Extreme thermopower anisotropy and interchain transport in the quasi-one-dimensional metal Li(0.9)Mo(6)O(17)
J. L. Cohn, S. Moshfeghyeganeh, C. A. M. dos Santos, J. J. Neumeier
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Extreme thermopower anisotropy and interchain transport in thequasi-one-dimensional metal Li . Mo O J. L. Cohn, ∗ S. Moshfeghyeganeh, C. A. M. dos Santos, and J. J. Neumeier Department of Physics, University of Miami, Coral Gables, FL 33124 Escola de Engenharia de Lorena - USP, P. O. Box 116, Lorena-SP, 12602-810, Brazil Department of Physics, Montana State University, Bozeman, Montana 59717
Thermopower and electrical resistivity measurements transverse to the conducting chains of thequasi-one-dimensional metal Li . Mo O are reported in the temperature range 5 K ≤ T ≤
500 K.For T ≥ K the interchain transport is determined by thermal excitation of charge carriers froma valence band ∼ . eV below the Fermi level, giving rise to a large, p -type thermopower thatcoincides with a small, n -type thermopower along the chains. This dichotomy – semiconductor-like in one direction and metallic in a mutually perpendicular direction – gives rise to substantialtransverse thermoelectric (TE) effects and a transverse TE figure of merit among the largest knownfor a single compound. Conducting materials with highly anisotropic Seebeckcoefficients (thermoelectric powers or TEPs) are poten-tially useful in transverse thermoelectric applications forenergy detection and cooling [1–3]. Bulk conductorsfor which the TEPs in different crystallographic direc-tions have opposite signs and yield a large magnitudefor their difference ( ∆ S ≥ µ V/K) are quite rare[3, 4], thus recent developments have focused on artifi-cial synthesis of stacked bulk materials [1, 2] or semi-conductor heterostructures [3] to achieve large Seebeckanisotropy. Here we present transport measurementson the quasi-one-dimensional (q1D) metal, Li . Mo O known as “lithium purple bronze” (LiPB), that reveala surprisingly simple mechanism for extreme Seebeckanisotropy in a bulk conductor. Direct electron trans-fer between the q1D metallic chains of this material issufficiently weak that interchain transport above 400 Kis predominated by thermal activation of valence bandstates ( ∼ . eV below E F ), yielding a large, p -typeinterchain Seebeck coefficient that coexists with n -typemetallic behavior confined along the q1D chains. A sub-stantial transverse Peltier effect is demonstrated. Theseingredients may exist in other materials or might possi-bly be engineered to develop transverse thermoelectricsbased on a single compound.A resistivity that is metallic at low-temperature anddecreases anomalously at high temperatures is a ubiqui-tous characteristic of transport transverse to the planesor chains of many q2D [5–8] and q1D metals [9–11], re-spectively. It is generally accepted that this behavior isdue to the onset of an additional conduction mechanismin parallel with band transport, possibly related to inter-plane or -chain defects (e.g. resonant tunneling) [12].Much less is known about the TEP transverse to theplanes or chains of such materials, partly because TEPmeasurements are difficult to perform in small singlecrystals for which the transverse transport directionshave very small dimensions (e.g., thin platelet or needle-like habits). In the few compounds where transverse TEP measurements have been reported [13–18], highanisotropy has not been observed.Li . Mo O known as “lithium purple bronze” (LiPB),is a low-temperature superconductor ( T c ≈ K) firstsynthesized and studied in the 1980s [19–21]. It has at-tracted interest more recently for its quasi-one dimen-sionality and Luttinger-liquid candidacy [22–28]. Crystalgrowth [20, 25] and transport properties along the chains(crystallographic b axis) for crystals similar to thosediscussed here have been presented elsewhere [29, 30].The resistivity anisotropy of LiPB is approximately [31] ρ b : ρ c : ρ a = 1 : 80 : 1600 . Single-crystal specimens wereoriented by x-ray diffraction and cut/polished into thinrectangular plates with the thinnest dimension (along the a axis) typically 40-80 µ m. The bc -plane dimensions weretypically . × . mm with the longest dimension coin-ciding with the transport axis ( b or c direction). Elec-trical contacts were made with Au leads attached withsilver epoxy. Current contacts covered the specimen endsand voltage contacts encircled the crystals across bothlarge faces and the sides. For thermopower measure-ments, specimens were suspended from a Cu heat sinkwith silver epoxy and affixed with a heater and 25- µ m-diameter differential chromel-constantan thermocouple,both attached with stycast epoxy. Separate radiation-shielded vacuum probes were employed for the cryogenicand high- T (> 320 K) measurements.Figure 1 (a) shows for two crystals the interchain re-sistivity and TEP, ρ c ( T ) and S c ( T ) , along with the in-trachain TEP, S b ( T ) , for two different crystals [29]. Ad-ditional c -axis data for two more crystals can be foundin the Supplementary Material [32]. The increase in allthree coefficients below 30 K has been discussed exten-sively elsewhere [29] and may be associated with local-ization, dimensional crossover or the development of un-conventional (e.g., electronically-driven) charge density-wave order [24, 25]. The focus of the present work is theinterchain TEP in the region T > K where it risessharply with increasing T , coincident with a deviation (K) S ( µ V / K ) Li Mo O ρ c ( m Ω c m ) A B (a) S c ( µ V / K ) Y-123Bi-2212 (b) cb FIG. 1. (color online) (a) LiPB interchain ( c -axis) ther-mopowers (left ordinate) and resistivities (right ordinate) fortwo crystals (labeled A and B), and intrachain ( b -axis) ther-mopowers for two different crystals. Solid curves through the c -axis data are fits to the parallel conduction model discussedin the text; parameters for the semiconducting component arelisted in Table I. Dashed line is a linear-least-squares fit to thelow- T ρ c ( T ) (see text). Inset: orientation of the crystallo-graphic axes with respect to the q1D Mo-O chains (2 per unitcell). (b) interplane ( c -axis) thermopowers for YBa Cu O (Y-123, Ref. 16) and Bi Sr CaCu O (Bi-2212, Ref. 17). of ρ c ( T ) from it’s low- T , linear- T behavior (dashed line,Fig. 1). The intrachain TEP is linear-in- T , modest inmagnitude, and becomes negative above 300 K, consis-tent with electron-like carrier diffusion as noted previ-ously [29], and extended here to 520 K.Several features of S c ( T ) are noteworthy. It remainspositive throughout the temperature range. In the linear- T regime of ρ c (40 K < T <
140 K), S c is nearly T -independent at ∼ µV /K and essentially the samefor all crystals measured. Near the maximum in S c at T ≃ K, ∆ S = S c − S b ≥ µ V/K. The interchaintransport in LiPB is incoherent, the metallic character of ρ c in the lower- T regime likely reflecting the intrachainscattering rate [33], consistent with very weak and indi-rect interchain hopping [28]. The nearly constant TEPat low- T is a characteristic of narrow-band hopping [34].As for the increase in S c to very large values at higher T , Energy (eV) E F q b * c * D C D V c * FIG. 2. (color online) Bands along main symmetry directions(green: along b ∗ , black: along c ∗ ) and projected Fermi sur-face (red and blue curves) in the b ∗ c ∗ plane for LiPB (adaptedfrom Ref. 28). ∆ V and ∆ C are excitation energies for valence-and conduction-band states, respectively, dispersing along c ∗ near the X-point of the Brillouin zone. Indirect transitions be-tween these states and the Fermi surface are possible throughthermal excitation and absorbtion of a phonon with sufficientmomentum along b ∗ (wavevectors for excitation of valence-band states are represented by pink arrows). it is instructive to compare with the behavior found forthe interplane ( c -axis) TEPs of the q2D metals [16, 17],YBa Cu O and Bi Sr CaCu O [Fig. 1 (b)]. The TEPsof the latter materials, truncated by the onset of super-conductivity at low- T , also rise for T & K, but theiroverall increases ( . µ V/K) are substantially smallerthan for LiPB and a tendency toward saturation is ev-ident at the highest T . Their upturns are plausibly at-tributed to the onset of interplane tunneling, though the-oretical work [35] has not yet treated the TEP within amodel that incorporates resonant tunneling through de-fects [12]. The upturn in the interchain TEP for LiPB isqualitatively and quantitatively different.We propose that this difference has its origin in theLiPB band structure which is distinguished from theseother compounds by the presence of valence and con-duction bands in close energy proximity to E F and withsufficient dispersion for interchain momentum so as tobecome increasingly important for interchain transportwith increasing T . Figure 2 shows the calculated bandstructure [21, 23, 27, 28] within the b ∗ - c ∗ plane (disper-sion along a ∗ is negligible). The projected Fermi surface(blue and red curves) is also shown. Electrons in valencebands dispersing along c ∗ (black curves) can make indi-rect, interband transitions to states at E F through ther-mal excitation ( ∆ V , Fig. 2) and absorption of phononswith sufficient momenta ( q ) along the b ∗ direction (pink rrows, Fig. 2). Similarly, electrons at E F in states dis-persing along b ∗ can be thermally excited ( ∆ C ) to thelowest-lying interchain conduction band above E F nearthe X-point by absorbing phonons with opposite mo-menta. These phonons, with momenta q . √ π/ b andenergies ~ qv . meV (assuming a dispersionless, acous-tic phonon with velocity v = 3 km/s), will be excited inlarge numbers at room temperature and above. The ac-tivation energies, from averaging the various band struc-ture calculations, are ∆ V = 0 . eV and ∆ C = 0 . eV.With ∆ V < ∆ C a p -type thermopower should result.These observations motivate an interpretation of theinterchain transport in LiPB that reflects parallel con-duction through band-like states ( lo ), predominant at T . K, and a thermally activated, semiconductorcontribution ( hi ), predominant at T > K, σ = σ lo + σ hi S = (cid:0) σ lo /σ (cid:1) S lo + (cid:0) σ hi /σ (cid:1) S hi . (1)The low- T TEP is taken as S lo = 32 µ V/K, inde-pendent of T as motivated by the S c ( T ) data, and σ lo = 1 / ( A + BT ) from linear-least-squares fits (dashedlines, Fig. 1) to the ρ c ( T ) data in the range 40 K ≤ T ≤ K. For the high- T contribution we first tried asingle semiconductor component to minimize the num-ber of free parameters: σ hi = C exp( − E σ /k B T ) and S hi = ( k B / | e | )( E S /k B T + D ) , where C and D are con-stants and E σ and E S are activation energies. The solidcurves through the ρ c ( T ) and S c ( T ) data (Fig. 1) demon-strate the agreement achieved with Eq. (1) throughoutthe range T ≥ K using these simple forms for σ hi and S hi (parameter values are listed in Table I). The dis-crepancy between the computed and measured TEPs inthe transition region (near 200 K) may reflect our ne-glect of a tunneling contribution, like that observed forthe cuprates [Fig. 1 (b)], but has negligible impact on thefitting at high- T if, as in the latter materials, this con-tribution adds a small constant. The average activationenergies found for the four crystals (with s.d. uncertain-ties) are: E σ = 0 . ± . eV and E S = 0 . ± . eV.The observation E S > E σ is incompatible with single-band ( E S = E σ ) and intrinsic two-band ( E S ≤ E σ )semiconductor conduction for the high- T component, but TABLE I. Fitting parameters for the semiconductor, high- T component ( hi ) of the parallel conduction model discussed inthe text.Specimen C (Ω − cm − ) E σ (eV) D E S (eV)A . × . × . × . × is naturally explained [32], consistent with the bandstructure, if both valence and conduction band statesare excited with differing activation energies, ∆ V and ∆ C . Analyzing the data with a three-component model,low- T metallic and two high- T semiconducting contri-butions [32], yields average energies ∆ V ≃ . eV and ∆ C ≃ . eV, i.e. nearly the same as E σ and E S inthe two-component model. The interband transitionsevidently serve as the predominant mechanism for in-terchain ( c -axis) transport above 400 K where the semi-conducting components represent more than 50% of thetotal conductivity [32]. In this regime LiPB thus behavessimultaneously as a p -type semiconductor and n -typemetal along mutually perpendicular directions, leadingto its high Seebeck anisotropy.To test for the transverse Peltier effect, a rectangularspecimen ( x × y × z = 1 . × . × . mm ) with its longaxis at an angle α = 32 ◦ to the b axis (Fig. 3, inset), wasmounted with one edge thermally anchored to a copperheat sink in vacuum; current was applied along the x direction. Cooling or warming at the free edge of thecrystal, monitored by a differential thermocouple withjunctions along the y direction (separation d ≃ . mm),was induced by forward or reverse current, respectively,and was linear in the current (Fig. 3). The averaged tem-perature gradient, dT /dy = [∆ T ( I +) − ∆ T ( I − )] / d , isshown as a function of the average specimen tempera-ture in Fig. 4 for fixed values of the current. The dashedcurves in Fig. 4 were computed using ∆ S [interpolated I (mA) -100 -50 0 50 100 d T / d y ( K / mm ) -8-6-4-202468
440 K330 Kcoolingwarming
FIG. 3. (color online) Current dependence of the transversePeltier-induced temperature gradient for a LiPB crystal at T = 330 K and
K. Error bars reflect uncertainty dueto slow oscillations in the Cu block temperature ( ≤ . K),particularly at the highest T . The inset shows the orientationof current and heat flow relative to the specimen ( x-y ) andcrystallographic ( b-c ) axes. (K)
300 350 400 450 500 d T / d y ( K / mm ) I=10 mA3050100 k yy ( W / m K ) FIG. 4. (color online) dT /dy (averaged for + , − current) v s. the average specimen temperature at fixed applied cur-rents (left ordinate), and κ yy ( T ) (right ordinate). The dashedcurves are computed (see text) using κ yy and ∆ S interpo-lated from Fig. 1 (a). Error bars for the data and computedcurves (12%) are determined by uncertainty in the thermo-couple junction separation. The κ yy data were corrected forradiation losses and conduction through the leads estimatedfrom direct measurements on similar specimens suspendedfrom their leads. These corrections amounted to 6% at 300 Kand 30% at 500 K. from data for crystal B, Fig. 1 (a)], the transverse ther-mal conductivity ( κ yy ) measured in a separate experi-ment with a heater attached to the free end of the spec-imen (Fig. 4, right ordinate), and the heat flux equation[2], dT /dy = ( T S xy /κ yy ) j , where S xy = (1 / S sin(2 α ) and j is the current density. The latter expression ig-nores a Joule heating term (varying as j ), justified bythe linearity of dT /dy with current noted above. BecauseLiPB has a rather low thermal conductivity along the c axis [30], the transverse thermoelectric figure of merit isamong the largest known for a single-phase material [36], Z xy T = T S xy / ( ρ xx κ yy ) ≃ . ± . at 450 K [37].LiPB may itself prove useful in converting waste heatto electrical power or in energy detection. Given that thefeatures underlying its extreme Seebeck anisotropy ap-pear fairly generic – low-dimensional, metallic electronicstructure and dispersing bands for the transverse direc-tion in close proximity to E F – the larger implicationfrom this study is that other materials with such proper-ties may yet to be revealed.The authors acknowledge M. Grayson and B. Cui forvery helpful comments. This material is based upon worksupported by U.S. Department of Energy (DOE)/BasicEnergy Sciences (BES) Grant No. DE-FG02-12ER46888(University of Miami), the National Science Founda-tion under grant DMR-0907036 (Montana State Univer-sity), and in Lorena by the CNPq (308162/2013-7) andFAPESP (2009/54001-2). ∗ Corresponding Author: [email protected][1] H. J. Goldsmid, J. Electron. Mater. , 1254 (2011).[2] C. Reitmaier, F. Walther, and H. Lengfellner, Appl.Phys. A , 717 (2010).[3] C. Zhou, S. Birner, Y. Tang, K. Heinselman, and M.Grayson, Phys. Rev. Lett. , 227701 (2013).[4] K. P. Ong, D. J. 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Lett. , 187003(2012).[32] For additional data and analysis, see Supplemen-tary Material at http://link.aps.org/supplemental/xx.xxxx/PhysRevLett.10x.xxxxxx, which includesRef.’s [38, 39].[33] N. Kumar and A. M. Jyannavar, Phys. Rev. B , 5001 , 2820 (1988).[35] T. W. Silk, I. Terasaki, T. Fujii, and A. J. Schofield,Phys. Rev. B , 134527 (2009).[36] Other materials with large Z xy T include CsBi Te andReSi . . D.-Y. Chung, et al. , Mat. Res. Soc. Symp.Proc., , S6.1.1 (2004); J-J Gu, et al. Mat. Res. Soc.Symp. Proc. , BB6.10.1 (2003); Yang Tang, BoyaCui, Chuanle Zhou, and M. Grayson, unpublished.[37] Since ρ xx was not measured for the Peltier crystal, wecomputed it using the comprehensive results for ρ b fromWakeham et al. [Nature Commun.
396 (2011)], lin- early extrapolated to 450 K, and the ρ c data for crystalB [Fig. 1 (a)]. Using the appropriate tensor expressions(Ref. 2), this gave ρ xx ≃ . × − Ω m . The linear-ity in ∆ T ( I ) (Fig. 3) implies that the actual ρ xx for ourspecimen cannot be much larger than this value (we esti-mate not by more than 30%); a deviation from linearity(suppressed ∆ T ) that is quadratic in j would be the sig-nature of Joule heating from the second term in the heatflux equation [2].[38] H. Fritzsche, Sol. St. Commun. upplementary Material: Extreme thermopower anisotropy and interchain transportin the quasi-one-dimensional metal Li . Mo O J. L. Cohn, S. Moshfeghyeganeh, C. A. M. dos Santos, J. J. Neumeier
Additional Data for S c and ρ c ρ c ( m Ω c m ) Li Mo O C D
T (K) S c ( µ V / K ) FIG. S1. LiPB inter-chain ( c -axis) thermopowers (left ordinate) and resistivities (right ordinate) for crystals C and D. Solidcurves through the data are fits to the parallel conduction model discussed in the text; parameters for the semiconductingcomponent are listed in Table I. Dashed lines are linear-least-squares fits to the low- T ρ c ( T ) . Differing excitation energies for valence and conduction bands yields E S > E σ Here we demonstrate that a single semiconductor component with E S > E σ , as emerged from the two-componentmodel (metal + semiconductor) discussed in the text, can be reproduced by a two-component semiconductor withdiffering hole and electron excitation energies, ∆ V = E F − E V and ∆ C = E F − E C , respectively, and ∆ V < ∆ C .Distinguishing electron ( e ) and hole ( h ) contributions, we have: σ = σ e + σ h = C C exp(∆ C /k B T ) + C V exp(∆ V /k B T ) S e = − k B e (cid:18) ∆ C k B T + A e (cid:19) ; S h = k B e (cid:18) ∆ V k B T + A h (cid:19) S = ( σ e /σ ) S e + ( σ h /σ ) S h . Figure S2 shows σ ( T ) and S ( T ) computed for the following parameters appropriate to crystal B: ∆ C = 0 . eV, ∆ V =0 . eV, C C = 1900 Ω − cm − , C V = 985 Ω − cm − , A e = 2 . , and A h = 2 . . The solid lines demonstrate thatthis set of parameters produces effective single-component parameters identical to those of the single semiconductorcomponent listed in Table I for specimen B and plotted in Fig. 1. The constants A e and A h represent weighted ABLE S2. Fitting parameters for the semiconducting components of the three-component model for three specimens.Specimen C C (Ω − cm − ) ∆ C (eV) C V (Ω − cm − ) ∆ V (eV) A e A h B . × . × . × . × . × . × averages over the charge carriers in the conduction and valence bands [38]. For example, A = 3 corresponds to adensity of states and mobility that increase linearly with energy, and A = 1 for constant density of states and mobility.Temperature dependent band energies, varying linearly in T to lowest order, can contribute to these constant termsand even alter their sign [39]. Three-component fitting
Incorporating two semiconductor components along with the low- T term described in the text constitutes a three-component model with appropriate weighting by the respective partial conductivities. This procedure producesfits that are indistinguishable from those of Fig. 1; Table S1 lists fitting parameters describing the semiconductingcomponents for three crystals. Crystal A was excluded because its data do not extend to high enough temperature tosufficiently constrain the semiconducting parameters. Note that this analysis yields values for ∆ C and ∆ V that areonly 5-10% smaller than E S and E σ , respectively, of the simpler (two-component) model (Table I). Figure S3 showsthe T -dependent weights (fractional conductivities) for each of the three components using the fitting parameters forcrystal B. -1 ) s ( W - c m - ) S ( m V / K ) S · (e/k B )=2804/T-2.681 s =1600exp(-1695/T) crystal B -1 ( - c m - S ( V / K ) (e/k )=2903/T-2.898=2410exp(-1845/T) crystal D FIG. S2. Conductivity and thermopower computed from the two-component semiconductor model having effective activationenergies (solid lines) matching those of the single-component semiconductor contribution to the fitting to specimen B discussedin the text (Table I). (K)
300 350 400 450 500 w e i g h t i n g o f T E P σ v /σ σ c /σ -1 S ( V / K ) S=( )S +( )S σ lo /σ FIG. S3. T -dependent weights (conductivity ratios) for each of the three components from three-component fitting to crystalB (parameters from Table S1). [38] H. Fritzsche, Sol. St. Commun.
226 (1953).226 (1953).