Theory of Strain-Controlled Magnetotransport and Stabilization of the FerromagneticInsulating Phase in Manganite Thin Films

Anamitra Mukherjee,1,2 William S. Cole,2 Patrick Woodward,3 Mohit Randeria,2 and Nandini Trivedi2

1Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada2Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

3Department of Chemistry, The Ohio State University, Columbus, Ohio 43210, USA(Received 11 October 2012; published 9 April 2013)

We show that applying strain on half-doped manganites makes it possible to tune the system to the

proximity of a metal-insulator transition and thereby generate a colossal magnetoresistance (CMR)

response. This phase competition not only allows control of CMR in ferromagnetic metallic manganites

but can be used to generate CMR response in otherwise robust insulators at half-doping. Further, from

our realistic microscopic model of strain and magnetotransport calculations within the Kubo formalism,

we demonstrate a striking result of strain engineering that, under tensile strain, a ferromagnetic charge-

ordered insulator, previously inaccessible to experiments, becomes stable.

DOI: 10.1103/PhysRevLett.110.157201 PACS numbers: 75.47.Lx, 68.60.Bs, 75.47.Gk, 77.80.bn

Introduction.—Transition metal oxides have long beenstudied for their surprising emergent behavior such as highTc superconductivity in the cuprates, ferroelectricity in thetitanates, and colossal magnetoresistance (CMR) in themanganites. However, very recent advances in heterojunc-tion growth [1–3] have opened the possibility of producingatomically perfect interfaces of oxide materials and there-fore applying precisely controlled strain to oxide thin films.In this Letter, we address the impacts of strain on orderedphases, temperature scales, and CMR in the manganites.As a specific example, we consider materials at ‘‘half-doping’’ which have a prototypical chemical formula ofA0:5A

00:5MnO3, where A is a rare earth element and A0 an

alkaline earth metal [4].At large bandwidths (BW), the half-doped manganites

are ferromagnetic metals (F-M), while narrow BW mate-rials are spin, charge, and orbitally ordered insulators(SCO-I), as described later in the text. CMR is known tooccur in F-M materials close to the metal-insulator phaseboundary and is the result of phase competition (betweenthe F-M and a charge-ordered insulator), which is tradi-tionally controlled by isovalent chemical substitutions atthe A site [4]. The unstrained material has been theoreti-cally studied extensively [5,6]. Most prior work on theeffects of strain (both theory [7–13] and experiment[14–22]) have focused on how it affects the magnetic andelectronic phases, with little emphasis on magnetotran-sport. We extend the usual model for manganites, previ-ously used to study magnetotransport without strain[5,6,23], to propose and solve, for the first time, a micro-scopic Hamiltonian that includes the effects of strain andobtain the following results. (i) Tensile strain provides aroute to stabilizing a ferromagnetic charge-ordered insula-tor (FC-I). This phase has not been conclusively observedin any half-doped manganite with tolerance factor varia-tions but should finally be observable with strain

engineering. (ii) We demonstrate that strain can inducephase transitions. As a consequence, we show that theCMR in F-M materials can be enhanced by tuning theproximity to the metal-insulator transition and that insulat-ing phases can be made metallic under strain and thereforealso exhibit a CMR response. This greatly expands thefamily of materials with potential device applications.(iii) We show that strain engineering also allows for controlover Tc in F-M manganites and can be used to control theCMR temperature in the CMR materials.Model.—We begin with the ‘‘standard model’’ for

the manganites. Because of the octahedral crystal field,the Mn eg levels have a higher energy than the t2g levels.

Combined with a large Hund’s coupling, which ensuresthat the electron spins align ferromagnetically, this local-izes three Mn 3d electrons in the t2g levels which form

local S ¼ 3=2moments (called ‘‘core spins’’). The remain-ing electrons, if any, are itinerant and occupy two bandsthat result from the hybridization of the Mn eg levels. The

model also includes an effective antiferromagnetic super-exchange J between neighboring Mn core spins, andfinally the eg electrons couple to Jahn-Teller phonons

with a coupling strength �.All energy scales are given in units of the unstrained

bandwidth t. This Hamiltonian yields an accurate descrip-tion of the physics of the manganites [24] and is discussedin detail in Sec. I of the Supplemental Material [25].We extend this model to incorporate the effects of strain.

Figure 1(a) schematically illustrates substrate-induced,in-plane tensile strain. Tensile (compressive) strain iscaused by growing the film on substrates with latticeparameters larger (smaller) than those of the unstrainedfilm. We assume that strain is applied parallel to the (a-b)plane, which we take to be the plane of the dx2�y2 orbital.

We quantify strain by the parameter ek ¼ ½ðas � aÞ=a� ¼�a=a, where as and a are the substrate and film lattice

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parameters. Here, a is the distance between two nearestneighbor Mn atoms. We consider the volume conservingstrain and use the relation e? ¼ �4�ek, where � is the

Poisson ratio. We choose � ¼ 0:375, consistent with pre-vious estimates [26,27]. In-plane compressive strain cor-responds to ek < 0, while ek > 0 for tensile strain. With

this in mind, we propose three microscopic effects of strainthat must be included in our model, extending previoustheoretical proposals [8,9]. (i) Strain modifies the hoppingmatrix elements. Strain affects the lattice parameter a,which in turn modifies both the Mn-O bond length d andthe Mn-O-Mn bond angle �, as seen in Fig. 1(b). In theSec. I(c) of the Supplemental Material [25], we show, byusing Slater-Koster [28] and Harrison [29] scaling, that theeffect on the hopping matrix elements due to the change in� can be neglected for a large class of half-doped man-ganites and that the hopping in the (a-b) plane scales withstrain as txy ! txyð1� 7ekÞ. We restrict our calculations to

a single-layer manganite film in the a-b plane [as depictedin Fig. 1(a)] and refer to the unstrained in-plane hoppingparameter txy as t and under strain as ~t. (ii) Strain modifies

the antiferromagnetic superexchange. The superexchangecoupling also scales with the hopping. From similarconsiderations, it can be shown that the in-plane super-exchange scales as Jxy ! ðJk=tÞð1� 14ekÞ with strain. We

refer to the unstrained in-plane superexchange as J and inthe strained case ~J. (iii) Strain generates an orbital bias.Because of the increase in the in-plane Mn-O bond length,tensile strain makes occupation of the dx2�y2 orbital ener-

getically favorable. In-plane compressive strain favors theout-of-plane d3z2�r2 orbital. This orbital bias induced by in-

plane compressive and tensile strains in La0:7Sr0:3MnO3

has been observed in x-ray absorption [30] as well as inangle resolved photoemission [31]. We incorporate thisin our model Hamiltonian with an extra term Hbias ¼P

i;���ni;�, where �� ¼ �=2 for � ¼ d3z2�r2 and �� ¼��=2 for � ¼ dx2�y2 .

From experiments, the eg splitting has been estimated to

be between 0:4t and 3t [30,32]. We make a conservativeestimate for the bias to be � � 10ekt, i.e., a splitting of

�0:2t for �2% strain. These values are consistent withdensity functional estimates [33,34]. Values of 2%–3% forstrain on manganite films, as we consider here, are easilyachievable in experiments [15,21].As mentioned, we perform our calculations in two

dimensions, describing a single-layer manganite film inthe a-b plane. Further, we assume strain to be uniform inthe layer. This is sufficient to bring out the importantfeatures of the phase diagram and in-plane transport. Wedescribe our method of solution in Secs. II and III in theSupplemental Material [25] and focus here on our results.Strain driven phase transitions.—Figure 2(b) shows the

T ¼ 0 �-J phase diagramwithout strain. On this, we denotetwo representative parameter points, F-M (blue dot) andSCO-I (red dot). The SCO-I is an insulator with planarcheckerboard charge order (CO), alternating dx2�r2=dy2�r2

orbital order (on the sites with larger charge density),and charge-exchange-type spin order (zigzag ferromag-netic chains coupled antiferromagnetically). Figures 2(a)and 2(c) show the effect of strain on these points.

FIG. 2 (color online). (a) The ferromagnetic metal (F-M), fer-romagnetic charge ordered insulator (FC-I), and paramagneticinsulator (P-I) in the temperature (T) vs strain (ek) plane.

Compressive strain (ek < 0) on F-M raises Tc, while tensile strain

initially suppresses Tc, but, beyond�1%, strain drives the systeminto an FC-I. (b) Unstrained T ¼ 0 phase diagram in the �=t andJ=t planes. This phase diagram also shows the spin-charge-orbitalordered insulator (SCO-I) and theA-type antiferromagnetic metal(A-M) [40]. The blue dot at ð�=t; J=tÞ ¼ ð1:6; 0:08Þ refers to theunstrained starting point from which we calculate (a), while thered dot at ð�=t; J=tÞ ¼ ð1:65; 0:09Þ refers to the unstrained pointfrom which we calculate the phase diagram (c). Under strain, thehopping parameter t is rescaled, which changes both the ratios�=tand J=t. The symbols at the arrow heads indicate such values for2% compressive (triangles) and tensile (squares) strain, respec-tively. (c) Tensile strain on the SCO-I enhances the charge-ordering temperature TCO while compressive strain causes atransition to F-M. TCE denotes the spin-ordering temperature. In(a) and (c), for clarity, we do not show strain-induced low tem-perature equilibrium phase separation [41].

FIG. 1 (color online). (a) Uniform tensile strain in the a-bplane is introduced by lattice mismatch with the substrate. Thevolume conserving tensile strain expands the in-plane Mn-Obonds while contracting bonds in the c direction. This causeshigher occupancy of the in-plane dx2�y2 orbital, a larger out-of-

plane hopping and smaller in-plane hopping compared to theunstrained values. Compressive strain, not shown, has the oppo-site effect. (b) Schematic of a generic Mn-O-Mn bond understrain. We indicate the Mn-O bond length d; the Mn-O-Mnbond angle �; and the shortest Mn-Mn distance, the latticeparameter a. An expansion of a along the green arrows causesa change in d and �.

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Starting from these parameters, in-plane compressivestrain favors F-M, as seen in both (a) and (c). There aretwo competing effects here, however. Compressive strainincreases the in-plane hopping, which in turn reduces �=~t,as seen by following the dashed black arrow in (b). Thisfavors a metallic state where double exchange promotesferromagnetism. On the other hand, compressive strainincreases ~J=~t, which tends to narrow the BW, while theorbital bias promotes occupancy of the out-of-plane d3z2�r2

orbital. Both of these latter effects work against the stabil-ity of the F-M, but we find the F-M to be dominant up to themaximum values of strain we have considered.

In-plane tensile strain stabilizes insulators that can haveeither ferromagnetic or antiferromagnetic spin textures, asseen in (a) and (c) respectively. The insulators have long-range checkerboard charge order and are stabilized by thereduced in-plane hybridization or increased �=~t, as seenby following the gray dashed arrow in (b). This effect tendsto localize the electrons. While a sufficient increase in �eventually turns the system insulating, regardless of theunstrained F-M parameter, the magnetic order dependscrucially on the value of ~J=~t.

This dependence of the magnetic and charge-orderingscales on strain has been seen in experiments, both awayfrom Ref. [17] and at half-doping [14,15]. They includeincrease of Tc with compressive strain in a La0:8Ba0:2MnO3,suppression ofTc with tensile strain inLa0:67Ca0:33MnO3, andenhancement of TCO with tensile strain on Pr0:5Ca0:5MnO3

[15,21]. We note that, while there is a dearth of experimentaldata on scaling of t and J formanganites, our results are robustto typical variations in the scaling [35].

Stability of the FC-I phase.—From Fig. 2(b), we see thatadequate tensile strain on F-M with J=t� 0:05–0:08 con-verts the system into a FC-I, just as that depicted for (�=t ¼1:6, J=t ¼ 0:08) by the gray dashed arrow. The unstrainedFC-I phase was predicted in theory [37] at �, J valuesas in Fig. 2(b). In small BW half-doped manganites, e.g.,La0:5Ca0:5MnO3, signatures of this phase coexisting withthe AF-CO phase were reported [38] at 90 K. This impliesthat in the half-doped manganites either FC-I is the trueground state only in a narrow �, J window or it is a meta-stable state with energy very close to the true ground state.We predict that tensile strain on an ordered F-M suppressesother phases and can stabilize the FC-I as the ground state.

Effect of strain on magnetotransport.—The maximumCMR temperature achievable by BW-tuned phase compe-tition is the TC of the unstrained material. Additionally, thebicritical nature of the phase diagram and proximity to themetal-insulator boundary needed for CMR keep the TC

quite low [39]. We show that, because strain affects differ-ent intrinsic energy scales differently, it not only tunesphase competition but also allows optimization of thecompetition between CMR temperature and %MR.

Figure 3(a) shows the resistivity �ðTÞ for various tensilestrain values on the unstrained F-M phase [blue dot in

Fig. 2(b)]. While CMR behavior has been reported before[5,6,9], our novelty is the use of strain as an external knob.Increasing tensile strain causes a rapid rise in the resistivitymaximum that occurs at T � TC, accompanied with reduc-tion of both the TC and the temperature at the resistivitymaximum (TCMR). The reduction in TC is due to the app-roach to the F-M–SCO-I boundary by increasing the tensilestrain, as depicted in the inset in (a). The reason for theincrease in the resistivity maximum is the strain-inducedenhancement of the metal-insulator coexistence at TC, asillustrated in (b).The color maps here depict the volume fraction of the

insulating regions (red patches) embedded in an otherwiseconducting background at TC. These insulating regionsgrow in volumewith increasing strain and have short-range(�, �) CO correlations, the same correlations that onefinds in the competing FC-I phase. The thermal fluctua-tions at TC are typically dominated by the nearest meta-stable minimum, in this case the FC-I phase. Further, sincestrain controls the proximity to the F-M–FC-I boundary,increasing tensile strain makes the FC-I state progressivelyapproach the energy of the F-M ground state, favoringgreater insulating regions with short-range (�, �) COcorrelations. If we start with other initial (unstrained)

FIG. 3 (color online). (a) Resistivity �ðTÞ for different valuesof tensile strain. Strain values are marked on the individualcurves, which are color coded with the arrows in the inset [whichis a schematic of Fig. 2(a) showing only the tensile strain part].Note that, while the temperature at which the peak occurs (whichis close to TC) decreases, the peak value of the resistivityincreases. (b) Real space snapshots of charge ordering at TC;the volume fraction of checkerboard charge-ordered regions(shown in red) grows systematically with tensile strain. Thelabels A, B, and C on them and in (a) denote the strain andtemperature values where these were calculated. The magneto-resistance (%MR) as a function of magnetic field for�1% strainon a system close to the F-M–FC-I phase boundary and �2% ona system close to the F-M–SCO-I boundary. The unstrainedvalues for both parent parameter points (triangles and diamonds)are shown for comparison.

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starting points, the tensile strain can result in a F-Mto SCO-I phase transition. The qualitative behavior ofmagnetotransport is the same as in the F-M to FC-I caseshown here. Magnetotransport data near the FM–SCO-Iphase boundary is shown in Sec. IV of the SupplementalMaterial [25].

In Fig. 3(c), we show the %MR, defined as 100½�ð0Þ ��ðhÞ�=�ð0Þ and calculated at T ¼ TCMR, as a function ofmagnetic field for two cases. One shows %MR close tothe F-M–FC-I phase boundary, with (circles) and without(diamonds) strain; the other shows %MR close to theF-M–SCO-I boundary. The amount of increase in theresistivity maximum with strain and the %MR dependson the domains of metastability of the competing phases,the type of the insulator, and the largest tensile strain thatcan be applied before the system becomes insulating.However, regardless of the nature of metal-insulator phasecompetition, applying tensile strain yields an enormousenhancement of %MR (circles and stars) over theunstrained values (diamonds and triangles).

Finally, we demonstrate that compressive strain candrive insulators across the metal-insulator transitioninto the CMR regime. Figure 4(a) shows �ðTÞwith increas-ing compressive strain on SCO-I. At 0.8% strain, theinsulator-to-metal transition is accompanied by a charge-exchange-to-ferromagnetic transition. Increasing the com-pressive strain further causes a monotonic increase in TC

[also seen in the inset in (a)]. The peak in the resistivitywith increasing strain is also systematically shifted tohigher temperatures. In (b), we plot %MR at the tempera-ture of the resistivity maximum as a function of strain. Wealso show the corresponding CMR temperatures (TCMR).We find that %MR is reduced upon increasing strain, asexpected, but there is in fact an optimal region in which

TCMR can be increased without substantially reducing%MR. We have checked that our results survive A sitedisordering.Conclusion.—We stress a major difference between the

strain engineering and isovalent substitution. While bothcan tune the bandwidth, uniform strain will not introducethe short-range disorder that naturally results from substi-tution. As a result, ferromagnetic TC’s and TCMR of strainedfilms should be higher than in their substitution-engineeredcounterparts. In turn, larger intrinsic bandwidthwould causea greater %MR at smaller magnetic fields since the externalfield required to align the core spins is reduced. Also, straineffects on manganites of any material composition anddoping can be directly studied in our approach. Our resultsindicate some promising future directions. First, we havedemonstrated that strain gives access to a large phase spaceof new and accessible states for a given unstrained material.Second, strain need not introduce disorder, in contrast tochemical substitution. Finally, strain directly impacts orbitaloccupancy in a tunable way and opens new possibilitiesfor orbital-state sensitive electronics.We gratefully acknowledge support from DOE DE-

FG02-07ER46423 (A.M.), DOE BES DE-SC0005035(M.R.), NSF DMR-0907275 (N. T.), and Center forEmergent Materials NSF MRSEC DMR-0820414(W. S. C. and P.W.).

[1] J. Mannhart, and D.G. Schlom, Science 327, 1607 (2010).[2] E. Dagotto, Science 318, 1076 (2007).[3] H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N.

Nagaosa, and Y. Tokura, Nat. Mater. 11, 103 (2012).[4] D. Akahoshi, M. Uchida, Y. Tomioka, T. Arima, Y.

Matsui, and Y. Tokura, Phys. Rev. Lett. 90, 177203 (2003).[5] C. Sen, G. Alvarez, and E. Dagotto, Phys. Rev. Lett. 105,

097203 (2010).[6] C. Sen, G. Alvarez, and E. Dagotto, Phys. Rev. Lett. 98,

127202 (2007).[7] K. H. Ahn, T. Lookman, and A. R. Bishop, Nature

(London) 428, 401 (2004).[8] A. Baena, L. Brey, and M. J. Calderon, Phys. Rev. B 83,

064424 (2011).[9] S. Dong, S. Yunoki, X. Zhang, C. Sen, J.-M. Liu, and

E. Dagotto, Phys. Rev. B 82, 035118 (2010).[10] J. H. Lee and K.M. Rabe, Phys. Rev. Lett. 104, 207204

(2010).[11] K. H. Ahn and A. J. Millis, Phys. Rev. B 64, 115103

(2001).[12] A. J. Millis, T. Darling, and A. Migliori, J. Appl. Phys. 83,

1588 (1998).[13] M. J. Calderon, A. J. Millis, and K.H. Ahn, Phys. Rev. B

68, 100401 (2003).[14] F. Yang, N. Kemik, M.D. Biegalski, H.M. Christen,

E. Arenholz, and Y. Takamura, Appl. Phys. Lett. 97,092503 (2010).

[15] D. Okuyama, M. Nakamura, Y. Wakabayashi,H. Itoh, R. Kumai, H. Yamada, Y. Taguchi, T. Arima,

FIG. 4 (color online). Compressive strain can induce F-M ininsulating manganites: (a) shows �ðTÞ of the resulting F-M as afunction of temperature for various values of compression [colorcoded with the double arrows in the inset, which is a schematicof Fig. 2(c)]. There is a pronounced peak which systematicallyshifts to higher temperatures with increasing compression.(b) Temperature at resistivity peak TCMR (left axis), whichincreases with increasing compression, and %MR (right axis),which decreases as compression is increased. The %MR does notdecrease drastically at first, and there is a region where TCMR canbe enhanced without losing the large %MR.

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M. Kawasaki, and Y. Tokura, Appl. Phys. Lett. 95, 152502(2009).

[16] J. Wang, F. X. Hu, R.W. Li, J. R. Sun, and B.G. Shen,Appl. Phys. Lett. 96, 052501 (2010).

[17] H. Chou, M.-H. Tsai, F. P. Yuan, S. K. Hsu, C. B. Wu,J. Y. Lin, C. I. Tsai, and Y.-H. Tang, Appl. Phys. Lett. 89,082511 (2006).

[18] C. K. Xie, J. I. Budnick, W.A. Hines, B. O. Wells, andJ. C. Woicik, Appl. Phys. Lett. 93, 182507 (2008).

[19] V. Pena, Z. Sefrioui, D. Arias, C. Leon, J. Santamaria, M.Varela, S. J. Pennycook, M. Garcia-Hernandez, and J. L.Martinez, J. Phys. Chem. Solids 67, 472 (2006).

[20] S. de Brion, G. Chouteau, A. Janossy, E. R. Buzin, andW. Prellier, J. Magn. Magn. Mater. 272–276, 450 (2004).

[21] Z. Q. Yang, Y. Q. Zhang, J. Aarts, M.-Y. Wu, and H.W.Zandbergen, Appl. Phys. Lett. 88, 072507 (2006).

[22] Y. Konishi, Z. Fang, M. Izumi, T. Manako, M. Kasai,H. Kuwahara, M. Kawasaki, K. Terakura, and Y. Tokura,J. Phys. Soc. Jpn. 68, 3790 (1999).

[23] R. Yu, S. Dong, C. Sen, G. Alvarez, and E. Dagotto, Phys.Rev. B 77, 214434 (2008).

[24] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1(2001).

[25] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.110.157201 for a de-tailed description of the unstrained Hamiltonian and strainmodelling, the technique applied to solve the problem andfor additional data supporting the conclusions in themanuscript.

[26] H. Yamada, M. Kawasaki, T. Lottermoser, T. Arima, andY. Tokura, Appl. Phys. Lett. 89, 052506 (2006).

[27] C. Adamo et al., Appl. Phys. Lett. 95, 112504(2009).

[28] J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498(1954).

[29] W. Harrison, Electronic Structure and the Propertiesof Solids: The Physics of the Chemical Bond (Dover,New York, 1989).

[30] C. Aruta, G. Ghiringhelli, A. Tebano, N. Boggio, N.Brookes, P. Medaglia, and G. Balestrino, Phys. Rev. B73, 235121 (2006).

[31] A. Tebano et al., Phys. Rev. B 82, 214407 (2010).[32] A. Tebano et al., Phys. Rev. B 74, 245116 (2006).[33] B. R. K. Nanda and S. Satpathy, Phys. Rev. B 78, 054427

(2008).[34] B. R. K. Nanda and S. Satpathy, Phys. Rev. B 81, 174423

(2010).[35] Using t� d�2 and J � d�4, known for La2CuO4 [36],

only causes quantitative changes. It shifts the metal-insulator transition in Fig. 2(a) to 2% tensile strain andthat in Fig. 2(b) to about 2.5% compressive strain. Thesechanges are small enough so that our results are still easilyaccessible to experiments.

[36] S. L. Cooper, G.A. Thomas, A. J. Millis, P. E. Sulewski,J. Orenstein, D. H. Rapkine, S.-W. Cheong, and P. L.Trevor, Phys. Rev. B 42, 10 785 (1990).

[37] S. Yunoki, T. Hotta, and E. Dagotto, Phys. Rev. Lett. 84,3714 (2000).

[38] J. C. Loudon, N.D. Mathur, and P.A. Midgley, Nature(London) 420, 797 (2002).

[39] Y. Tokura, Colossal Magnetoresistive Oxides (Gordon andBreach, Amsterdam, 2000).

[40] K. Pradhan, A. Mukherjee, and P. Majumdar, Phys. Rev.Lett. 99, 147206 (2007).

[41] At low temperatures (T � 0:005t), strain causes phaseseparation between F-M and SCO-I or FC-I, as seen inexperiments [19,20,32] and theory [7]. However, withincreasing temperature, such phase separated states rap-idly evolve either into a global F-M or SCO-I phase and donot affect our finite temperature results.

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