Frequency dependent transport in heavy fermion systems

Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 207-212 North-Holland, Amsterdam

207

INVITED PAPER

FREQUENCY DEPENDENT TRANSPORT IN HEAVY FERMION SYSTEMS W.P. BEYERMANN, A.M. AWASTHI, J.P. CARINI and G. G R O N E R * Department of Physi{~ and Solid State Science Center, UniversiO, of California, Los Angeles. USA Frequency dependent conductivity measurements, performed in the millimeter wave spectral range on several heavy fermion materials, are analyzed in terms of a renormalized Drude theory. The experiments indicate an enhanced relaxation time with an enhancement factor approximately the same as that for the thermodynamic quantities. The crossover from the renormalized to unrenormalized response with increasing frequency is discussed, and a comparison is made with the various models of transport in highly correlated Fermi liquids.

1. Introduction Many thermodynamic and transport properties of so-called "heavy fermion" materials are well understood in terms of a gradual crossover from uncorrelated to coherent electron states with decreasing temperature [1]. The (loosely defined) coherence temperature T~ is roughly inversely related to the low temperature specific heat coefficient and to the coefficient A of the temperature dependent resistivity which is given by o(T) = Oo + A T 2, where O0 is the residual resistivity [2]. In the coherent regime, the density of states is expected to have a sharp peak near the Fermi level e v in analogy to the Kondo resonance for single impurities. The width of this feature is expected to be narrower for materials with smaller Tc. Much less is known about the frequency dependent response of these materials. As for the temperature-driven excitations and renormalizations, it is expected that the appearance of coherence depends on the frequency associated with the external probes; at low frequencies many-body renormalizations play a significant role, while at high frequencies these are not important. In this paper we briefly review several important features of the optical properties of these materials from the microwave to visible spectral range. Particular emphasis will be devoted to the low frequency (h~0 < kT~) properties where renorrealization effects are essential. We analyze our experimental results in terms of a renormalized Drude theory and compare the renormalization of the relaxation time with the renormalized thermodynamic quantities.

2. Models and theoretical predictions Various models [3,4] that take the s - f hybridization into account lead to a "hybridization gap" at an energy approximately kBT¢ (or the " K o n d o temperature" kBTk) above {V" A narrow manybody resonance also develops at {F. The conductivity G ( ~ ) and dielectric constant {r(¢~) resulting from an auxiliary boson calculation of the Anderson Hamiltonian are displayed in fig. 1. The essential aspects of fig. 1 are an unrenormalized plasma frequency 2

~Oo= (4~rnce /rnb)

1/2

(1)

where rn~, is the bandmass, and n c is the number of conduction electrons. A renormalized plasma frequency ~0~' is given by ~0]~ = [6(1 + nf/nc) ] '/2T*

(2)

within the framework of this theory. In the equation above nf is the f electron number. ~0~ is different from the frequency

w~ = (4~rne2/m* )t/2

(3)

which is the square root of the oscillator strength obtained from a simple Drude response with a renormalized heavy electron mass m*. The reason for the difference between wi~ and ~0~ is the important modification of the Coulomb screening in the coherent regime. The narrow resonance at zero frequency can be described by a renormalized Drude expression of the form O'r( ~ )

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//e2~- *

l

m*

1 -J- ~2T'2

(4)

14dP. Bevermann et al. / Transport in heaqv fermion systems

208

b°10~ 1"2 0 O l -S

contribution from these transitions to the dielectric constant. We note, that ~(o~) in eq. (4) follows from a particular mode of heavy fermion systems, the overall behavior is not model specific, and is expected for significant electron-electron or electron-phonon interactions.

~

3. Experimental details

b

While standard optical methods are used from infrared to visible frequencies, waveguide configuration measurements are employed in the millimeter wave spectral range. In general, a resonant cavity configuration is employed, where one wall of the (typically copper) cavity is replaced by the material to be measured. F r o m the change in the resonant frequency f0 and the quality factor Q, the real and imaginary parts of the surface impedance [7]

' ~",...__L~-"

Z~=R~ +iX~ I

0

/

I

2

I

I

4

6

t~w/k eTK Fig. 1. Expected frequency dependent conductivity or(w) and dielectric constant %(~o) for heavy fermions, in the coherent state. Or(~) is taken from ref. [3]. Note the different notations for the p l a s m a f r e q u e n c i e s in the text.

with a renormalized relaxation time that is conjectured to be [5] (S)

*/~ = m*/mh.

r is the high frequency unrenormalized Drude relaxation time, and m b is approximately equal to the free electron mass. With m * / m b ~ 10 3 and r-10 is s, r * is of the order of 10 12 s; the characteristic frequency 1/r* where Or(~O) drops to 1 / 2 of its zero frequency value is in the millimeter wave spectral range. For frequencies comparable to l / r * the high frequency properties are not expected to influence Or(~O), however, they do effect ~(to), and one expects .

~ ( , o ) - ~,~ - ( , o ~ / , o )

2

(6)

in the region l / r * < ¢0 < ~ . The b r o a d h u m p in the real conductivity at high frequencies represents interband transitions, due to the development of the hybridization gap A and e~b is the

(7)

can be evaluated. Using R~ and X~, the real and imaginary c o m p o n e n t s of the conductivity o(a0) = Or(~ ) + ioi(o0 ) are extractable, and c(~0) = oi(~)/q)~0. For a metal with ~o < 1 / r

Z, = (/toto/2o)w2(1 + i).

(8)

That is, the real and imaginary parts of Z~ are each equal to (/~0a~fl2o) 1 / 2 . The drawback of the technique lies in its cost and the fact that o can be evaluated only at individual frequencies. The important advantage is that two c o m p o n e n t s are measured, enabling an evaluation of both or(c0) and o~(~) without resorting to a K r a m e r s - K r o n i g analysis, which can be influenced by uncertain extrapolations. 4. CePd 3, a prototype example

The temperature dependence of the dc resistivity, and 2R2,//~o~o are shown [81 in fig. 2. Note, that for a simple metal 2 R ~ / / , 0 ~ o - O, and consequently fig. 2 indicates that the conductivity is strongly frequency dependent below 130 K, where 0de has a maximum. Various arguments suggest that the coherence sets in at this point as the temperature is lowered. Above about 130 K, both R~ and X~ can be described by eq. (8), and deviations from this equation occur at low tempera-

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Infrared and millimeter wave absorptivity in CePd3, measured at 4 K: infrared measurements are from ref. [9].

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This value is in close agreement with the enhancement of the specific heat [10]. Therefore, it appears that in CePd 3 the enhancement is the same for the thermodynamic quantities and the relaxa-

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conductivity of CePd s. The optical data are taken from Webb et al., ref. [9].

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tion time (i.e. eq. (5) is approximately valid for this material). We have also evaluated the temperature dependence of the relaxation time, by measuring R, and X at different temperatures. The inverse relaxation time together with the dc resistivity are displayed in fig. 5 as a function of temperature. The similar temperature dependence of the two parameters gives evidence that n/m* is independent of temperature below the resistivity maximum.

1

i

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g 5. Other heavy fermion systems Experiments, similar to those shown for CePd~ in fig. 2, have been conducted [8,11,12] on UPt:~, CeAI> CeCu~, and UBe]3. These materials span a wide range of thermodynamic mass enhancemerits. The conductivity Re o ( ~ ) measured at 7" = 5 and 300 K for UPt~ is displayed in fig. 6. together with the optical measurements of Marabelli et al. [13]. Again, at T = 5 K a renormalized

25

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W.P. Beyermann et al. / Transport in heavv fermion systems

Drude behavior is evident at low frequencies, with ~-* >> ~" as expected. In contrast, at room temperature Re(~0) is independent of frequency in the millimeter to infrared spectral range. As for CePd 3, the temperature dependence of 1/~-* closely follows the dc resistivity above 5 K. As a further example, ,Odc and 2R2/IXo w measured in UBe13 are shown in fig. 7. Even though the conductivity is frequency independent at high temperatures where o ( T ) increases with decreasing temperature, in the coherent regime when o ( T ) decreases, a substantial frequency dependence develops. Similar behavior has been found in CeA13 and in CeCu 6, and o(~o) is strongly frequency dependent also in these materials at low temperatures. We have used a renormalized Drude expression; eq. (4), to analyze our data and from the fit, the parameter 1/~'* can be extracted. This in turn was used to evaluate ~0~' with eq. (3). A comparison with the unrenormalized plasma frequencies obtained by optical methods enables us to evaluate the effective transport mass enhancements using the relation

0o)

Wo/W ~ = ( m * / m b ) a/2.

These values are displayed in table 1, along with the relevant parameters, ¢00, ~0~, and ~-*. The renormalized parameters are taken from data at T = 1.4 K. The above analysis assumes that at all

Table 1 Parameters for several heavy fermion systems. F o r the definitions of the various quantities see the text 1/'r* (1011S CePd 3 UPt 3 CeCu6 CeAI~ UBel3

4.2 6.7 13.9 5.9 12 11

w~ ~0 m*/mb 1) (1014 S-1 ) (1015 S 1) 4.7 15 7.4 3.3 8.4 2.8

3.5 5.3 5.3 4.8 b} 21 ~) 9.0

56 12a) T = I . 4 K 51 T = 5 K 208 640 1000

~) The b a n d m a s s for UPt 3 is between 10 and 20, ref. [11 and C.S. Wang et al., Physica B 135 (1985) 34. h} Estimation based on the assumed n u m b e r of carriers n = 9 / f o r m u l a unit. ~} Estimation based on the assumed n u m b e r of carriers n = 3 / f o r m u l a unit. For the unrenormalized plasma frequencies, see: P.C. Eklund et al., Phys. Rev. B35 (1987) 4250. Marabelli et al., Solid State C o m m u n . 59 (1986) 381. Webb et al., Phys. Rev. Lett. 46 (1981) 862.

211

,

lO 3

~

UBe,3 ~ e ~

eCeCu~

E

I0 zCePd~

10 I0 "~

1,0

ILO

. . . .

A(~-~"cm / \ Kz / Fig. 8. Mass enhancement versus the coefficient A of the low temperature resistivity D d ~ = A T 2 in various heavy fermion systems. The full line corresponds to A - (m */m~ )2.

frequencies oi(o0 ) << Or(~0), a condition valid for CePd3. and that the number of electrons entering both the renormalized and unrenormalized plasma frequencies are the same (i.e. n - n o ) . Further experiments where both Re o(~o) and E(o0) are measured are required to test the validity of the above assumptions. The measured effective masses m * / m e are shown in fig. 8 versus A, the coefficient for the T 2 dependence of the dc resistivity in the coherent regime. This is expected to scale as ( r n * / m e ) 1/2. The good correlation between A and m * / m e suggests a universal picture for all heavy fermions. We also note, that the optical and thermodynamic mass enhancements are approximately the same, supporting the conjecture expressed by eq. (5). 6. Conclusion

The spectral response of heavy fermion systems in their low temperature, coherent state is characterized by a narrow resonance at {F- The spectral width is probably determined by the enhanced relaxation time, which appears to be the same as the enhancement of the thermodynamic quantities. The dielectric constant also indicates a renormalized plasma frequency, c0~' << ~00, but further experiments are required to establish that this feature comes from electron electron correlation effects and not from features of the band structure

212

W.P. Bevermann et al, / Transport in heaw.ferrnion ,~3,stems

or electron-phonon interactions in the materials. While optical properties of both CePd~ [9] at UPt 3 [13] show a zero crossing of ~ well below that corresponding to the unrenormalized plasma frequency, further experiments on other materials as a function of temperature are required to clarify this point. While the interpretation of the low energy dynamics in terms of a renormalized Drude response leads to results in accord with theories, other contributions to the features observed are possible. A narrow many-body resonance (similar to the Kondo resonance) also leads to strong frequency dependence, and a Fermi-liquid behavior due to electron-electron interactions gives [14]

p(oo, t)= A(kBr)-+ B( hao)"

(11)

indicating a frequency dependent electrical conduction at low temperatures. Comparison with tunneling experiments may answer the first possibility, and impurity studies would help distingish between renormalized Drude and Fermi-liquid behavior. Finally, recent experiments on the high T~ material Y B a : C u 3 0 7 , have also been interpreted in terms of theories similar to those advanced here. Both the thermodynamic [15] and transport [16] parameters are renormalized with approximately the same factor of 10, and an unrenormalized mass is recovered at high frequencies. We are grateful to M.B. Maple, Y. Dalichaouch, Z. Fisk, and J. Smith for providing the samples used in this study. We have enjoyed useful discus-

sions with D. Pines. Z. Fisk, P. Fulde, and A.J. Millis. This research was supported by the NSF grant D M R 8620340.

References [1] See for example, P.A. Lee, T.M. Rice, J.W. Serene, l,.,1, Sham and J.W, Williams, C o m m e n t s Cond. Matt. Phys. 12 (1986) 99. [2] A. Auerbach and K. Levin, Phys. Rev. B34 (1986) 3624. [3] A.J. Millis and P.A. Lee, Phys. Rev. B35 (1986} 3624. P. Coleman, Phys. Rev. Len. 59 (1987) 1026. P. Fulde, J, Keller and G. Zwickhagl, Solid State Phys. 41 (1988) 1. D.1. Cox and N. Greve (to be published}. [4] D. Pines, (private communication). [5] H. Fukuyama, in: T h e o w of Heavy Fermions and Valence Fluctuations, eds. T. Kasuya and T. Saso (Springer-Verlag, Berlin. 1985). [6] C.M. Varma in ref. [5], and Phys. Rev. t,en. 55 (1985) 2733. [7] J.R. Waldram, Advan. Phys. 13 (1964) 1. [8] W.P. Beyermann, G. Gruner, Y. Dalichaouch and B.M. Maple, Phys. Rev. Lett. 60 (1988) 216. [9] B.C. Webb, A.J. Sievers and T. Mihalism, Phys, Rev. Lett. 57 (1986) 1951, [10] T. Mihalisin, P. Scoborea and J.A, Ward, Phys, Rev. Lelt. 46 (1981) 862. [11] W.P. Beyermann, G. Gruner, Y. Dalichaouch and M.B. Maple. Phys. Rev. B37 (1988) 10353, and A.M. Awasthi, W.P. Beyermann. J. Carini and G. Gruner (to be published). [12] W.P. Beyermann, A.M. Awasthi, G. Gruner. Y. Dalichaouch, M.B. Maple, J.L. Smith and Z. Smith, Bull. Am. Phys. Soc. 33 (1988) 624 (also to be published). [13] Marabelliet al.,Solid S t a t e ( ' o m m u n . 59(1986) 381. [14] J. Serene and J. Wilkins (to be published). [15] G.A. T h o m a s et al. (to be published). [16] ('heong et al., Phys. Rev. B36 (1987) 3913.