Anisotropy of transport properties in the Kondo compound CePt 2Si2 : experiments and theory

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Anisotropy of transport properties in the Kondo compound CePt 2Si2 : experiments and theory

A.K. Bhattacharjee, B. Coqblin, M. Raki, L. Forro, C. Ayache, D. Schmitt

To cite this version:

A.K. Bhattacharjee, B. Coqblin, M. Raki, L. Forro, C. Ayache, et al.. Anisotropy of transport prop-

erties in the Kondo compound CePt 2Si2 : experiments and theory. Journal de Physique, 1989, 50

(18), pp.2781-2793. �10.1051/jphys:0198900500180278100�. �jpa-00211102�

Anisotropy ^of transport properties in the Kondo compound CePt2Si2 : experiments ^and theory

A. K. Bhattacharjee (1), ^B. Coqblin (1), M. ^Raki (2), ^{L. Forro} (2, *),

C. Ayache (2) and D. Schmitt (3)

(1) Laboratoire de Physique ^{des Solides, Bât. 510,} Université Paris-Sud, ^Centre d’Orsay,

91405 Orsay, ^France

(2) Service des Basses Températures, DRFG/CENG, 85 X, 38041 Grenoble Cedex, ^France

(3) Laboratoire Louis Néel, CNRS, 166X, 38042 Grenoble Cedex, ^France (Reçu ^{le 29 mars 1989,} accepté ^sous forme définitive le 19 mai 1989)

Résumé.

Nous présentons ^{ici des mesures} de la résistivité électrique, ^du pouvoir thermoélectri- que ^et de la conductivité thermique d’un monocristal de CePt2Si2 ^{de 1,5 à} 300 K. Nous présentons

aussi un modèle théorique, ^{basé sur} l’Hamiltonien d’échange ^{effectif et} prenant ^en compte l’anisotropie ^du temps de relaxation des électrons de conduction, pour expliquer ^{la forte} anisotropie observée dans les propriétés ^de transport ^du composé Kondo du cérium CePt2Si2.

Abstract.

Measurements of the electrical resistivity, thermoelectric power and thermal

conductivity ^{from 1.5 to 300 K in} single crystal CePt2Si2 ^are reported ^{here. A} theoretical model, based on the effective exchange Hamiltonian and taking ^{into account the} anisotropy ^{of the}

conduction electron relaxation time, is also presented ^to explain ^the strong anisotropy observed in the transport properties of the cerium Kondo compound CePt2Si2.

Classification

Physics ^Abstracts

72.15E - 72.15J

72.15Q

71.70C

75.20H

1. Introduction.

Cerium Kondo compounds ^{have been} extensively studied in the last years from both ^an

experimental ^{and a} theoretical point of view. At sufficiently high temperatures, ^{i.e. for} temperatures larger than the Kondo temperature Tk and the overall crystal-field splitting,

these compounds ^{have a} magnetic susceptibility which follows a Curie-Weiss law with a

magnetic ^moment corresponding roughly ^{to the Ce 3,} trivalent ion or to the 4f1 configuration

and show a decrease of the magnetic resistivity ^with increasing temperature, generally ⁱⁿ Log ^{T in a} given température range. At very low temperatures compared ^to Tk, ^cerium

Kondo compounds ^are characterized by ^a heavy-fermion ^behaviour giving ^{rise to enormous}

values of the magnetic susceptibility and electronic specific ^{heat term as} experimentally

observed in CeA13 [1], CeCu6 [2], CeCu2Si2 [3], ^CePtSi [4], ^CePtln [5], CeInCu2 [6], CePd3B [7], CeCu4Ga [8], CeCu4Al [9] and many other cerium compounds. ^{Three different}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180278100

behaviours have been found ^at very low temperatures ^{in these} compounds : many of them,

such as CeAl2 [10], CeB6 [11], CeInAg2 [12] ^and roughly thirty other cerium compounds,

order magnetically, generally ^{in an} antiferromagnetic ^{or modulated} magnetic order. On the other hand, CeCu2Si2 ^becomes superconducting ^below roughly ^{0.6 K} [3]. Finally, ^{the other} compounds, ^{such as} CeAl3 [1], CeCu6 [2], CeCu2Si2 [3], CeInCu2 [6], CeRu2Si2 [13] ^or CePt2Si2 [14] ^{were initially} thought ^{to exhibit no} long-range magnetic ^{order and to become}

non magnetic ; ^the problem ^{is now more} controversial since a very weak magnetic ^{order has}

been detected in some compounds ^{such as} CeAl3 [15] ^{and moreover} strong short-range magnetic correlations have been any way observed in ^most of these compounds.

The magnetic ^and transport properties of cerium Kondo alloys ^and compounds ^{have been}

also extensively ^studied experimentally ^and theoretically in both the ^« high-temperature ^{» and}

« low-temperature » limits, ^{i.e. for} temperatures respectively larger and smaller than the Kondo temperature Tk. ^The transport properties, namely the electrical resistivity [16, 17], ^the magnetoresistivity [17], the Hall effect [18], the thermoelectric power [19] and the thermal

conductivity [20] ^{of cerium Kondo} alloys ^and compounds ^{have been} computed ^{in detail} by ^the Orsay group within ^a third-order calculation performed in the framework of the effective

exchange Hamiltonian [21] (the ^so-called « Coqblin-Schrieffer Hamiltonian »), ^{which de-}

scribes the large ^resonant scattering connected with the Kondo effect and takes into account the crystalline field effect of cerium. A good agreement ^with experimental data has been found in a broad temperature domain, ^{i.e. for} temperatures larger ^than Tk ^{where the} perturbation theory ^is valid, ^{for most of the} transport properties in many cerium systems.

First of all, ^the previous calculation [16] ^{of the} magnetic resistivity explains ^the experimental

behaviour of roughly forty ^cerium compounds ^at sufficiently high temperatures, ^{i.e. an} increase of the resistivity ^with increasing temperature, ^{then a maximum} corresponding roughly ^{to the} crystalline ^field splitting ^and finally ^a decrease in Log ^{T at} higher temperatures.

The ^« high-temperature ^» Log ^T decrease of the resistivity ^{can be} in fact considered ^now as a

signature of the Kondo effect in cerium compounds. This calculation has been extended to low temperatures [17] ; ^it yields ^a peak ^{at the} ordering temperature ^{with a} sharp ^decrease

below it and generally another slow decrease in approximately log ^T above it in the case of

compounds which order magnetically (such ^as CeAl2 ^or CeS). ^{On the} contrary, ^{a continuous} decrease with decreasing temperature ^{below the} crystal field maximum and a T2 behaviour at very low temperatures ^is generally obtained in compounds which become non magnetic (such

as CeAl3 ^or CeCu6) ; ⁱⁿ CeCu2Si2, coherence effects which develop when Cerium atoms are

demagnetized, give ^{rise to a} second weak maximum before the low temperature ^decrease

[22].

The other transport properties ^{have been} computed within the same model in the « high temperature ^{» limit. A} negative magnetoresistivity ^{has been} found in the temperature ^domain around the Néel temperature, ⁱⁿ good agreement ^with experiments ⁱⁿ CeAl2 ^or CeB6 [17].

The temperature dependence of the thermoelectric power exhibits ^{one or} two, positive ^or negative, peaks ^{which can reach} large absolute values and are located at a temperature corresponding ^{to a fraction 1/6 to} 1/3 of the crystal-field splitting [19]. ^A qualitative agreement with experiments ^can be found in most cerium compounds. ^Moreover the calculations have

quantitatively accounted for thermoelectric power data in Ce1-xLaxAl3 alloys [19]. Finally,

the thermal conductivity ^{has been} recently computed ^{in the same model} [20] ; this theoretical

electronic contribution to the thermal conductivity ^behaves linearly ^{at low and} high

temperatures, while it exhibits an inflexion and even a maximum followed by ^{a minimum at a}

temperature approximately given by ^{a fraction 1/3 to} 1/2 of the crystal-field splitting. ^A

quantitative agreement is obtained with experimental ^{data in} CeAl2, CeCu2Si2 ^and

CePt2Si2 polycrystalline samples [20].

The purpose of thé présent ^{paper is to} study ^both experimentally ^and theoretically ^{a new}

feature observed in the transport properties of cerium Kondo compounds, namely, ^the anisotropy of the electrical resistivity, the thermoelectric power and the thermal conductivity

found in CePt2Si2 single crystals ^{and to} present ^a theoretical explanation ^{of this} anisotropy,

based on the effective exchange Hamiltonian of references [16, 21], by taking ^{into account the} anisotropy of the conduction electron relaxation time. Experimental evidence for anisotropy

effects has been previously found in the resistivity ^of CeCu2Si2 [23], CeA13 [24] ^and CePt2Si2 [25] single crystals and in the thermoelectric power of CeRu2Si2 single crystals [26].

The resistivity ^of CeCu2Si2 ^and CeAl3 is smaller along the c-direction than perpendicular ^to it,

in contrast to the case of CePt2Si2 [25].

In the present paper, ^we describe the experimental ^{results on the} transport properties ^of CePt2Si2 single crystals ^{in the next section,} the theoretical model in section 3 and the

comparison ^between experiment ^and theory in section 4.

2. Measurements of transport properties ^of single crystal CePt2Si2.

The ternary compound CePt2Si2 ^{has been} recently recognized as a new cerium Kondo

compound, which shows no sign ^of magnetic ordering ^or superconductivity ^{down to 50 mK.}

Let us summarize firstly the main known properties ^of tetragonal CePt2Si2. ^The magnetic resistivity ^of polycrystalline CePt2Si2 ^increases regularly up ^{to a} maximum at 76 K and then decreases logarithmically ^{down to room} temperature ; ^{at low} temperatures ^below roughly

15 K, ^the resistivity ^{follows a law} given by ^{AT 2} [27]. ^The magnetic susceptibility ^of CePt2Si2 ^{follows a} Curie-Weiss law above 150 K with a magnetic ^{moment close to that of} Ce3+, then goes through ^a maximum around 60 K and is almost constant below 20 K reaching

a value of 3.6 x 10-3 emu/mole at 1.5 K, after correction of impurity ^effects [14, 27].

Measurements of the specific ^{heat C} yield ^an extrapolated ^{value of} C/T ^{down to 0 K} equal ^to

80 [14]-86 [27] ^{mJ/mole K2 and a} maximum of C/T ^{close to 120 mJ/mole K2 at} 2 K. Both the low temperature value of the magnetic susceptibility and the A coefficient of the low temperature T2 law of the resistivity ^decrease rapidly with pressure [27].

Recent measurements on CePt2Si2 single crystals have shown the strong ^{effect of} anisotropy

on the magnetization [28] and electrical resistivity [29]. ^{The low} temperature magnetization ^is larger along ^the (100) direction than along ^the (001) direction and the observed strong decrease of the initial susceptibility is associated with the possible existence of a metamagnetic

transition at about 30 kOe in the basal plane [28]. These results differ from the magnetization

measurements in CeRu2Si2 which show clearly ^an important metamagnetic ^transition occurring along the c-direction at about 80 kOe [13]. ^The analysis ^of magnetization ^and

inelastic neutron scattering measurements in single crystals [30], consistent with resistivity ^and specific ^heat data, ^{has led to an} evaluation of the crystal-field splitting scheme : the ground

state is the doublet ± 1/2 and the two doublets ± 3/2 and ± 5/2 lie at approximately ^{80 and}

230 K above it.

We report ^here measurements of the transport properties ^of CePt2Si2 single crystals along

the c-direction and along ^{one direction} (110) in the basal plane ^{of the} tegragonal ^structure.

Resistivity measurements were already presented ^{in a short} paper [25] ^{and we} report ^here

new data on the thermoelectric power and the thermal conductivity.

The two samples used in the present study ^{were cut from the same} single crystal ; ^{the effect}

of heat treatment is reported ^elsewhere [31]. ^{For the} present measurements, ^the samples ^were equiped ^{with an} Au-Fe/Chromel thermocouple for the determination of the thermal gradients

while the thermopower ^{was obtained} by using copper wires. Experiments ^{were carried out} by

sweeping continuously ^the heating power through ^the sample in both directions following ^a

procedure described in reference [31].

In figure 1, ^we reproduce the results for the resistivity [25], which shows stronger ^{values for} the (001) ^{direction over the whole} investigated temperature range. The thermoelectric power is shown in figure ^{2 and the} anisotropy appears ^to be again very strong. Along ^the (110) direction, ^the thermoelectric _{power is} always positive ; ^at 300 K its value is 5.4 tV/K ^{and it}

increases logarithmically ^with decreasing temperature ^{down to 50-55 K} where it reaches the maximum value of 31 f.L V /K. ^{At lower} temperatures, ^the thermoelectric power decreases

monotonically ^with decreasing temperature ^{down to zero. At 300} K, the thermoelectric power along ^the (001) direction is negative and its absolute value, 12.3 f.L V/K, ^is higher ^{than in}

the former case. Around and below the room temperature, ^the thermoelectric power varies in

a logarithmic manner ; it crosses zero near 180 K and it reaches ^a maximum of 8.8 f.LV/K

around 85 K. Below this maximum, ^{it exhibits an} oscillatory behaviour, crossing ^{zero around}

56 K and 22 K and going through ^a minimum of - 7 f.L V /K ^at 35 K. At 4 K, ^it joins ^{the value}

of the first orientation (110) after another maximum at 8.8 K. The thermoelectric power of

CePt2Si2 presents, therefore, ^a strong anisotropy ^with larger values for the component perpendicular ^{to the} c-axis ; ^{the same} tendency is observed in CeRu2Si2 [26] ^{but the} thermopower ^{curves have a different} temperature dependence ^{in the two cases.}

Fig.l.

Fig. 2.

Fig. ^1.

Experimental plots of the electrical resistivity along ^the (001) ^and (110) directions for a single crystal CePt2S’2 compound-.

Fig. ^2.

Experimental plots of the thermoelectric power along ^the (001) ^and (110) directions for a

single crystal CePt2Si2 compound.

Figure ³ gives ^{the two curves} of the thermal conductivity K ^versus temperature ^{for the two} directions (110) ^and (001) ^{and once} again ^{we can observe a} strong anisotropy which increases

steadily ^{from 4 to} 300 K. The thermal conductivity ^is larger along ^the (110) direction, ^as expected from the electrical resistivity ^{data. The} temperature dependence ^{of both} components of K shows a smooth oscillation separating ^two quasi-linear regions ; the inflexion point ^{of the}

two curves is located approximately ^{at 50 K. Our} previous investigation ^{on a} polycrystal [32]

has shown a similar behaviour which appears ^to be typical of the thermal conductivity ^for

most cerium Kondo compounds. ^{Inset of} figure 3 shows the Wiedemann-Franz ratio

L = Kp /T ^versus temperature. ^{Between room} temperature ^and roughly ¹⁵⁰ K, ^the

Wiedemann-Franz ratio L is constant for both components ^{and is a little} larger ^{than the}

Sommerfeld value Lo

^{2.45 x} 10- 8 WOK- 2 of the Lorenz number ; ⁱⁿ fact, ^this discrepancy

Fig. ^3.

Experimental plots ^{of the thermal} conductivity along ^the (001) ^and (110) directions for a single crystal CePt2Si2 compound. ^Inset gives the reduced Lorenz number along ^{the two} directions.

with Lo above 150 K is slight ^{and not} far from the uncertainty ^{on the absolute values.} But, below 150 K, the ratios defined along ^{the two} directions have opposite behaviours. That

corresponding ^{to the} (001) ^{direction increases} significantly and reaches a plateau ^{between 4}

and 50 K corresponding ^{to a value of} roughly ^1.5 Lo. ^{On the} contrary, ^{the ratio} L along ^the (110) direction decreases steadily ^{down to} roughly ^0.7 Lo ^{at 4 K, after} crossing Lo ^{at 35 K.}

Thus, experimental measurements of the electrical resistivity, ^thermal conductivity ^and

thermoelectric power show ^a strong anisotropy between the directions (110) ^and (001) [33]

and we will present ^a model able to explain it in the next section.

3. Theoretical model.

We shall compute here the three transport coefficients in single crystals, i.e. the electrical

resistivity ^{p, the thermal} conductivity K ^and the thermoelectric power S along ^the c-direction and along ^{a direction} in the basal plane ^{of a} tegragonal ^or hexagonal crystal structure, ^{as in}

CePt2Si2 ^or CeAl3 respectively. ^{In the} present paper, ^we do not consider the anisotropy ^within

the basal plane ; this effect will be described elsewhere [34].

There are certainly ^several origins ^{for the} anisotropy, ^{but we} will show here that the effective exchange Hamiltonian of reference [21], including crystal-field ^{effects as in}

reference [16], yields ^a large anisotropy ⁱⁿ transport properties ^{of cerium Kondo} compounds.

A first calculation using the self-consistent ladder approximation by applying Abrikosov’s

pseudo ^fermion technique ^{has been} recently performed by Kashiba et al. [35] ^{in order to}

compute numerically ^the electrical resistivity. Here, ^we present ^{the main} points ^{of the}

derivation of the third-order calculation of the transport properties, ^as in references [16, 19, 20], ^{but now in the case of a} single crystal. ^Detailed calculations will be published ^elsewhere [34].

The effective resonant-scattering Hamiltonian can be written as :

with the usual notations and definitions of references [16, 21] : ctM represents the creation operator ^{of a} conduction electron of energy Ek in the partial-wave ^{state M} (defined by ^the quantum numbers Q = 3, s = 1/2, j

5/2), ^while cù represents the creation operator ^{of a} localized 4f electron in the state M of the Ce atom. M denotes, ^as usual, ^the quantum ^number characterizing ^an eigenstate of energy EM in presence of the crystalline ^field (CF), i.e. either any ^state of the doublet fy ^{or the} quartet T 8 for the cubic symmetry (as ⁱⁿ CeAl2), ^or any

eigenvalue ±1/2, ^± 3/2, ± ^{5/2 for the} hexagonal ^or tetragonal symmetry (as ⁱⁿ CePt2Si2). ^The energies Ek ^and EM ^are defined with respect ^to the Fermi energy. The exchange integrals ^are given by :

defined with a cut-off D. In equation (2), Vu ^{is the} hybridization parameter between 4f and conduction electrons. The VMM parameters, ^which represent pure direct scattering, ^{are all}

taken equal ^{to a common value V.}

Our previous calculations [16, ^19, 20] ^{of the} transport properties ^{have been} always performed ^for polycrystals ^{and we} have there approximated the relaxation time of a

conduction electron by ^an isotropic average ^over the different k directions of the conduction electrons. In the present ^{case of a} single crystal, ^we compute ^the transport properties along

different crystallographic directions i (defined by the direction of the electrical field or that of the temperature gradient) ^{and we must} calculate the relaxation time _Tku for a conduction electron plane ^{wave of} wavevector k and spin ^{(T, which} turns out to be highly anisotropic.

The classical formulae for the electrical conductivity a

1 /p , ^thermal conductivity

K and the thermoelectric power S can then be written, ^{for each} direction i, ^{as :}

where the integrals Kn (written ^{in the} following Kn for each direction i) ^{are defined} by :

where the conduction-electron energy Ek is defined by (1) ^and f k is the Fermi-Dirac distribution function. The relaxation time _Tku is given ^here by :

In the expression (7), Cm’ (f2 k) represents ^the weight ^{of the} partial ^wave 1 kM> ^{inside the} plane ^wave 1 ko, > [21]. The coefficients Cm’ (f2k) ^are given by [21] :

where Yf({1k) ^{are the} spherical harmonics of order 3.

The partial ^wave relaxation time _TkM is given by :

RkM ^and SkM ^are, respectively, the second- and third-order terms and they ^are given by ^the expressions (38) ^and (39) of reference [16] without summation on M, ^{i.e. :}

In the expression (9), ^m, vo ^{and c are} respectively ^{the mass} of the conduction electron, ^the

considered volume and the cerium concentration.

As usual in the calculation of the transport properties, ^{we need to} invert the relaxation times in order to compute ^the integrals Kn ^{and we use} the third-order perturbation approximation ^which implies that the third-order terms are much smaller than the second- order terms. Thus, ^{we can write :}

We will consider now the k-dependence ^of both LeM ^RkM and LeM ^{SkM. Let us explain}

M M

how we proceed : the average value RkM ^of RkM ^can be written as :

where Rk = E ^{Rkm is} obtained from the expression (10).

M

However, ^{in the} expression (12), ^{if we} replace RkM by its average value Rk/ (2 j ⁺ 1 ), ^we

obtain :

The isotropic approximation of référence [16] implies taking ^an average value of

CM equal ^to 2 j 1+ 1 ’ ^{2 ’+1 i.e. equal to - . ô However, the correct average given by (14) is found to}

be 1, which leads to a discrepancy of factor 3 between the coefficients of the electrical 2

resistivity and thermal conductivity obtained below and those in références [16] ^and [20]

respectively.

Then, ^{we write} Rkm ^as RkM ⁺ (Rkm - RkM) ^{and we} expand ^{the first term of} (12) up ^to the first order in (RkM - RkM) / Rk. ^This corresponds ^{to an} expansion ⁱⁿ (J MM’ /91)2 ^since RkM is of order 912 and (RkM - Rx,,) ^{of order} 1 jMM@ 12 ^this expansion is valid in the present

case where 1 ’U 1 ^is larger than 1 J MM’ 1.

^This expansion ^{was not} considered in our previous

short paper [25]. Finally, ^{we have} replaced RkM by its average value Rkm in the denominator of the second term of (12) ^to be consistent with the third-order approximation.

Thus :

The integrals Kri given by (6) ^become equal ^{to :}

where mi ^{is the effective mass} along ^the i-direction, ki ^the i th _component of k and

kF ^{the Fermi wave number.}

In the expression (16), ^the integral ^over f2k gives different results for the different directions. In order to perform ^the integration ^over f2k, ^{we write} ki and kx ^{as a} function of

spherical harmonics, ^{as follows :}

and, ^after performing ^the integration ^{of the} products ^of spherical harmonics, ^we get ^different results for the integrals Kn ^and Kn ^and finally ^we obtain different expressions ^{for the} transport properties along ^{the z-} and x-directions.

We present ^the results here only ^{in the case} of three doublets with M values equal ^to

± 1/2, ^± 3/2, ^± 5/2 ; thus, according ^to (10) ^and (11), RkM

Rk - M ^and SkM

Sk - M. ^{In this}

case, there is no anisotropy in the basal plane ^{and the} transport coefficients are equal ^{to each}

other along the x-and y-axes. The situation corresponds ^{to the case of} CePt2Si2 ^{which has a}

± 1/2 ground ^{state and two excited states ± 3/2 and + 5/2} [28, 30].

The expressions (16) ^for Kn become, therefore :

with :

and

Using ^{the different} energy integrals given in references [16, ^19, 20] for the three transport properties, ^{we can} finally write them down for the z- and x-directions. We have seen in the

previous calculations [16] that the so-called « fk

^{1/2 »} approximation yields results very close to the exact ones and that it simplifies very much the calculations. Thus, ^{we will use it}

here and, ^when f k ^{in the} expression (9) takes the f k

^{1/2 value at} the Fermi energy,

Rk ^and A k(2) ^become independent ^{of k} and will be called respectively R ^and A (2) in the

following. ^{If we} call also :

the resistivity p along the i-direction is given by :

The numerical coefficient of the resistivity ^is here three times larger ^{than the} coefficient

previously derived in reference [16], ^as already explained. However, apart ^{from this} numerical discrepancy, ^{the first two terms of} (25) give exactly ^the previous calculation of reference [16], while the last terms represent ^the anisotropy ^{of the} resistivity.

Using ^{the same} approximations ^{as for the} electrical resistivity, the thermoelectric power

Si along the i-direction is given by :

The first term of (26) gives ^{the same result as} previously ^{derived in reference} [19] ^with exactly

the same coefficient, while the second term yields ^the anisotropy of the thermoelectric power.

Finally, the thermal conductivity ^K’ along ^the i-direction is given by :

where we find again the thermal conductivity previously derived in reference [20], apart ^from the numerical factor of 3, ^{and the two} anisotropy ^{terms in} A (2) and A (3).

Thus, the formulae (25), (26), ^and (27) give ^the expressions of the electrical resistivity pi, thermoelectric power Si and thermal conductivity Ki along the i-direction and we will use

them in the next section for the specific ^{case of} CePt2Si2. ^{Let us} finally remark that the electrical resistivity ^{behaves as} mf ^{and the thermal} conductivity ^as llm?, ^{while the}

thermoelectric _{power is} independent ^{of the effective mass} mi along ^the i-direction.

4. Application ^to CePt2Si2 compound.

We shall apply ^{now the} preceding theoretical results to the experimental measurements

presented in section 2. In fact we have computed only ^the electronic contributions to the

transport properties, ^via « impurity » scattering. ^The phonon contribution must be also

considered for heat transport, ^{as well as the} phonon scattering of electrons. In the case of the electrical resistivity, ^{the usual} procedure ^{is to take the} phonon scattering contribution equal ^to

the resistivity ^{of the} equivalent ^Lanthanum compound. But, ^{in the} present ^{case of} CePt2Si2, ^no resistivity is available in single crystal LaPt2Si2. Moreover, the electrical resistivities of polycrystalline LaPt2Si2 ^and CePt2Si2 ^{show an} increase with temperature ^which is much larger ^{than that} presented ⁱⁿ figure ^1. So, it ^{is at} present impossible ^{to have a direct}

estimation of the phonon contribution to the electrical resistivity. ^The problem ^of separating

the phonon and electronic contributions is even more difficult for the thermal conductivity

and thermoelectric power.

However, ⁱⁿ spite ^{of these} difficulties, according ^to previous analyses ^{of the} transport

properties, ^we think that the Kondo scattering electronic contribution is the dominant one

and on this basis we derive here theoretical curves for the three transport coefficients with the

same set of parameters. ^{Let us discuss now} the values used here for the different parameters.

First, ^we take thé crystal-field ^scheme recently ^derived by magnetization and inelastic neutron

scattering irr single crystal CePt2Si2 [28, 30] : ^the ground ^{state is the 1/2 doublet and the two}

doublets ± 3/2 and ± ^{5/2 lie at} respectively 80 and 230 K above. The exchange parameter

JMM for the ground ^{state doublet ±} 1/2 is chosen here equal ^{to a} typical ^value Jl,

^{0.1 eV}

and we take also the same parameters ^as previously ^used for cerium Kondo compounds [16] :

the hybridization parameter Vu entering (2) ^is equal ^to Vu

^0.07 eV, ^the density ^{of states of}

the conduction band at the Fermi energy for ^one spin ^direction n (EF) ^{= 2.2} statesleV.at., ^the

cut-off D

850 K, the number of conduction electrons per ^{atom z} = 3, the atomic volume vo

²³⁴ atomic units and kF

^0.72 atomic unit. Finally, the direct potential parameter cU and the two effective masses mx and mz ^are chosen in the present work, ^{in order to obtain}

an optimal fit of the anisotropic transport properties ^of CePt2Si2. First, ^{we take}

CU = - 0.26 eV, ⁱⁿ agreement ^with previous results, ⁱⁿ particular for the thermal conductivity [20]. ^{Then, the two effective masses} have been chosen as follows : if we adjust the theoretical

curves to the experimental ^ones for the electrical resistivity ^{at its} experimental ^maximum value, i.e. 60 K along the z-direction and 90 K along ^the x-direction, ^we get ^{the two values}

mi = ^{21.3 and} mx

8.2 in atomic units ; if, ^{on the} contrary, ^{we make this} adjustement ^{for the}

thermal conductivity ^{at 100} K, ^we get ^{the two values} mi = 10.8 and mx

^{4.7. Thus, we take}

here the mean values for mi ^i.e. mi ^{= 16 and} mx

^6.5, (or ^mz

⁴ and mx

2.55 in atomic

units).

Figures 4, ^{5 and 6} give the theoretical plots ^versus temperature ^of respectively ^{the electrical} resistivity, thermoelectric power and thermal conductivity. ^When comparing figures ^{1 and 4}

on the one hand and figures ^{3 and 6 on the} other, ^{we see a} relatively good agreement ^between theory ^and experiment for both the electrical resistivity and the thermal conductivity. ^{In the}

first case (p ), ^the shape ^{of the} curves, the magnitude, ^the ratio of the values along x ^{and z,}

which is indeed fixed by the choice of mx/mz, ^and finally ^the respective positions ^{of the}

maxima are similar in theory ^and experiment, except ^at very low temperatures ^where coherence effects develop ^and single-impurity third-order perturbation calculations are no

longer valid. In the second case (K), there is also a good agreement for the ratio and the shape

of the two curves, in particular ^{with two} inflexions at approximately ^the right temperatures and a more pronounced ^inflexion along ^the Oz-axis ; ^this good agreement ^is clearly ^connected

with our choice of mx/mz ^{and CU.}

On the other hand, ^the agreement between 2 and 5 for the thermoelectric power appears ^to

be relatively good ^for Sx and poor for Sz, ^{but we must note} that the theoretical thermoelectric

power does ^not depend ^{on the} adjustable ^{effective masses} mx and mz. For Sx, ^the shape, ^the

position of the maximum at roughly 50 K and the positive values obtained by theory agree

with experiment ^{and there is a} disagreement only ^with respect ^{to the} magnitude ^of

Fig. 4.

Fig. 5.

Fig. ^4.

Theoretical plots of the electrical resistivity along ^{the z} 01 c) direction and along the direction x or y perpendicular ^{to it. The following} parameters have been used : III

0.1 eV for the ground ^{state ;}

the ground ^state is the ± 1/2 doublet and the ^two excited doublets ± 3/2 and ± 5/2 lie ^at respectively ⁸⁰

and 230 K ; V kf

^{0.07 eV ; n(EF)}

2.2 states/eV. at. for one spin direction ; D

850 K ; cU=-0.26eV; ^z

³ electron/atom ; ^v o

234 atomic units ; kF

0.72 atomic unit ; mz

4 and mx

2.55 atomic units.

Fig. ^5.

Theoretical plots of the thermoelectric power along ^{the z} 01 c) direction and along ^{the direction}

x or y. The parameters ^{are the same as} those used in figure ^4.

Fig. ^6.

Theoretical plots of the thermal conductivity along ^{the z} 01 c) direction and along the direction

x or y. The parameters ^{are the same as} those used in figure ^4.

Sx. ^{On the} contrary, ^{we obtain} only two extrema instead of three for Sz, but Sz ^is generally negative and much smaller than Sx, ⁱⁿ agreement ^with experiment.

In conclusion, ^we have obtained a semi-quantitative agreement for the three transport properties along ^two directions with the same set of parameters. ^In particular, ^the positions ^of

the maxima for _p and Sx and of the inflexions for K, ^{as well as} the relative values of

Sz ^and Sx, ^indicate clearly that the values used here for the crystal-field scheme and in

particular for the overall splitting ^are certainly ^{correct for} CePt2Si2. ^The values used here for

JMM, ^{and ’U, which} play obviously ^a very important ^{role for} determining ^the shape ^{of the}

curves, are quite reasonable and in good agreement ^with previous determinations [16, 19, 20].

Finally, ^{we have} taken mZ

4 and _mx

2.55, which allow a good interpretation ^{of the} anisotropy ^{of p} and K. At this point, these values are rather phenomenological ; ^{to our} knowledge, ^{no direct} study of the Fermi surface has been carried out and we have ^not really

considered the anisotropy of the Fermi surface in the integrals.

5. Conclusion.

We have presented ^{in the} present paper ^a complete ^{set of} measurements for the transport properties ^of single crystal CePt2Si2 along ^the c-axis of the tetragonal ^{structure and} perpendicular ^{to it.} The observed anisotropy ^is quite large ⁱⁿ CePt2Si2, ^as previously ^shown

for the resistivity ^of CeAl3 [24] and the thermoelectric power of CeRu2Si2 [26]. On the other

hand, ^{we have} presented ^an explanation ^{of this} anisotropy ^{in terms of the} anisotropic

relaxation time resulting ^{from the} Coqblin-Schrieffer Hamiltonian (1) appropriate ^{for cerium} impurities. ^{The most} striking point ^concerns the theoretical calculation of the thermoelectric power which shows ^a strongly anisotropic behaviour, ^{with even values of different} signs according ^to the considered directions, ^{and this} anisotropy ^{is due} only ^{to the} anisotropic

character of the Hamiltonian (1) ^since the thermoelectric power does ^not depend ^{on the}

effective masses. We must add that this intrinsic effect of the anisotropic Hamiltonian (1)

exists also obviously for the electrical and thermal conduction, although there is also an

anisotropy ^effect phenomenologically ^described by ^{a ratio} mz/mx taken here different from 1.

Further studies of transport properties ⁱⁿ single crystals would be necessary ^to better understand the anisotropy in cerium Kondo compounds.

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