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The electronic thermal transport in the EuBa2Cu3Oy superconductor
Physica C 432 (2005) 25–34 www.elsevier.com/locate/physc
The electronic thermal transport in the EuBa2Cu3Oy superconductor M. Matusiak a
a,*
, Th. Wolf
b
Department of Superconductivity Investigations, Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. Okolna 2, P.O. Box 1410, 50-950 Wrocław, Poland b Forschungszentrum Karlsruhe, Institut fu¨r Festko¨rperphysik, 76021 Karlsruhe, Germany Received 11 April 2005; received in revised form 28 June 2005; accepted 13 July 2005 Available online 22 August 2005
Abstract The heat transport properties in the normal state of a high quality EuBa2Cu3Oy single crystal in two oxygen configurations were studied. Measurements of the longitudinal and transverse transport coefficients were performed for an optimally doped sample (y 7), then the oxygen content in the material was reduced to y 6.65, and the set of measurements was repeated. The object of investigations was still literally the same crystal. We found the Hall–Lorenz numbers (Lxy) for both specimens behave almost identical in a temperature range 300–160 K, i.e. Lxys depend weakly on temperature and are about two times larger than the Sommerfelds value of the Lorenz number (L0). Below T 160 K the Hall–Lorenz number for the optimally doped sample slowly drops, while the value of Lxy for the oxygen reduced sample begins to rise. A phenomenological model of the electronic structure containing a pseudo-gap was proposed to explain the observed behaviors. The determined temperature dependences of the Lorenz number have been used to separate the total thermal conductivities for electronic and phonon parts and some conclusions about the scattering mechanisms are withdrawn. 2005 Elsevier B.V. All rights reserved. PACS: 74.25.Fy; 74.72.Bk Keywords: High-Tc superconductors; Thermal transport; Lorenz number; Righi–Leduc effect
1. Introduction *
Corresponding author. Tel.: +48 71 3435021; fax: +48 71 3441029. E-mail address: [email protected] (M. Matusiak).
Unusual properties of an electronic system in the high-temperature superconductors manifest themselves in many aspects. Among others, the transport coefficients behave atypically in the
0921-4534/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.07.006
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superconducting, as well as in the normal state of these compounds. An example of such a behavior can be a growth of the thermal conductivity (j) that occurs below the critical temperature (Tc). It is a very characteristic feature of high-Tc cuprates [1], but there is still no agreement on the reason for the appearance of this maximum. Difficulties in determination whether the rise of the thermal conductivity is of electronic [2–4] or phonon [5,6] origin refer to the more general problem. Since heat in solids is usually carried by both electrons and phonons, the measured total thermal conductivity is the sum of the electronic (jel) and phonon (jph) components: j = jel + jph. Therefore, a contribution from each component has to be established for a clear interpretation of the experimental data. This separation could be theoretically done on the basis of a relation between thermal and electrical conductivities known as the Wiedemann–Franz (WF) law. It can be written at given temperature as jel L rT
2 e ; kB
ð1Þ
where jel is the electronic thermal conductivity, r is the electrical conductivity, kB is the Boltzmanns constant, and L is the Lorenz number. But as was mentioned above, high-temperature superconductors usually do not behave like regular metals and some deviations from this law can be expected. Indeed, a violation of the WF law at low temperatures has been reported in one of cuprate superconductors: (Pr, Ce)2CuO4 [7]. To determine a real value of the Lorenz number Zhang et al. [6] have proposed to use the Righi– Leduc effect (called also the thermal Hall effect), which consists in the appearance of a transverse temperature gradient $yT, when a longitudinal temperature gradient $xT and a magnetic field are applied to the sample. The method allows one to measure the electronic component separated from the total thermal conductivity using the fact that in the external magnetic field a Lorentz force acts only on the electronic component of the heat current, while the phonon component remains unaffected. Thus, the transverse $yT refers to the heat transport by the charge carriers only
and the transverse thermal conductivity jxy = $yT Æ jxx/$xT is regarded as a direct source of information about the electronic heat current [8]. In this paper we study the thermal Hall effect in the normal state of optimally doped, and underdoped single crystals of EuBa2Cu3Oy (Eu-123) superconductor. Results of these studies, along with data on the electrical resistivities, thermal conductivities and Hall coefficients, are used to determine the temperature dependences of the Hall–Lorenz numbers (Lxy) for both samples. The values of Lxys were find to be almost identical and weakly temperature dependent in a range 300– 160 K. Below this temperature Lxy for the optimally doped sample slowly drops, while Lxy for the underdoped sample begins to rise. The values of both Hall–Lorenz numbers significantly exceed the Sommerfelds value (L0 = p2/3). To explain such a behavior of the transport coefficients we suppose that a pseudogap opens at the Fermi level in the optimally doped sample, while two different pseudogaps appear in the oxygen reduced one. The Lxy(T) dependences are also used to calculate the electronic and phonon thermal conductivities, and these results are interpreted in terms of relaxation times of different scattering processes.
2. Experimental Single crystals of EuBa2Cu3Oy were grown from CuO–BaO flux in ZrO2 crucibles by the slow cooling method [9]. To avoid any substitution of Ba by Eu the growth was carried out in a reduced atmosphere of 100 mbar air. The crystal selected for measurements was about 2 mm long, 1.1 mm wide and 0.18 mm thick (along the crystallographic c-axis). Its superconducting transition was magnetically measured to have an onset of 94.5 K and a midpoint of 94.0 K. The oxygen content y was estimated to be close to 7 (on the basis of the value of the critical temperature (Tc) and the in-plane thermoelectric power value at room temperature Sab(300 K) 2 lV/K [10– 12]). The crystal was twinned, therefore the coefficients measured along the ab-plane should represent values averaged over the a and b crystallographic directions.
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As described below a second set of measurements was performed on the same specimen after annealing in air for 12 days at T = 500 C to reduce its oxygen content. The reduction treatment caused a drop of Tc to about 61 K (measured magnetically) and the thermoelectric power became equal to Sab(300 K) 30 lV/K. Both values suggest [10–12] that the oxygen content fell to y 6.65. The experimental techniques for measuring the electrical resistivity, Hall coefficient, thermal longitudinal and transverse conductivities were the same as described elsewhere [13,14].
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(a)
(b)
3. Results and discussion Our studies are based on an analysis of the longitudinal and the transverse transport coefficients. The longitudinal thermal and electrical conductivities (jxx and rxx) were measured in absence of a magnetic field, while for the measurement of the transverse thermal and electrical conductivities (jxy and rxy) a magnetic field was applied. In Fig. 1 we show the temperature dependences of the electrical coefficients: rxx(T) and rxy(T). The first quantity is presented as the electrical resistivity (q = 1/rxx) in the upper panel (a). The second one is shown in the inset as a result of multiplication the Hall coefficient (RH) by square of the longitudinal electrical conductivity: rxy/B = RH(rxx)2. RH(T) is shown in the bottom panel (b). As mentioned above, all curves relate to the same Eu-123 single crystal: at the beginning it could be regarded as an optimally doped (solid lines), but after annealing in air it became underdoped (dashed lines). As expected, the electrical resistivity rises (about three times) at room temperature in the reduced sample. The critical temperature, defined as a point where curvature of q(T) changes sign, drops from 94.5 K (optimally doped) to 62.5 K (underdoped). The Hall coefficient in the underdoped sample is about six times larger (at T = 300 K) than it was before oxygen reduction. However, RH as well as q for the oxygen reduced sample still show similar temperature dependences to those observed for the optimally doped one. There are also no qualitative differences between thermal conductivities presented in Fig. 2.
Fig. 1. Temperature dependences of the electrical transport coefficients for the EuBa2Cu3O7 (optimal—solid lines) and EuBa2Cu3O6.65 (reduced—dashed lines) single crystal. The longitudinal electrical resistivities are present in the upper panel (a), and the Hall coefficients are shown in the bottom panel (b). The inset shows the transverse electrical conductivities calculated as rxy/B = RH(rxx)2.
For both specimens the longitudinal thermal conductivities in the normal state slowly rise with decreasing temperature—see panel (a). jxx for the reduced crystal is almost two times smaller (at room temperature) than it was for the optimal one. Also the transverse thermal conductivity jxy is lower by a factor of two at room temperature in the reduced sample. The jxy(T) dependences for both samples can be fitted well by a power function aTb, where the parameter b is equal to 1.14 for the optimally doped, and 1.21 for the underdoped sample. The above presented transverse transport coefficients were used to calculate the Hall–Lorenz number from the modified WF law [8]
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aL
(a)
(b)
Fig. 2. Temperature dependences of the thermal conductivities for the EuBa2Cu3O7 (optimal—solid lines, solid points) and EuBa2Cu3O6.65 (reduced—dashed lines, open points) single crystal. The longitudinal coefficients are presented in the upper panel (a), and the transverse coefficients are placed in the bottom panel (b); the dash–dot lines, which are drawn along jxy points show functions aTb being best power fits with exponents equal to bopt = 1.14 and bred = 1.21.
jxy Lxy rxy T
2 e ; kB
L2 ; Lxy
ð3Þ
where aL is expected to be nearly constant, since L hlsi/hlei and Lxy hls2i/hle2i [8,15]. In Fig. 3 the temperature dependences of the Hall–Lorenz number for the optimally doped (solid points) and oxygen reduced (open points) samples are presented. As can be seen both curves are practically the same at high temperatures. That is, Lxy for the optimally doped, as well as underdoped specimen decreases slowly from the room temperature down to T 160 K. Below this temperature the Hall–Lorenz number for the optimally doped sample continuously drops, while Lxy in the oxygen reduced sample increases to significantly higher values. The Lxy(T) dependence found for the optimally doped Eu-123 specimen is very similar to that previously observed for the optimally doped YBa2Cu3Oy (YBCO) single crystal [14]. Though both our results differ significantly from those presented by Zhang et al. in Ref. [8]—they found Lxy linear dependent on temperature and smaller than L0. Since we report almost identical data for jxy/B as Zhang and coworkers do (jxy/B(300K) 7.5 · 104 W m1 K1 T1 and the best power fit (aTb) with exponent equal to b = 1.14 in the
ð2Þ
the Hall–Lorenz number is a purely electronic factor, as it is related only to the phonon-unaffected thermal and charge conductivities. If charge carriers scatter elastically, then Lxy is identical to the regular Lorenz number [15], and both numbers are equal to the Sommerfelds value L0 = p2/3. Otherwise, i.e. when the inelastic scattering disturbs more effectively the heat current than the charge current [16], different mean free paths for the transport of entropy (ls) and charge (le) may be defined. In such a case Lxy may be compared to L by the ratio:
Fig. 3. Temperature dependences of the Hall–Lorenz numbers for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystal; the dashed lines are guides for the eye; the dash–dot line shows the Sommerfelds value of the Lorenz number (L0 = p2/3).
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present paper, while jxy/B(300 K) 6.7 W m1 K1 T1 6.7 and b = 1.19 in Ref. [8]) we think that a possible reason for the discrepancy in the Lxy data is using two different crystals in Ref. [8], one for jxy and the second for rxy measurements. Unequal amounts of out-of-plane impurities existing in the two YBa2Cu3O6.95 crystals may lead to serious inconsistencies, as it has been discussed in detail by Mei-Rong Li in Ref. [17]. The presented Lxy(T) data have been used to find temperature dependences of the regular Lorenz number. Because the values of the Hall–Lorenz numbers for both optimally doped and reduced samples tend to saturate at high temperatures we suppose that room temperature is high enough to assume that charge carriers are scattered mostly elastically. Thus it is possible to write Lxy(300) L(300) and evaluate the temperature dependences of the Lorenz numbers using Eq. (3). Results of these calculations are shown in Fig. 4. The values of both Ls are about two times larger at room temperature than Sommerfelds value of the Lorenz number L0 = p2/3, which is expected for a regular Fermi-liquid system. Such an enhancement of the Lorenz number was already reported for high-Tc superconductors [7,14,18], and it could appear due to a pseudogap that opens at the Fermi level [14,18]. This is not a new idea, since very similar consideration was carried out by Goff in articles from year 1970 [19,20] to explain the anomalies in an L(T) dependence [21–23] in Chromium. Goff has suggested that the unusual high value of the Lorenz number of Cr is a result of an atypical distribution of the electronic states about the Fermi level (eF). Namely, he proposed a model of the electronic structure, which contains a depletion of density of states at eF. It was supposed to be a parabolic-shaped well in a single band model [19] or a parabolic and BCS-like well in model of two electronic subsystems with two different gaps [20]. For the calculation the Klemens moments method [24] was used, where it is assumed that the same relaxation times describe both electrical and thermal transport, the phonon system is in equilibrium and the Fermi level is chosen as the reference energy. The transport coefficients are expressed using moments of the function r(2) called specific conductivity [24]
29
Fig. 4. Temperature dependences of the Lorenz numbers for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystal; the solid lines present results of the Ls calculations, which are based on the assumption that a depletion of the reduced electrical resistivity appears at the Fermi level. In the inset (a) the shape of the depletion assumed for EuBa2Cu3O7 is shown, while in the inset (b) the shapes of two wells assumed in two temperature regions for EuBa2Cu3O7 are presented: for T > T it is the solid line, and for T 6 T it is the dashed line.
(2 is the reduced energy: 2 = E/kBT). The nth moment is defined as Z þ1 of 0 d 2; ð4Þ rð2Þ2n Mn ¼ o2 1 where f 0 is the equilibrium Fermi–Dirac distribution function: f 0(2) = (e2 + 1)1. In this approach the Lorenz number is written as L ¼ ðk B =eÞ2 ðM 2 =M 0 Þ S 2 .
ð5Þ
Since the relaxation time of individual scattering mechanisms may be separated for an energy and temperature dependence: s1(E, T) = f(E)g(T), a temperature independent reduced specific conductivity: rr(E) = r(E)/r(0) may be defined [19]. Therefore, the transport coefficients can be rewritten in terms of new integrals Gn(T), where: M n ¼ rð0ÞGn ðT Þ; and Z þ1 of 0 d2. rr ð2Þ2n Gn ðT Þ ¼ o2 1
ð6Þ ð7Þ
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Now the calculated quantities become the products of two factors: r(0), containing the temperature dependences of the scattering mechanism and density of states at the Fermi level, and Gn(T) which gives the temperature dependence from anomalies in the energy dependence of r(2). If the retardation effect of the internal electric field on the thermal conductivity is negligible (it is, at most, a few percent in our samples), the S2 term in Eq. (5) may be ignored and the Lorenz number becomes L ¼ ðk B =eÞ2 ðG2 =G0 Þ.
ð8Þ
Then, to calculate an L(T) dependence there has to be proposed a model of irregularity occurring in the electronic structure. One of the considered by Goff anomalies in distribution of the electronic states is a parabolic-like depletion of the reduced specific conductivity about the Fermi level [19], which may be related to a reduction of the electron density of states. Therefore it is a structure very similar to the pseudogap observed in high-Tc superconductors [25,26], and the described model has been used to illustrate our experimental data. The L(T) data of the optimally doped sample, presented in Fig. 4, has been fitted with a numerically calculated function G2(T)/G0(T), which has only two free parameters describing a shape of the assumed parabolic-like pseudogap. A model of the electronic structure used in the calculations is shown in the inset (a) in Fig. 4. Despite the shape of the well was arbitrarily chosen and it is assumed to be temperature independent we have obtained a satisfying agreement between the fit and the experimental data presented in Fig. 4. Moreover, the estimated width of the supposed pseudogap (2D = 170 meV) is of the same order of magnitude as reported for, so called, large pseudo-gap in YBCO [27]. While in the optimally doped crystal we observe an influence of this large pseudo-gap only, the growth of the L(T) value in under-doped sample below the temperature T 160 K suggests an opening of the second, narrower, pseudo-gap. An appearing of two distinct pseudogaps seems to be on of characteristic for high-Tc superconductors phenomena and it has been observed in many experiments [25–29]. To respect this feature we have assumed in our calcula-
tions that the large pseudo-gap is simply replaced by the small pseudogap at T = 160 K. The shapes of both pseudogaps are presented in the inset (b) in Fig. 4. The estimated width of the small pseudogap (2D = 65 meV) and the temperature where it opens are similar to those observed previously in underdoped YBCO single crystals with Tc 60 K [30]. It has to mentioned that other processes can be also considered as a reason for the enhancement of the Lorenz number [23], but regardless of the used approach the electronic part of the thermal conductivity may be evaluated from the experimentally determined L(T) dependence. Then, the phonon contribution jph can be calculated from the total thermal conductivity as jph = j jel. Effects of these evaluations are presented in Fig. 5 along with the fitting curves (dashed lines) described below. It is noticeable that both, i.e. electronic and phonon, thermal conductivities are suppressed by the reduction treatment. The value of jel for the underdoped crystal at room temperature is about three times smaller than jel for the optimally doped one. Most likely this is because of a reduced concentration of free charge carriers. On the other hand, a reason of suppression of the jph compo-
Fig. 5. Temperature dependences of the longitudinal thermal conductivities for the EuBa2Cu3O7 (optimal: solid points— electronic contribution and solid diamonds—phonon contribution) and EuBa2Cu3O6.65 (reduced: open points—electronic contribution and open diamonds—phonon contribution) single crystal; the dashed lines show fits described in the text.
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nent is probably an increased number of oxygen defects. This mechanism was suggested by Minami et al. [18], who studied jph in a series of YBa2Cu3Oy single crystals. The estimated, in assumption that the Lorenz number is larger than L0, values of the phonon thermal conductivity for samples with oxygen contents corresponding to those estimated for our Eu-123 specimens were jph,opt(100 K) 9.5 and jph,red(100 K) 5. These values are close to the observed by us jph,opt(100 K) 7 and jph,red(100 K) 4.5. The strong dependence of the phonon thermal conductivity on the oxygen content would mean that oxygen vacancies within Cu–O chains and planes play an important role in scattering of phonons. On the other hand, the significant drop of jph in the underdoped sample, occurring despite of significantly lower concentration of free charge carriers, suggests that the phonon–electron scattering does not play an important role in disturbing phonon thermal transport. To find the mechanisms which determine the temperature dependence of the thermal conductivity we started with redrawing in Fig. 6 the electronic thermal conductivities along with Hall concentrations calculated as nH = 1/(eRH). Because changes of the real charge carriers concen-
Fig. 6. Temperature dependences of the electronic thermal conductivities (left axis) for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystal compared to the temperature dependences of the Hall concentration (right axis) evaluated for the same samples, i.e. EuBa2Cu3O7 (solid line) and EuBa2Cu3O6.65 (dashed line).
31
tration n would affect the values of the Hall concentration as well as electronic thermal conductivity, the visible similarities between nH(T) and jel(T) dependences persuade us to suppose that n is also temperature dependent. If nH is proportional to the effective concentration of the mobile charge carriers, then jel is related to nH as [16] jel ¼
p2 k 2B T ns TnH s. 3 m
ð9Þ
Then the electronic thermal resistivity W el ðW el ¼ j1 el Þ, thanks to the Matthiessens rule, may be divided between main scattering processes: 1 W el T 1 n1 H s 1 1 1 ¼ T 1 n1 H ðse–d þ se–p þ se–e Þ;
ð10Þ
where se–d, se–p, se–e are relaxation times of electron–defect, electron–phonon and electron– electron scattering, respectively. At high temperatures they depend on temperature like [16]: s1 e–d ¼ ae ;
s1 e–p ¼ be T ;
2 s1 e–e ¼ ce nH T .
ð11a; b; cÞ The dependence s1 e–e on carrier concentration nH was explicitly retained, since the number of free electrons (scattering center) is supposed to vary with temperature. The results of fitting the experimental points with a function comes from Eqs. (9) and (11a–c) are shown in Fig. 7. The data can not be fitted well if either electron–phonon or electron–electron scattering is neglected—the dashed lines show best fits for such cases. On the other hand, electron–defect scattering seems to be insignificant for both samples in the temperature region, where our studies are performed—the fitting procedure cancels the ae parameter out. In the optimally doped crystal the weight of e–e processes grows with temperature and becomes comparable to the weight of e–p processes at T = 300 K (the dash–dot–dot lines). While in the reduced sample, as could be expected for a sample with a lower concentration of free charge carriers, e–p processes are most prominent in the whole temperature range. A similar procedure of separation of the thermal resistivity for distinct processes was used for
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(a)
(b)
Fig. 8. Temperature dependences of the phonon thermal resistivities for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystals; the dashed lines show linear fits to the experimental data.
In the considered temperature region a relaxation time of phonon–phonon scattering is the only process which is explicitly temperature dependent [32]: Fig. 7. Temperature dependences of the electronic thermal resistivities for the EuBa2Cu3O7 (optimal—upper panel (a)) and EuBa2Cu3O6.65 (reduced—upper panel (b)) single crystal. For both panels the solid lines show the best fits, if the electron– phonon and electron–electron scatterings were taken into considerations—the weights of the particular processes (e–p or e–e) are presented by the dash–dot–dot lines. The dashed lines show the best fits if electron–defect and electron–phonon (e–d + e–p), or electron–defect and electron–electron (e–d + e– e) scatterings are only present.
the phonon thermal conductivities, presented as thermal resistivities (W ph ¼ j1 ph ) in Fig. 8. At high temperatures Wph is proportional to s1 only [16], what can be written as 1 1 W ph ðs1 p–others þ sp–e þ sp–p Þ;
ð12Þ
1 where s1 p–e and sp–p are relaxation times of phonon–electron and phonon–phonon scattering, respectively; the s1 p–others coefficient describes any other scattering processes (e.g. phonon-boundaries, phonon-defect, phonon-sheetlike faults [31]).
s1 p–others ¼ ap ;
s1 p–e ¼ nH bp ;
s1 p–p ¼ cp T . ð13a; b; cÞ
The phonon–electron scattering is dependent on temperature implicitly, since the concentration of scattering centers (electrons) is supposed to be a function of T. Unfortunately, for our samples 1 s1 p–e and sp–p vary with temperature very similarly and it is hard to distinguish between participation of p–e and p–p scattering in the phonon thermal resistivity on the basis of results of a fitting procedure. However, as it was already written, we do not expect that phonon–electron processes play a significant role in the heat transport of the lattice. Furthermore, almost the same slope of Wph(T) dependences for specimens with dramatically different nH, suggests once again that p–e scattering may be neglected. Therefore the dashed lines presented in Fig. 8 show simply linear fits (ap + cpT) that match the experimental data very well. The very similar cp parameter in the Wph(T) = ap + cpT dependences suggest that mechanisms of phonon–phonon interaction are the same for both
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crystals. This should be expected, as it is still the same crystal before, and after oxygen reduction.
vacancies, although phonon–electron scattering is not very important for phonon heat transport.
4. Summary
Acknowledgments
In summary, we have investigated transport properties in the normal state of a EuBa2Cu3Oy single crystal. The sample was examined in two different oxygen content conditions: first as an optimally doped one (y 7), next, after oxygen reduction treatment, as an underdoped one (y 6.7). In both cases the sample was superconducting, where the critical temperatures were equal to 94.5 K (optimally doped) and 62.5 K (underdoped). The measured temperature dependences of the longitudinal and transverse electrical, as well as thermal conductivities have been used to calculate the temperature dependences of the Hall– Lorenz number (Lxy). The found Lxy(T)s are practically identical from room temperature down to T 160 K. Below this temperature Lxy for the optimally doped crystal still slowly decreases, while the value of Lxy for the underdoped sample significantly increases. For both samples the Hall–Lorenz numbers are enhanced—at room temperature they are about two times larger than the Sommerfelds value of the Lorenz number. The Lxy(T) dependences were used to find the temperature dependences of the regular Lorenz numbers. To explain their behaviors a phenomenological model of the electronic structure was proposed. The main feature of this model is a pseudogap present at the Fermi level. We believe there is seen an influence of one pseudogap, for the optimally doped crystal, and two different pseudogaps for the underdoped one. The experimentally determined L(T) was used to separate the total thermal conductivity for the electronic (jel) and phonon (jph) components. The temperature dependences of jel suggest that the electron– electron scattering plays a significant role in disturbing the electronic thermal current, while the electron–defect scattering for both samples may be neglected. On the other hand, phonons seem to be effectively scattered by oxygen
The authors are grateful to Dr. T. Plackowski, Dr. C. Sułkowski and Dr. A.J. Zaleski for cooperation.
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The electronic thermal transport in the EuBa2Cu3Oy superconductor M. Matusiak a
a,*
, Th. Wolf
b
Department of Superconductivity Investigations, Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. Okolna 2, P.O. Box 1410, 50-950 Wrocław, Poland b Forschungszentrum Karlsruhe, Institut fu¨r Festko¨rperphysik, 76021 Karlsruhe, Germany Received 11 April 2005; received in revised form 28 June 2005; accepted 13 July 2005 Available online 22 August 2005
Abstract The heat transport properties in the normal state of a high quality EuBa2Cu3Oy single crystal in two oxygen configurations were studied. Measurements of the longitudinal and transverse transport coefficients were performed for an optimally doped sample (y 7), then the oxygen content in the material was reduced to y 6.65, and the set of measurements was repeated. The object of investigations was still literally the same crystal. We found the Hall–Lorenz numbers (Lxy) for both specimens behave almost identical in a temperature range 300–160 K, i.e. Lxys depend weakly on temperature and are about two times larger than the Sommerfelds value of the Lorenz number (L0). Below T 160 K the Hall–Lorenz number for the optimally doped sample slowly drops, while the value of Lxy for the oxygen reduced sample begins to rise. A phenomenological model of the electronic structure containing a pseudo-gap was proposed to explain the observed behaviors. The determined temperature dependences of the Lorenz number have been used to separate the total thermal conductivities for electronic and phonon parts and some conclusions about the scattering mechanisms are withdrawn. 2005 Elsevier B.V. All rights reserved. PACS: 74.25.Fy; 74.72.Bk Keywords: High-Tc superconductors; Thermal transport; Lorenz number; Righi–Leduc effect
1. Introduction *
Corresponding author. Tel.: +48 71 3435021; fax: +48 71 3441029. E-mail address: [email protected] (M. Matusiak).
Unusual properties of an electronic system in the high-temperature superconductors manifest themselves in many aspects. Among others, the transport coefficients behave atypically in the
0921-4534/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2005.07.006
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superconducting, as well as in the normal state of these compounds. An example of such a behavior can be a growth of the thermal conductivity (j) that occurs below the critical temperature (Tc). It is a very characteristic feature of high-Tc cuprates [1], but there is still no agreement on the reason for the appearance of this maximum. Difficulties in determination whether the rise of the thermal conductivity is of electronic [2–4] or phonon [5,6] origin refer to the more general problem. Since heat in solids is usually carried by both electrons and phonons, the measured total thermal conductivity is the sum of the electronic (jel) and phonon (jph) components: j = jel + jph. Therefore, a contribution from each component has to be established for a clear interpretation of the experimental data. This separation could be theoretically done on the basis of a relation between thermal and electrical conductivities known as the Wiedemann–Franz (WF) law. It can be written at given temperature as jel L rT
2 e ; kB
ð1Þ
where jel is the electronic thermal conductivity, r is the electrical conductivity, kB is the Boltzmanns constant, and L is the Lorenz number. But as was mentioned above, high-temperature superconductors usually do not behave like regular metals and some deviations from this law can be expected. Indeed, a violation of the WF law at low temperatures has been reported in one of cuprate superconductors: (Pr, Ce)2CuO4 [7]. To determine a real value of the Lorenz number Zhang et al. [6] have proposed to use the Righi– Leduc effect (called also the thermal Hall effect), which consists in the appearance of a transverse temperature gradient $yT, when a longitudinal temperature gradient $xT and a magnetic field are applied to the sample. The method allows one to measure the electronic component separated from the total thermal conductivity using the fact that in the external magnetic field a Lorentz force acts only on the electronic component of the heat current, while the phonon component remains unaffected. Thus, the transverse $yT refers to the heat transport by the charge carriers only
and the transverse thermal conductivity jxy = $yT Æ jxx/$xT is regarded as a direct source of information about the electronic heat current [8]. In this paper we study the thermal Hall effect in the normal state of optimally doped, and underdoped single crystals of EuBa2Cu3Oy (Eu-123) superconductor. Results of these studies, along with data on the electrical resistivities, thermal conductivities and Hall coefficients, are used to determine the temperature dependences of the Hall–Lorenz numbers (Lxy) for both samples. The values of Lxys were find to be almost identical and weakly temperature dependent in a range 300– 160 K. Below this temperature Lxy for the optimally doped sample slowly drops, while Lxy for the underdoped sample begins to rise. The values of both Hall–Lorenz numbers significantly exceed the Sommerfelds value (L0 = p2/3). To explain such a behavior of the transport coefficients we suppose that a pseudogap opens at the Fermi level in the optimally doped sample, while two different pseudogaps appear in the oxygen reduced one. The Lxy(T) dependences are also used to calculate the electronic and phonon thermal conductivities, and these results are interpreted in terms of relaxation times of different scattering processes.
2. Experimental Single crystals of EuBa2Cu3Oy were grown from CuO–BaO flux in ZrO2 crucibles by the slow cooling method [9]. To avoid any substitution of Ba by Eu the growth was carried out in a reduced atmosphere of 100 mbar air. The crystal selected for measurements was about 2 mm long, 1.1 mm wide and 0.18 mm thick (along the crystallographic c-axis). Its superconducting transition was magnetically measured to have an onset of 94.5 K and a midpoint of 94.0 K. The oxygen content y was estimated to be close to 7 (on the basis of the value of the critical temperature (Tc) and the in-plane thermoelectric power value at room temperature Sab(300 K) 2 lV/K [10– 12]). The crystal was twinned, therefore the coefficients measured along the ab-plane should represent values averaged over the a and b crystallographic directions.
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As described below a second set of measurements was performed on the same specimen after annealing in air for 12 days at T = 500 C to reduce its oxygen content. The reduction treatment caused a drop of Tc to about 61 K (measured magnetically) and the thermoelectric power became equal to Sab(300 K) 30 lV/K. Both values suggest [10–12] that the oxygen content fell to y 6.65. The experimental techniques for measuring the electrical resistivity, Hall coefficient, thermal longitudinal and transverse conductivities were the same as described elsewhere [13,14].
27
(a)
(b)
3. Results and discussion Our studies are based on an analysis of the longitudinal and the transverse transport coefficients. The longitudinal thermal and electrical conductivities (jxx and rxx) were measured in absence of a magnetic field, while for the measurement of the transverse thermal and electrical conductivities (jxy and rxy) a magnetic field was applied. In Fig. 1 we show the temperature dependences of the electrical coefficients: rxx(T) and rxy(T). The first quantity is presented as the electrical resistivity (q = 1/rxx) in the upper panel (a). The second one is shown in the inset as a result of multiplication the Hall coefficient (RH) by square of the longitudinal electrical conductivity: rxy/B = RH(rxx)2. RH(T) is shown in the bottom panel (b). As mentioned above, all curves relate to the same Eu-123 single crystal: at the beginning it could be regarded as an optimally doped (solid lines), but after annealing in air it became underdoped (dashed lines). As expected, the electrical resistivity rises (about three times) at room temperature in the reduced sample. The critical temperature, defined as a point where curvature of q(T) changes sign, drops from 94.5 K (optimally doped) to 62.5 K (underdoped). The Hall coefficient in the underdoped sample is about six times larger (at T = 300 K) than it was before oxygen reduction. However, RH as well as q for the oxygen reduced sample still show similar temperature dependences to those observed for the optimally doped one. There are also no qualitative differences between thermal conductivities presented in Fig. 2.
Fig. 1. Temperature dependences of the electrical transport coefficients for the EuBa2Cu3O7 (optimal—solid lines) and EuBa2Cu3O6.65 (reduced—dashed lines) single crystal. The longitudinal electrical resistivities are present in the upper panel (a), and the Hall coefficients are shown in the bottom panel (b). The inset shows the transverse electrical conductivities calculated as rxy/B = RH(rxx)2.
For both specimens the longitudinal thermal conductivities in the normal state slowly rise with decreasing temperature—see panel (a). jxx for the reduced crystal is almost two times smaller (at room temperature) than it was for the optimal one. Also the transverse thermal conductivity jxy is lower by a factor of two at room temperature in the reduced sample. The jxy(T) dependences for both samples can be fitted well by a power function aTb, where the parameter b is equal to 1.14 for the optimally doped, and 1.21 for the underdoped sample. The above presented transverse transport coefficients were used to calculate the Hall–Lorenz number from the modified WF law [8]
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aL
(a)
(b)
Fig. 2. Temperature dependences of the thermal conductivities for the EuBa2Cu3O7 (optimal—solid lines, solid points) and EuBa2Cu3O6.65 (reduced—dashed lines, open points) single crystal. The longitudinal coefficients are presented in the upper panel (a), and the transverse coefficients are placed in the bottom panel (b); the dash–dot lines, which are drawn along jxy points show functions aTb being best power fits with exponents equal to bopt = 1.14 and bred = 1.21.
jxy Lxy rxy T
2 e ; kB
L2 ; Lxy
ð3Þ
where aL is expected to be nearly constant, since L hlsi/hlei and Lxy hls2i/hle2i [8,15]. In Fig. 3 the temperature dependences of the Hall–Lorenz number for the optimally doped (solid points) and oxygen reduced (open points) samples are presented. As can be seen both curves are practically the same at high temperatures. That is, Lxy for the optimally doped, as well as underdoped specimen decreases slowly from the room temperature down to T 160 K. Below this temperature the Hall–Lorenz number for the optimally doped sample continuously drops, while Lxy in the oxygen reduced sample increases to significantly higher values. The Lxy(T) dependence found for the optimally doped Eu-123 specimen is very similar to that previously observed for the optimally doped YBa2Cu3Oy (YBCO) single crystal [14]. Though both our results differ significantly from those presented by Zhang et al. in Ref. [8]—they found Lxy linear dependent on temperature and smaller than L0. Since we report almost identical data for jxy/B as Zhang and coworkers do (jxy/B(300K) 7.5 · 104 W m1 K1 T1 and the best power fit (aTb) with exponent equal to b = 1.14 in the
ð2Þ
the Hall–Lorenz number is a purely electronic factor, as it is related only to the phonon-unaffected thermal and charge conductivities. If charge carriers scatter elastically, then Lxy is identical to the regular Lorenz number [15], and both numbers are equal to the Sommerfelds value L0 = p2/3. Otherwise, i.e. when the inelastic scattering disturbs more effectively the heat current than the charge current [16], different mean free paths for the transport of entropy (ls) and charge (le) may be defined. In such a case Lxy may be compared to L by the ratio:
Fig. 3. Temperature dependences of the Hall–Lorenz numbers for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystal; the dashed lines are guides for the eye; the dash–dot line shows the Sommerfelds value of the Lorenz number (L0 = p2/3).
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present paper, while jxy/B(300 K) 6.7 W m1 K1 T1 6.7 and b = 1.19 in Ref. [8]) we think that a possible reason for the discrepancy in the Lxy data is using two different crystals in Ref. [8], one for jxy and the second for rxy measurements. Unequal amounts of out-of-plane impurities existing in the two YBa2Cu3O6.95 crystals may lead to serious inconsistencies, as it has been discussed in detail by Mei-Rong Li in Ref. [17]. The presented Lxy(T) data have been used to find temperature dependences of the regular Lorenz number. Because the values of the Hall–Lorenz numbers for both optimally doped and reduced samples tend to saturate at high temperatures we suppose that room temperature is high enough to assume that charge carriers are scattered mostly elastically. Thus it is possible to write Lxy(300) L(300) and evaluate the temperature dependences of the Lorenz numbers using Eq. (3). Results of these calculations are shown in Fig. 4. The values of both Ls are about two times larger at room temperature than Sommerfelds value of the Lorenz number L0 = p2/3, which is expected for a regular Fermi-liquid system. Such an enhancement of the Lorenz number was already reported for high-Tc superconductors [7,14,18], and it could appear due to a pseudogap that opens at the Fermi level [14,18]. This is not a new idea, since very similar consideration was carried out by Goff in articles from year 1970 [19,20] to explain the anomalies in an L(T) dependence [21–23] in Chromium. Goff has suggested that the unusual high value of the Lorenz number of Cr is a result of an atypical distribution of the electronic states about the Fermi level (eF). Namely, he proposed a model of the electronic structure, which contains a depletion of density of states at eF. It was supposed to be a parabolic-shaped well in a single band model [19] or a parabolic and BCS-like well in model of two electronic subsystems with two different gaps [20]. For the calculation the Klemens moments method [24] was used, where it is assumed that the same relaxation times describe both electrical and thermal transport, the phonon system is in equilibrium and the Fermi level is chosen as the reference energy. The transport coefficients are expressed using moments of the function r(2) called specific conductivity [24]
29
Fig. 4. Temperature dependences of the Lorenz numbers for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystal; the solid lines present results of the Ls calculations, which are based on the assumption that a depletion of the reduced electrical resistivity appears at the Fermi level. In the inset (a) the shape of the depletion assumed for EuBa2Cu3O7 is shown, while in the inset (b) the shapes of two wells assumed in two temperature regions for EuBa2Cu3O7 are presented: for T > T it is the solid line, and for T 6 T it is the dashed line.
(2 is the reduced energy: 2 = E/kBT). The nth moment is defined as Z þ1 of 0 d 2; ð4Þ rð2Þ2n Mn ¼ o2 1 where f 0 is the equilibrium Fermi–Dirac distribution function: f 0(2) = (e2 + 1)1. In this approach the Lorenz number is written as L ¼ ðk B =eÞ2 ðM 2 =M 0 Þ S 2 .
ð5Þ
Since the relaxation time of individual scattering mechanisms may be separated for an energy and temperature dependence: s1(E, T) = f(E)g(T), a temperature independent reduced specific conductivity: rr(E) = r(E)/r(0) may be defined [19]. Therefore, the transport coefficients can be rewritten in terms of new integrals Gn(T), where: M n ¼ rð0ÞGn ðT Þ; and Z þ1 of 0 d2. rr ð2Þ2n Gn ðT Þ ¼ o2 1
ð6Þ ð7Þ
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Now the calculated quantities become the products of two factors: r(0), containing the temperature dependences of the scattering mechanism and density of states at the Fermi level, and Gn(T) which gives the temperature dependence from anomalies in the energy dependence of r(2). If the retardation effect of the internal electric field on the thermal conductivity is negligible (it is, at most, a few percent in our samples), the S2 term in Eq. (5) may be ignored and the Lorenz number becomes L ¼ ðk B =eÞ2 ðG2 =G0 Þ.
ð8Þ
Then, to calculate an L(T) dependence there has to be proposed a model of irregularity occurring in the electronic structure. One of the considered by Goff anomalies in distribution of the electronic states is a parabolic-like depletion of the reduced specific conductivity about the Fermi level [19], which may be related to a reduction of the electron density of states. Therefore it is a structure very similar to the pseudogap observed in high-Tc superconductors [25,26], and the described model has been used to illustrate our experimental data. The L(T) data of the optimally doped sample, presented in Fig. 4, has been fitted with a numerically calculated function G2(T)/G0(T), which has only two free parameters describing a shape of the assumed parabolic-like pseudogap. A model of the electronic structure used in the calculations is shown in the inset (a) in Fig. 4. Despite the shape of the well was arbitrarily chosen and it is assumed to be temperature independent we have obtained a satisfying agreement between the fit and the experimental data presented in Fig. 4. Moreover, the estimated width of the supposed pseudogap (2D = 170 meV) is of the same order of magnitude as reported for, so called, large pseudo-gap in YBCO [27]. While in the optimally doped crystal we observe an influence of this large pseudo-gap only, the growth of the L(T) value in under-doped sample below the temperature T 160 K suggests an opening of the second, narrower, pseudo-gap. An appearing of two distinct pseudogaps seems to be on of characteristic for high-Tc superconductors phenomena and it has been observed in many experiments [25–29]. To respect this feature we have assumed in our calcula-
tions that the large pseudo-gap is simply replaced by the small pseudogap at T = 160 K. The shapes of both pseudogaps are presented in the inset (b) in Fig. 4. The estimated width of the small pseudogap (2D = 65 meV) and the temperature where it opens are similar to those observed previously in underdoped YBCO single crystals with Tc 60 K [30]. It has to mentioned that other processes can be also considered as a reason for the enhancement of the Lorenz number [23], but regardless of the used approach the electronic part of the thermal conductivity may be evaluated from the experimentally determined L(T) dependence. Then, the phonon contribution jph can be calculated from the total thermal conductivity as jph = j jel. Effects of these evaluations are presented in Fig. 5 along with the fitting curves (dashed lines) described below. It is noticeable that both, i.e. electronic and phonon, thermal conductivities are suppressed by the reduction treatment. The value of jel for the underdoped crystal at room temperature is about three times smaller than jel for the optimally doped one. Most likely this is because of a reduced concentration of free charge carriers. On the other hand, a reason of suppression of the jph compo-
Fig. 5. Temperature dependences of the longitudinal thermal conductivities for the EuBa2Cu3O7 (optimal: solid points— electronic contribution and solid diamonds—phonon contribution) and EuBa2Cu3O6.65 (reduced: open points—electronic contribution and open diamonds—phonon contribution) single crystal; the dashed lines show fits described in the text.
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nent is probably an increased number of oxygen defects. This mechanism was suggested by Minami et al. [18], who studied jph in a series of YBa2Cu3Oy single crystals. The estimated, in assumption that the Lorenz number is larger than L0, values of the phonon thermal conductivity for samples with oxygen contents corresponding to those estimated for our Eu-123 specimens were jph,opt(100 K) 9.5 and jph,red(100 K) 5. These values are close to the observed by us jph,opt(100 K) 7 and jph,red(100 K) 4.5. The strong dependence of the phonon thermal conductivity on the oxygen content would mean that oxygen vacancies within Cu–O chains and planes play an important role in scattering of phonons. On the other hand, the significant drop of jph in the underdoped sample, occurring despite of significantly lower concentration of free charge carriers, suggests that the phonon–electron scattering does not play an important role in disturbing phonon thermal transport. To find the mechanisms which determine the temperature dependence of the thermal conductivity we started with redrawing in Fig. 6 the electronic thermal conductivities along with Hall concentrations calculated as nH = 1/(eRH). Because changes of the real charge carriers concen-
Fig. 6. Temperature dependences of the electronic thermal conductivities (left axis) for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystal compared to the temperature dependences of the Hall concentration (right axis) evaluated for the same samples, i.e. EuBa2Cu3O7 (solid line) and EuBa2Cu3O6.65 (dashed line).
31
tration n would affect the values of the Hall concentration as well as electronic thermal conductivity, the visible similarities between nH(T) and jel(T) dependences persuade us to suppose that n is also temperature dependent. If nH is proportional to the effective concentration of the mobile charge carriers, then jel is related to nH as [16] jel ¼
p2 k 2B T ns TnH s. 3 m
ð9Þ
Then the electronic thermal resistivity W el ðW el ¼ j1 el Þ, thanks to the Matthiessens rule, may be divided between main scattering processes: 1 W el T 1 n1 H s 1 1 1 ¼ T 1 n1 H ðse–d þ se–p þ se–e Þ;
ð10Þ
where se–d, se–p, se–e are relaxation times of electron–defect, electron–phonon and electron– electron scattering, respectively. At high temperatures they depend on temperature like [16]: s1 e–d ¼ ae ;
s1 e–p ¼ be T ;
2 s1 e–e ¼ ce nH T .
ð11a; b; cÞ The dependence s1 e–e on carrier concentration nH was explicitly retained, since the number of free electrons (scattering center) is supposed to vary with temperature. The results of fitting the experimental points with a function comes from Eqs. (9) and (11a–c) are shown in Fig. 7. The data can not be fitted well if either electron–phonon or electron–electron scattering is neglected—the dashed lines show best fits for such cases. On the other hand, electron–defect scattering seems to be insignificant for both samples in the temperature region, where our studies are performed—the fitting procedure cancels the ae parameter out. In the optimally doped crystal the weight of e–e processes grows with temperature and becomes comparable to the weight of e–p processes at T = 300 K (the dash–dot–dot lines). While in the reduced sample, as could be expected for a sample with a lower concentration of free charge carriers, e–p processes are most prominent in the whole temperature range. A similar procedure of separation of the thermal resistivity for distinct processes was used for
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(a)
(b)
Fig. 8. Temperature dependences of the phonon thermal resistivities for the EuBa2Cu3O7 (optimal—solid points) and EuBa2Cu3O6.65 (reduced—open points) single crystals; the dashed lines show linear fits to the experimental data.
In the considered temperature region a relaxation time of phonon–phonon scattering is the only process which is explicitly temperature dependent [32]: Fig. 7. Temperature dependences of the electronic thermal resistivities for the EuBa2Cu3O7 (optimal—upper panel (a)) and EuBa2Cu3O6.65 (reduced—upper panel (b)) single crystal. For both panels the solid lines show the best fits, if the electron– phonon and electron–electron scatterings were taken into considerations—the weights of the particular processes (e–p or e–e) are presented by the dash–dot–dot lines. The dashed lines show the best fits if electron–defect and electron–phonon (e–d + e–p), or electron–defect and electron–electron (e–d + e– e) scatterings are only present.
the phonon thermal conductivities, presented as thermal resistivities (W ph ¼ j1 ph ) in Fig. 8. At high temperatures Wph is proportional to s1 only [16], what can be written as 1 1 W ph ðs1 p–others þ sp–e þ sp–p Þ;
ð12Þ
1 where s1 p–e and sp–p are relaxation times of phonon–electron and phonon–phonon scattering, respectively; the s1 p–others coefficient describes any other scattering processes (e.g. phonon-boundaries, phonon-defect, phonon-sheetlike faults [31]).
s1 p–others ¼ ap ;
s1 p–e ¼ nH bp ;
s1 p–p ¼ cp T . ð13a; b; cÞ
The phonon–electron scattering is dependent on temperature implicitly, since the concentration of scattering centers (electrons) is supposed to be a function of T. Unfortunately, for our samples 1 s1 p–e and sp–p vary with temperature very similarly and it is hard to distinguish between participation of p–e and p–p scattering in the phonon thermal resistivity on the basis of results of a fitting procedure. However, as it was already written, we do not expect that phonon–electron processes play a significant role in the heat transport of the lattice. Furthermore, almost the same slope of Wph(T) dependences for specimens with dramatically different nH, suggests once again that p–e scattering may be neglected. Therefore the dashed lines presented in Fig. 8 show simply linear fits (ap + cpT) that match the experimental data very well. The very similar cp parameter in the Wph(T) = ap + cpT dependences suggest that mechanisms of phonon–phonon interaction are the same for both
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crystals. This should be expected, as it is still the same crystal before, and after oxygen reduction.
vacancies, although phonon–electron scattering is not very important for phonon heat transport.
4. Summary
Acknowledgments
In summary, we have investigated transport properties in the normal state of a EuBa2Cu3Oy single crystal. The sample was examined in two different oxygen content conditions: first as an optimally doped one (y 7), next, after oxygen reduction treatment, as an underdoped one (y 6.7). In both cases the sample was superconducting, where the critical temperatures were equal to 94.5 K (optimally doped) and 62.5 K (underdoped). The measured temperature dependences of the longitudinal and transverse electrical, as well as thermal conductivities have been used to calculate the temperature dependences of the Hall– Lorenz number (Lxy). The found Lxy(T)s are practically identical from room temperature down to T 160 K. Below this temperature Lxy for the optimally doped crystal still slowly decreases, while the value of Lxy for the underdoped sample significantly increases. For both samples the Hall–Lorenz numbers are enhanced—at room temperature they are about two times larger than the Sommerfelds value of the Lorenz number. The Lxy(T) dependences were used to find the temperature dependences of the regular Lorenz numbers. To explain their behaviors a phenomenological model of the electronic structure was proposed. The main feature of this model is a pseudogap present at the Fermi level. We believe there is seen an influence of one pseudogap, for the optimally doped crystal, and two different pseudogaps for the underdoped one. The experimentally determined L(T) was used to separate the total thermal conductivity for the electronic (jel) and phonon (jph) components. The temperature dependences of jel suggest that the electron– electron scattering plays a significant role in disturbing the electronic thermal current, while the electron–defect scattering for both samples may be neglected. On the other hand, phonons seem to be effectively scattered by oxygen
The authors are grateful to Dr. T. Plackowski, Dr. C. Sułkowski and Dr. A.J. Zaleski for cooperation.
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