Thermoelectric power of overdoped Tl2201 crystals: charge density waves and T¹ and T² resistivities

Above $1.1~T_c$ the TEP of OD cuprates generally varies approximately as $a + b(T/300)$, where $a \simeq 1$ to $2~\mu$V K⁻¹ and b changes from ${\simeq}-1$ to $-10~\mu$V K⁻¹ as p is increased from 0.16, the value for maximum T_c , to the upper edge of the superconducting dome [5] while for UD samples it is usually positive and much larger [5]. TEP data were taken in an earlier, independent research project before the RIXS studies, using the method described in the appendix. One crystal was used for the TEP work and re-annealed to give successive T_c values of 26.5 K, 29 K, 67 K and 88 K, annealing conditions are given as supplementary information [14]. A small correction $\simeq +0.1\mu$V K⁻¹ below 50 K has been made for the T_c = 29 K and 26.5 K data using the phosphor bronze results shown in the appendix. AC susceptibility data for various crystals of Tl2201 with small AC fields parallel and perpendicular to the CuO₂ layers after different annealing treatments are reported as supplementary information [14]. They reveal an unexpectedly complex behavior and give evidence for two oxygen diffusion processes with very different time scales. It is probable that one or both of these are influenced by the concentration of Cu ions on Tl sites. This data could be useful for future studies. TEP data for polycrystalline samples [5] with T_c = 84 K, 82 K, 59 K, 29 K and less than 4 K, provide important confirmation of the present work.

Figure 1(a) shows that the in-plane TEP $S_a(T)$ for the same Tl2201 crystal with T_c values of 88 K and 67 K has a small upturn below ∼150 K, where there is also a clear change in the derivative $dS_a/dT$, as shown in figure 1(b), but there is no such effect for $T_c = 26.5$ K. For $T_c = 29$ K there is marginal evidence for a CDW with an onset near 40 K as indicated in figure 3. All this correlates well with RIXS measurements on Tl2201 crystals from the same preparation batch for which CDWs were detected for T_c values of 56 K and 45 K but not for 22 K [3]. TEP data for T_c = 88 K, on an expanded scale in figure 1(c), show a particularly clear anomaly near 145 K. The CDW onset temperature for T_c = 45 K is 150 K as shown by the RIXS data on the right hand scale of figure 1(b) but unfortunately the onset temperature of 160 K shown in [3] for T_c = 56 K is only an upper bound. Although better overlap of T_c values would have been desirable it is very probable that the changes in $S_a(T)$ and $dS_a/dT$ near 150 K shown in figures 1(a)–(c) as well as at various temperatures for the polycrystalline data in [5] are caused by a CDW. This agreement is potentially useful because it shows that the TEP is a sensitive probe of the CDW and could be used to study its pressure dependence and possibly its magnetic-field dependence. Data for the OD88 crystal and the OD84 polycrystalline sample [5] show that the CDW is still present at lower p, which is a new finding.

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Resistivity data [1] for five different crystals from the same preparation batch are shown in figure 2(a). As noted in [3] there appears to be no effect of the CDW on the in-plane resistivity $\rho_a(T)$. This is unusual since other CDW materials do show resistivity anomalies, but it is reinforced by the derivative plots in figure 2(b). Although there is a change in slope of $d\rho_a(T)/dT$ near 160 K, we do not believe that this is caused by a CDW. It is also present for the crystal with $T_c =$ 26.5 K and a crystal from another batch with $T_c =$ 17 K as well as for older data scanned in from [15].

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Such derivative plots [16, 17] are a convenient way of applying the formula

$$\begin{equation} {\rho_a\left(T\right) = \alpha_0 +\alpha_1T+\alpha_2T^2} \end{equation} \tag{ 1 }$$

because the range of fit is easily seen by inspection. They suggest that there are two distinct values for the T¹ and T² terms, below and above 160 K where both the slopes and the intercepts in figure 2(b) change. For the crystals with T_c = 65 K, 40 K and 26.5 K, α₁ and α₂ were found from the intercepts at T = 0 and the slopes of the straight lines in figure 2(b), while the residual resistivity α₀ was then found by fitting the data in figure 2(a) to a second order polynomial in which α₁ and α₂ were fixed. The coefficients α are given in table 1. (A small correction for a measurement offset of ${\sim}1\mu\Omega\cdot\mathrm{cm}$ below T_c has been included where appropriate). Here we focus on the values below 160 K, expecting them to be more closely related to superconductivity. The coefficient α₁ is presently of great interest because of the suggestion that the Planckian scattering rate $\hbar/\tau \approx k_BT$ [6] is the maximum allowed for any material [8], and that it is an important intrinsic property of superconducting cuprates [7] and many other correlated metals [8]. The Drude formula for the in-plane conductivity of an isotropic cylindrical Fermi surface, $\sigma = ne^2\tau/m^*$, which is a reasonable first approximation for Tl2201, shows that $\alpha_1 =$ 1 $\mu\Omega\cdot\mathrm{cm}\,\mathrm{K}^{-1}$ corresponds to $\hbar/\tau \approx 3.3 k_BT$, if the carrier density n = 1.3 holes per unit cell and $m^* = 5.2m_e$. Although this value of $m^*$ has been measured by QO only for OD Tl2201 [2], specific heat data [9] show that it only varies weakly with p. So $\alpha_1 =$ 0.3 $\mu\Omega\cdot\mathrm{cm}\,\mathrm{K}^{-1}$ in figure 2(b) corresponds to $\hbar/\tau = k_BT$. For crystals with $T_c\lt$ 50 K, $\hbar/\tau$ falls below this value because, as found previously [16, 17], the size of the T¹ term falls linearly to zero as p approaches the edge of the superconducting dome. This could be an indication that the anomalous scattering becomes confined to certain parts of the Fermi surface, in contrast to the results for some other cuprates [7]. The value of α₀ for T_c = 26.5 K in table 1, gives $\hbar/(k_B\tau) = 19$ K and for a Fermi velocity $v_F \equiv\hbar k_F/m^* = 1.7 \times 10^7$ cm s⁻¹, a mean free path of 68 nm. Specific heat data for Tl2201 [9] show a large residual linear term which increases as p approaches the edge of the superconducting dome, as expected for a d-wave superconductor when $\hbar/(k_B\tau)$ becomes comparable with T_c .

Table 1. Values of α given by the straight lines below 160 K in figure 2(b) plus a constrained fit to the data in figure 2(a) (see text), i.e. $\rho(T) = \alpha_0 + \alpha_1T +\alpha_2T^2$. The carrier mean free path (mfp) is obtained from α₀. $T_c^{\textrm{TEP}}$ values are from figure 1(a). S₀, S₁ and S₂ are from fits to equation (2), but cannot be determined independently (see text). The errors in α values are $\pm10\%$ from geometrical uncertainties.

Tc	α0	α1	$\alpha_2 \times 1000$	mfp	$T_c^{\textrm{TEP}}$	S$_0/T$	S$_1/T$	S$_2/T$
(K)	($\mu \Omega\cdot\mathrm{cm}$)	($\mu \Omega\cdot\mathrm{cm}\,\mathrm{K}^{-1}$)	($\mu \Omega\cdot\mathrm{cm}$) K−2)	(nm)	(K)	(µV K−2)	(µV K−2)	(µV K−2)
90	$(6.4\pm0.4)$	1.37	0.125	—	88	0.132	$-0.0115\pm6\%$	—
65	6.3	0.313	1.82	65	67	0.128	—	$-0.030\pm10\%$
40	6.9	0.17	1.34	60	—	—	—	—
26.5	6.0	0.053	1.6	68	26.5	0.016	—	$- 0.033\pm3\%$

Previously [16] the deviations from a straight line above 160 K for the three most overdoped crystals in figure 2(b). were interpreted in terms of a parallel conducting channel and the maximum resistivity of 1800 $\mu\Omega\cdot\mathrm{cm}$ obtained when the carrier mean free path is equal to the in-plane lattice spacing a. However they are still clearly visible for the two most overdoped crystals in figure 2(a) that have resistivities as low as 60 $\mu\Omega\cdot\mathrm{cm}$ at 160 K and as mentioned already for another crystal from a different preparation batch with $T_c = 17$ K. Therefore we have no clear interpretation of this effect. It is worth noting that α₂ is rather insensitive to p as also shown in [16]. At 160 K it corresponds to a T² term in the resistivity of 40 $\mu\Omega\cdot\mathrm{cm}$ and, making the same assumptions as before, a scattering rate $\hbar/\tau = k_BT$, the Planck condition [6]. This observation could have ramifications for interpretations of the T¹ term arising from the Planck condition. Perhaps $\hbar/\tau \unicode{x2A7E} k_BT$ might cause the deviations from linearity in figure 2(b) above 160 K. Alternatively the deviations might be connected with the effect of lattice vibrations on electron–electron Umklapp scattering. Electron–electron scattering is rarely studied up to such high temperatures. It is known that the Debye–Waller factor, which reduces the intensity of x-ray diffraction lines, causes the energy gap of semiconductors such as Si to fall as T is increased. It is possible that Umklapp scattering between Brillouin zones is reduced by a similar T-dependent factor that becomes more significant as the amplitude of the lattice vibrations increases.

Data for all hole-doped cuprates clearly show that the curvature of in-plane $\rho(T)$ plots increases on the OD side. It is widely accepted that there is a crossover line on the T − p phase diagram [18], which although ill-defined, increases linearly with p from zero for $p\unicode{x2A7E}0.19$. It can be interpreted as a coherence temperature $T_{\textrm{coh}}(p)$ for the charged carriers and when $T\lesssim T_{\textrm{coh}}$ the T² term dominates. The results for $T\unicode{x2A7D}$ 160 K in figure 2(b) and table 1 are qualitatively consistent this is because α₁ falls with p while α₂ stays constant. For $T\unicode{x2A7E}$ 160 K this requires the slope of $d\rho_a/dT$ to become constant which is the case above 200 K for $T_c =$ 30 K [16] and for data in [15] above 250 K for several T_c values between 0 and 80 K. It is puzzling that this does not hold for T_c = 65 K in figure 2(b) suggesting that more data are needed to clarify this point. We note that there can be complications from Cu ions on the Tl sites affecting p [14] as well as time dependent changes in T_c when a Tl2201 crystal is held at room temperature [19] that were also seen in our TEP data for T_c = 26.5 K after one month at room temperature after the first anneal.

There seems to be two possible ways of explaining the variation of α₁ and T_c with p, which we will refer to as scenarios A and B. The more standard one, scenario A, based on the usual d-wave form for the pairing interaction, $V_{kk^{{\prime}}}$ $\propto$ $(\cos k_x - \cos k_y)( \cos k^{{\prime}}_x - \cos k^{{\prime}}_y)$, has been thoroughly analyzed and discussed in [10, 11] and related papers where the importance of small angle scattering was emphasized. Here T_c falls as p is increased because the pairing interaction falls [12, 13] and hence $(\hbar/\tau)/\Delta$ where Δ is the superconducting energy gap, is larger. The theory is very similar to that used to understand the effect of magnetic impurities in classical superconductors. Experimental evidence for the relation between T_c and ρ_res is summarized in [20] for both UD and OD cuprates. A second, less conventional one, scenario B, involves a different variation of the pairing interaction around the Fermi surface and is considered later. Scenario A could be referred to as 'pair-breaking' and B as ʼnot-pairing'. For our data set, the values of α₀ for the OD65, OD40 and OD26 crystals in table 1 are very close to that in [11] for their OD25 crystal measured using microwaves.

In standard transport theory, when there are two or more independent scattering mechanisms, for example electron-impurity and electron-phonon scattering, giving contributions ρ₀ and ρ₁ to the resistivity, the total resistivity $\rho = \rho_0 + \rho_1$, which is known as Matthiessen's rule (MR). For ordinary metallic alloys MR is a good first approximation although there are systematic deviations from MR that are well-documented [21]. There is considerable experimental evidence for Zn-doped YBCO and LSCO that MR applies to the T—independent impurity scattering and the T¹ scattering terms in the cuprates [22–24]. The equivalent way of calculating the TEP is via the NG [25] rule:

$$\begin{equation} {S_{\textrm{NG}} = \left( \rho_0S_0+\rho_1S_1+\rho_2S_2 + \cdots\right) /\left(\rho_0 +\rho_1 +\rho_2 + \cdots\right)} \end{equation} \tag{ 2 }$$

where S₀ and S₁ etc are the corresponding contributions to the TEP. If the electron diffusion TEP is dominant, and there are no small energy scales, these will be proportional to T, but with different signs and magnitudes governed by the energy dependence of the particular scattering mechanism. As shown in table 1 the T_c values in figure 1 are close enough to some of those in figure 2(a) so the coefficients α can be used directly to calculate the TEP via equation (2).

For $T_c\unicode{x2A7D}$ 67 K the parameters S₀, S₁, and S₂, all assumed to be proportional to T, were found by fitting the TEP data in figure 1(a) to equation (2). The range of fit was very limited, to below 200 K because of deviations from equation (1) at higher T and for the two higher T_c values to above 150 K, because of the onset of the CDW. Partly as a result of this, only the parameter S₂ representing the contribution to the TEP from the T² resistivity term is well-defined, although the term S₀ from impurity scattering must be positive. This procedure did not work for $T_c =$90 K where it gave the very small value of α₂ shown in table 1 and a negative value for α₀. For this crystal we therefore set $\alpha_0 = 6.4\pm 0.4\ \mu \Omega\cdot\mathrm{cm}$, the average for the other three T_c values, giving the values of S₁ and S₀ shown in table 1. For T_c = 88 K and 67 K S₀ must be at least 0.1µV K⁻² which is unexpectedly large. In figure 3 it can be seen that the calculated S_NG curves for T_c = 67 K and 26.5 K, agree well with the data giving $S_2 = -0.033T\mu$V K⁻¹, of the same sign and approximately 3 times larger than S₁ for T_c = 90 K. We can therefore conclude that the energy dependence of the T¹ scattering term has the same sign as that of the T² term, which is generally attributed to electron–electron scattering, but is significantly smaller. Measurements in which the superconductivity is suppressed by a magnetic field and on crystals with different residual resistivity, possibly controlled by doping with in-plane impurities such as Zn, would provide a further test of this picture. The magnitude of S₂ is discussed below. Because of momentum conservation the electron–electron scattering contribution to $\rho(T)$ usually depends on Umklapp scattering processes which will generally tend to be stronger for larger energies. This implies that for electrons, the electron–electron scattering term S₂ will be positive and for holes it will be negative in agreement with experiment for Tl2201. Within this NG picture the changes in TEP below 160 K in figure 1(a) as p is increased are caused by the increase in the ratio $\alpha_2T^2/\alpha_1T$ with doping p and the fact that $-S_2$ is larger than $-S_1$.

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In the previous sections it was assumed implicitly that the T¹ and T² scattering rates do not vary appreciably around the Fermi surface, so that their contributions to $\rho(T)$ can be added in the same way as for electron-phonon and electron-impurity scattering. This is only approximate because there is experimental evidence from angle-dependent magnetoresistance (ADMR) of the c-axis resistivity [26] that for crystals with T_c = 20 K, 17 K and 15 K, the anisotropy, (the parameter α in [26]) increases from 15$\%$ at 20 K to 40$\%$ at 110 K. Recently, new short-range CDW signals have been identified in RIXS studies of many cuprate families [27]. These are present for a wide range of p, including the 'strange metal' region, p > 0.19, that is relevant for Tl2201 although such short range signals have not yet been observed for Tl2201. They are short-ranged in real space, and therefore broader in reciprocal space and are visible to higher T than the CDWs that have reasonably well-defined, often incommensurate, wave vectors $\underline{Q}$. It has been argued that [28] that scattering from these short-range charge density fluctuations (CDF) are responsible for the T¹ term in $\rho(T)$ and because the CDF have a wide range of $\underline{Q}$ vectors they would indeed give isotropic scattering. So if CDF could be observed in Tl2201 crystals a good experimental test of this approach would be to compare their intensities with the size of the T¹ terms in $\rho(T)$ for different values of p.

Calculations by Rice et al [29] are more consistent with scenario B. These are based on the theory of Yang et al [30] and show that Umklapp scattering between hot spots gives rise to a T¹ scattering rate in a very limited region of $\underline{k}$-space and a corresponding T¹ term in $\rho(T)$. They are not based on the usual Boltzmann transport equation. Somewhat paradoxically, in an earlier paper using the variational principle [31] to solve the Boltzmann equation, Hlubina and Rice [32] showed that the T² term will dominate, although of course at low enough T any non-zero T¹ term must become larger than a term going as T². In this scenario $1/\tau$ at or near the hot spots is likely to be close to the Planck value because for $p\simeq$ 0.19 where any T² term is very small, as shown for example in table 1, $1/\tau$ is somewhat larger than the Planck value. It may be possible to distinguish between scenarios A and B by measuring the resistivity and London penetration depth on several crystals with similar T_c values. Scenario B provides a more direct link between the T¹ term and the superfluid density, emphasized recently in [17], than scenario A. However unlike scenario A it would not necessarily account for MR, the NG rule or the ADMR results. One possible test would be to see if either scenario can account for the 'vertical scaling' of the electronic specific heat [33] in applied magnetic fields.

The low T TEP of many correlated electron materials, including some where the T² term in $\rho(T)$ is dominant, has been shown [34] to obey the following simple empirical formula:

$$\begin{equation} \frac{S}{T} = \frac{\gamma q}{N_{\textrm{AV}}e}. \end{equation} \tag{ 3 }$$

Here γ is the Sommerfeld coefficient, N_AV Avogadro's number and e the magnitude of the electronic charge. The Faraday number $N_{\textrm{AV}}e = 9.6\times{}10^4$ Coulombs/mole. For the simple case of free electron dispersion and 1 electron per formula unit, $q = -1$ so taking $\gamma = 0.6\ \mathrm{mJ}\,(\mathrm{gm}\cdot \mathrm{at})^{-1}\,\mathrm{K}^{-2}$ [9] gives $S = -13~\mu$V K⁻¹ at 160 K, the end of the linear regions in figure 2(b). This will be reduced by a factor $1+p\simeq1.3$ by the larger carrier concentration and by a factor 2/3 for a constant carrier mean free path [34] giving $-6.7~\mu$V K⁻¹ at 160 K. The value of $-S_2 = 0.033T\mu$V K⁻¹ in table 1 gives a TEP of $-5.3~\mu$V K⁻¹ at 160 K. So the T² electron–electron scattering term in our TEP data for Tl2201 crystals agrees well with findings [34] for many other correlated electron materials, while the contribution from the T¹ term is approximately a factor 3 smaller.

The basic method used for the TEP is similar to that originally used for sintered polycrystalline bars in [5], but because of the small size of the crystals, and the fact that thicker crystals or thin films on a crystalline substrate have high thermal conductance, Au reference wires and Chromel wires forming two differential thermocouples are attached to Au pads on the ends of the crystal as shown in the photograph in figure A1. These wires extend for a few centimeters before being soldered to Cu wires heat sunk to a high thermal conductivity Cu block. A temperature difference ($\Delta T\lesssim1$ K) is produced by alternately heating two sapphire blocks to which the thermocouple wires are attached with General Electric GE-7031 low temperature varnish and the Seebeck coefficient of the sample S_sample is given by eliminating $\Delta T$ from the equations:

$$\begin{equation} {V_{\textrm{Au-Au}} = \left(S_{\textrm{sample}}-S_{\textrm{Au}}\right)\Delta T} \end{equation} \tag{ A.1 }$$

$$\begin{equation} {V_{\textrm{Ch-Ch}} = \left(S_{\textrm{sample}}-S_{\textrm{Ch}}\right)\Delta T} \end{equation} \tag{ A.2 }$$

Here the voltages across the Au wires, $V_{\textrm{Au-Au}}$ and the Chromel wires, $V_{\textrm{Ch-Ch}}$ are measured using a Keithley Instruments digital nanovoltmeter and a home-built nanovolt amplifier respectively, and their ratio obtained by making a least squares linear fit to a plot of $V_{\textrm{Ch-Ch}}$ vs $V_{\textrm{Au-Au}}$ for $\pm\Delta T$ and two points with $\Delta T = 0$. S_Ch is the Seebeck coefficient of Chromel taken from published standards and shown in figure A2. The same reel of $50~\mu$m diameter Au wire was always used and S_Au, also shown in figure A2 was obtained by measuring a strip of pure Pb which has a low TEP [35].

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In the past further checks have been made by measuring a superconducting cuprate, Bi2212 with $T_c\simeq$ 90 K, and a piece of 127 µm diameter phosphor bronze alloy wire purchased from Lake Shore Cryotronics which has a conveniently low TEP. In the present work a piece of the same wire was measured again and compared with earlier data, as shown in figures A3(a) and (b). Small changes in the TEP of Au can be induced by bending (cold working) the wire which will change the phonon drag component and the electron diffusion component from any Fe impurities. Measuring the phosphor bronze wire is a convenient method of correcting for such changes. Figure A3(c) shows the TEP data from figure 1(a) after making a small correction of ${\sim}0.1~\mu$V K⁻¹ corresponding to the difference in the TEP of the phosphor bronze wire at low T in earlier and more recent measurements shown in figure A3(a). It can be seen that T_c rises from $26 \pm{}1$ to $29 \pm{}1$ K after one month at room temperature after the first anneal.

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