit) is 16 kOe. H~(0) values in excess of the paramagnetic limit is a characteristic of HFS systems. For example, in CeCu2Si 2 it is two times and in UBe13 it is more than six times as large.

Magnetic impurities suppress superconductivity and nonmagnetic impurities weakly influence T~ in classical superconductors. In H F S systems even nonmagnetic impurities suppress superconductivity. For example, the higher the concentration of Ce 3 + ions in the alloy CexLa~ _xCu2Si2, the higher is T~ (Buzdin and Moshchalkov 1986). So, a highly unusual situation arises: a decrease of the number of magnetic centres leads to a sharp decrease of To.

Can one obtain from the thermal conductivity some independent information about the nature of superconductivity in HFS systems? It turns out that from the temperature dependence of ~c~ at T ~ To one can decide if there is triplet or singlet coupling in the HFS system. For triplet coupling a vanishing energy gap occurs only at points on the Fermi surface, whereas for singlet coupling vanishing energy gaps can form entire lines on the Fermi surface (Volovik and Gor'kov 1984, 1985, Varma 1985, Schmitt-Rink et al. 1986). This is the only way to determine the coupling type at the present time (Volovik and Gor'kov 1984, 1985). One can obtain indirectly information about the behaviour of the gap, A, from the temperature dependences of the heat capacity and the thermal conductivity of HFS systems (see table 6 and fig. 105) (Varma 1985).

Let us consider the experimental data on the thermal conductivity of CeCu2Si 2 (Sparn et al. 1985, Steglich et al. 1985a, b, Franz et al. 1978, 1979, Schneidner et al.

TABLE 6

Influence of the superconducting-gap shape on the thermal properties of semiconductors.

Properties at T ~< T~ Quasi-particle spectrum has a gap over the entire Fermi surface

(fig. 105a)

Gap vanishes at points on the Fermi surface (fig. 105b)

Gap vanishes at lines on the Fermi surface (fig. 105c)

Heat capacity o~ e x p ( - A / T ) o~(T/A) 3 o c ( T / A ) 2

Thermal conductivity 1¢~ oc e x p ( A / T ) o c ( T / A ) 3 o c ( T / A ) 2

186 I.A. S M I R N O V and V.S. OSKOTSKI

K z Kz Kz

K x Ky K x

a b c

Fig. 105. Schematic shape of the superconducting gap A(k) in momentum space (Buzdin and Moshchalkov 1986) (a) in a standard superconductor; and in a heavy-fermion superconductor with (b) vanishing gaps at some points and (c) on lines on the Fermi surface.

1983), UBe13 (Sparn et al. 1985, Jaccard and Flouquet 1987, Flouquet et al. 1986, Jaccard et al. 1985a, b, Ravex et al. 1987, Alekseevski et al. t986) and UPt3 (De Visser et al. 1987, Floquet et al. 1986, Steglich et al. t985a, Jaccard et al. 1985b, Sulpice et al. 1986, Franse et al. 1985). At T ~ T~ all three compounds have ~[ ~ ~c~

and the ~c~ dependence on T has the form

tc~ = ~ T 2 + f i T (51)

The values of ~ and fl are given in table 7. The temperature dependence tc~ oc T 2 which is observed in all three compounds, corroborates the presence of singlet coupling (see tables 8 and 6), however, the temperature dependence of the heat capacity C oc T 3 (see table 8) in CeCu2 Si2 and UBe13 contradicts this conclusion (see table 6). Experimental data on the absorption coefficient of ultrasound in UPt3 (Bishop et al. 1984) and on the reciprocal time of the spin relaxation in CeCu2Si2 (MacLaughlin et al. 1984) and UBe13 (Clark et al. 1984) confirm the conclusion about singlet coupling.

At present there is no clear understanding of the nature of the linear term in ~c~.

In a number of works the appearance of this term is connected with the presence of impurities in the samples (or the nonsuperconducting phase), because the contribution

TABLE 7

Experimental values of the coefficients c~ and fl in the dependence tc~ = f i T + c~T 2 for T < T~.

Compound c~ Refs.* /? Refs.*

(mW/K 3 cm) (mW/K 2 cm)

CeCu 2 Si 2 2.8 [1] 0.7 [1] a

1.8 [2]

UBe13 0.38 [1] 0.03 [4]"

0.53 [3]

UPt3 19.5 [1, 5, 6] 0.55 [1, 6-8]

20 [7, 8]

U R u 2 Si2 0.96 [9] 0.2 [9]

aThe contribution fiT for CeCu2Si 2 and UBe13 has also been observed by Franse et al. (1984) and Jaccard and Flouquet (1987).

*References: [1] Jaccard et al. (1985a), [2] Steglich et al. (1985b), [3] Alekseevski et al. (1986), [4] Ravex et al. (1987), [5] Steglich et at. (1984), [6] Jaccard et al. (1985b), [73 Sulpice et al. (1986), [8] Schmitt- Remi et al. (1986), [9] Lopez de la Torre et al. (1988).

187

Property CeCu 2 Si 2 Refs.* U P t 3 Refs.* UBe13 Refs.*

C T 3 [1] T 2 [1, 2] T 3 [3]

T 2 [ 4 - 6 ] T 2 [1, 4, 7] T z [4, 8-10]

K e

*References: [1] Steglich et al. (1985a), [2] Sulpice et al. (1986), [3] Ott et al. (1984b), [4] Jaccard et al.

(1985a), [5] Steglich et al. (1985b), [6] Franse et al. (1984), [7] Jaccard et al. (1985b), [8] Sparn et al.

(1985), [9] Varma (1985), [10] Alekseevski et al. (1986).

of the linear term to the thermal conductivity decreases with increasing purity of the material. A linear term is also observed in the heat capacity at T ~ T~ in a number of works [e.g., in UBe13 (Ravex et al. 1987) and UPt3 (Sulpice et al. 1986)]. Figures 106 and 107 show, e.g., data on x[ of UPt3 (Schmitt-Rink et al. 1986, Clark et al.

[ I

/ /

f,./

m e / , J

N 0 I I

0 100 200

T¢

1 I t

300 400 500

T , m K

=L N

Fig. 106. The temperature dependence of x~/T for UPt3 (Schmitt-Rink et al. 1986, Clark et al. 1984).

5 - -

- - T ] ] - - I

2 5 -

2 0 -

1 5 -

1 0 -

/ /

I I I I

0 . 0 5 0.10 0.15 0.20

T, K

T H E R M A L C O N D U C T I V I T Y O F RECs TABLE 8

Experimental temperature dependences of C and xso for T < T~.

Fig. 107. The temperature dependence of x~/T for UBe13 (Volovik and Gor'kov 1984, 1985).

188 l.A. SMIRNOV and V.S. OSKOTSKI

1984) and UBe13 (Volovik and Gor'kov 1984), which illustrate the T 2 and 7' depen~- dences in the low-temperature region.

8.3. Thermal conductivity of rare earth compounds in which superconductivity and magnetism coexist

For a long time it was thought that superconductivity and magnetism could not coexist (Ginzburg 1957). A small amount of magnetic impurities (up to ~ 1%) destroys superconductivity (Matthias et al. 1958, 1959, Suhl et al. 1959). Superconductivity is destroyed also during a phase transition from the superconducting to the magnetic state. However, at the beginning of the 1970s two classes of ternary compounds, RxMo6X8 (x = 1 or 1.2, X = S, Se) and RRh4B4, in which magnetism and superconductivity coexist, were discovered by Hamaker et al. (1979) and Fischer et al. (1979).

Several dozens of magnetic superconductors are known at the present time. In the system RxMo6Xs (rhombohedral-hexagonal crystal structure with rhombohedral angle near 80 °) all compounds with R from La to Lu (excluding Ce and Eu) are superconductors (Fischer et al. 1975, Shelton et al. 1976). Coexistence of magnetism and superconductivity is observed in R1.2Mo6S8 (R = Gd, Tb, Dy, Er) (Ishikawa and Fischer 1977, Moncton et al. t978) and RxMo6Se 8 ( x = 1, 1.2; R = Gd, Er) (McCallum et al. 1977a, b). In the system RRh4B4 (tetragonal crystal structure) superconductivity is observed in compounds with R = Y, Nd, Sm, Er, Tm, Lu and superconductivity coexists with magnetism in compounds with R = Nd, Sm, Er, Tm (Matthias et al. 1977, Vandenberg and Matthias 1977).

Different electrons are responsible for magnetism and superconductivity in the case of their coexistence. Superconductivity arises because of the formation of Cooper pairs of outer shell electrons, while magnetism is connected with the magnetic moments of the inner electron shells of the lanthanide ions. As has been shown by Buzdin et al. (1984), competition between superconductivity and ferromagnetism leads to the appearance of a new type of magnetic order in the superconducting phase a magnetic domain structure, which is an average between ferromagnetism and antiferromagnetism (Buzdin and Moshchalkov 1986).

Since then extensive investigations of the physical properties of systems with coexistence of superconductivity and magnetism have been carried out. The thermal conductivity of the RRh4B 4 system (R = Sm, Er, Tm, Lu) (Ott et al. 1980, Hamaker et al. 1981a, b, Odoni and Ott 1979, Odoni et al. 1981) has been investigated in detail. Table 9 gives some parameters of these materials which are necessary for a discussion of the results on to. T~I and T~ 2 are, respectively, the high and the second critical superconducting temperatures. As one can see from table 9, the width of the temperature interval of coexistence of superconductivity and magnetism is small:

~0.5 1 K for TmRh4B4 and SmRh4B4 and ~0.05 K for ErRh4B4. The last interval has been fixed only by means of precise measurements of the magnetic susceptibility and the linear expansion coefficient ( O t t e t al. 1978, Woolf et al. 1979). Therefore one can try to search for an influence of this effect on ~c only in TmRh4B4 and SmRh4B4. Nevertheless, we will consider in this section, to make the picture com- plete, all four compounds from table 9. LuRh4B~ can be considered a reference

T H E R M A L C O N D U C T I V I T Y O F RECs TABLE 9

Data on T~a, T~z , Tc and TN for some compounds RRe,~B,~.

189

Compound To1 Refs.* T~2 Refs.* T c Refs.* TN Refs.*

(K) (K) (K) (K)

SmRh4 B 4 2.72 I l l 0.87 [1, 2]

2.68 [2]

ErRh4B 4 8.7 [3-5] 0.98 [~3, 4] 0.93 [3, 4, 6]

TmRh~B~ 9.8 [7, 8] 0.4 [7, 8]

LuRh 4 B 4 1 t.6 [4]

*References: [1] Hamaker et al. (1979), [2] Ott et al. (1980), [3] O t t e t aI. (1978), [4] Woolf et al. (1979), [5J Odoni and Ott (1979), [6] Fertig et al. (1977), [7] Hamaker et al. (1981a), [8] Hamaker et al. (1981b).

material, because it undergoes only the superconducting transition. Figure 108 shows the temperature dependence of ~Cto t of LuRh4B~ (Odoni et al. 1981). tel and 1£ e are separated into the normal and superconducting state contributions by calculation.

At T ~ To a sharp increase of tc~ (tc[) occurs due to decreasing p h o n o n scattering by the electrons, and a decrease of tc2 (to s) occurs. At the lowest temperatures the thermal conductivity in the superconducting state 0d) is only due to ~c[. At T < 2 . 5 K td = tc[ and is proportional to T a, which is typical for boundary phonon scattering.

ErRh4B ~ is a prototype of compounds with two superconducting transition temperatures (T~ and

To2)

- "reentrant" superconductors (see table 9 and fig. 109). Figures 110 and 111 show the K(T) dependences. Unlike LuRh4B4, the contribution of ~c L

t o /('tot is small near T~. However, at low temperatures/£tot becomes equal to tc[ and

proportional to T 2"s (as in the case of LuRh4B4 (see figs. 110 and 11 la)). At T < 1.2 1.4 K lqo t increases with decreasing temperature and at T ~ T~ (T¢2) it has a maximum.

At T < To (To2) the behaviour of the thermal conductivity changes radically. Now the main heat carriers are not phonons (~c L decreases sharply again due to strong p h o n o n - e l e c t r o n scattering), but electrons, ~c L ~ ~c e and/£tot ⁼ Ne' This conclusion is proved by the temperature dependence of tgto t = /£e OC r~ which is typical for electron scattering by static defects and impurities. On extrapolation of T to zero, tro(T) tends

0 0

Tc x n

f f ~ e

! ....

5 1 0 1 5

T,K

Fig. 108. The temperature dependence of 1¢

for LuRh4B,~ (Odoni et al. 1981). Solid line is the experimental iCto t at H = 0, points give the experimental lgtot at H = 5 kOe, dashed lines show calculated values x2, 1~~, Jc~ and 1~,

190 I.A. SMIRNOV and V.S. OSKOTSKI

180 I" Tc2 T C 12kOe Tel

c~:a_ 120600~~

H= 0 ROe

o I 1 i I I I i i

0 1 2 3 4 5 6 7 8

T,K

Fig. 109. The temperature dependence of the electrical resistivity, R, for ErRh4B~ without a magnetic field and in a longitudinal magnetic field H = 12 kOe (Ott et al.

1978).

~c fi

Tq Xt°t / / ^-

Tc~ I ~/~I//- x e

t

t /

I ,~ 4 I

ilJ....'(, , a / i "i'-'?- ~.__l.~__e_l__gr- 7- q--

0 2 4 6 8 10 12 14

T.K

Fig. 110. The temperature dependence of ~ for ErRh4B 4 (Odoni et al. 1981).

The solid line is the experimental value.

The dashed line gives the calculated

~", 1<~, t¢[ and t~.

1.0 E

O.1

0.02

0.03 10

f m I '

X n // ~,S

T 2 . B 2 , X S

,(, Lt, ,o.

0.1 1.0

T,K

f

0 0.2 0,4 0.6 0.8 1.0 1.2 T , K

Fig. 111. (a) The temperature dependence of 1¢ for ErRh4B 4 in region I of fig. 110 (Odoni and Ott 1979). (b) A more detailed picture near the phase transition.

T H E R M A L C O N D U C T I V I T Y O F R E C s 191

to zero (fig. 11 lb). Besides, ~qot(T) can be described well by the Wiedemann-Franz law with L = Lo. Near the phase transition to(T) of ErRh4B4 shows hysteresis (fig. 11 lb), similar to the case of R(T) (fig. 109).

Figures 112-115 show ~c(T) and R(T) dependences for SmRh4B4 and TmRh4B 4 with the coexistence of superconductivity with antiferro- and ferromagnetic ordering, respectively (Hamaker et al. 1979, 1981a, b, Ott et al. 1980). At H = 0 in the regions of superconductivity (II), of superconductivity and magnetism coexistence (I), and also at the points Tc (0.4 K for TmRh4B4) and TN (0.87 K for SmRh4B4) no anomalies in ~c s behaviour are observed except for the decrease of the value n in the dependence

~c s oc T" in the region of superconductivity and magnetism coexistence from _~2.8 (typical for tc[) to ~-1 0doc T) with decreasing temperature. Ott et al. (1980) have connected this effect with the appearance of a magnon contribution against a back- ground of small tcL.

In the normal state [which requires a field ~ 2 kOe for SmRh4B 4 (fig. 112) and

~ 3 kOe for TmRh4B 4 to quench superconductivity (fig. 114)] ~cnoc T, as in the case

10 °

10 -1

10-2

i0-3 0.01

t I I

T I

/ / ~ s I

r/ N T: I

/ I t l /

0.1 1.0

T,K

Fig. 112. The temperature dependence of 1~ for SmRh4B 4 without a magnetic field and in a longitudinal magnetic field. The numbers by the curves are the values of H in kOe (Hamaker et al. 1981b).

100 - ~

8 0

6 0

4 0

2 0

0 0

J / / / / / / L /

i/1.7

^~1.5

/ I

I I

/ i I

I I

! I

/ I

0.5 N

- - - - T - - "

~

^{f F} ^{/ r}

11.0

I I /

1.0 1.5

T,K

I I

I f

I I

I f J

~

^{0 . 4}

J I H=O kOe I I I I

/J I

2.0 2.5

-4 Fig. 113. The temperature dependence of the electrical resistivity, R, for SmRh4B 4 without a magnetic field and in a longitu- dinal magnetic field. The numbers by the 3.0 curves are the values of H in kOe

(Hamaker et al. 1979, O t t e t al. 1980).

Dalam dokumen Handbook on the Physics and (Halaman 190-200)