Heat capacity of type I superconductivity in the Dirac semimetal PdTe_2
HHeat capacity of type I superconductivity in the Dirac semimetal PdTe M. V. Salis, ∗ Y. K. Huang, and A. de Visser † Van der Waals - Zeeman Institute, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The Netherlands (Dated: February 18, 2021)Type I superconductivity has recently been reported for the Dirac semimetal PdTe ( T c ≈ . I. INTRODUCTION
Recently, layered transition metal dicalchogenideshave sparked great interest by virtue of their exotic elec-tronic properties, especially the possibility of realizingnovel quantum states stemming from the topologicalnon-trivial band structure as uncovered by densityfunctional theory . A generic coexistence of type I andtype II 3-dimensional Dirac cones has been proposed tobe at play in these materials . PdTe is interesting inparticular because of the appearance of superconduc-tivity at T c ≈ . , as well as its classification as atype II Dirac semimetal. The latter is extracted from acombination of ab initio electronic structure calculationsand angle resolved photoemission spectroscopy . ADirac cone with a tilt parameter k > .The Dirac point then forms the touching point of theelectron and hole pockets, possibly resulting in a nearlyflat band adjacent to the Fermi level. This prompts thequestion whether superconductivity is bolstered by thepresence of the nearly flat band .The superconducting properties of PdTe have beenextensively investigated. Transport and magneticmeasurements carried out on single crystals of PdTe revealed the existence of bulk type I superconductivity,an uncommon feature for a binary compound . Dcmagnetization data showed the appearance of theintermediate state, the hallmark of type I superconduc-tivity in an applied magnetic field. This was furthercorroborated by the differential paramagnetic effectobserved in ac magnetization measurements. A bulkcritical field B c = 13 . B surfc = 34 . . This led the authors ofRef. 6 to suggest surface superconductivity to have atopological nature. Moreover, an even higher criticalfield of 0.3 T was observed in resistance data. Thetheoretical possibility of type I superconductivity in PdTe was analyzed within a microscopic pairing theoryexploring the tilt parameter k of the Dirac cone . Therealization of type I superconductivity was establishedfor k = 2.Evidence for the weak-coupling Bardeen-Cooper-Schrieffer (BCS) nature of superconductivity in PdTe was obtained through measurements of the specificheat , penetration depth , scanning tunnelingmicroscopy and spectroscopy (STM and STS) andtunneling spectroscopy on side junctions . Surprisingly,distinct and fairly large critical fields were observedin STM/STS measurements , and their spatialdistribution on the surface was attributed to a mixtureof type I and type II superconductivity. This providedthe motivation for further experimental work to unravelthe nature of the superconducting phase. Additional ev-idence for type I superconductivity was inferred from thelocal electronic behavior necessary to properly analyzethe magnetic penetration depth data . Evidence on themicroscopic scale was obtained from transverse muonspin relaxation measurements in an applied magneticfield, that unambiguously demonstrated the presenceof the intermediate state . Similarly, scanning squidmagnetometry provided evidence for type I supercon-ductivity on the macroscopic scale . Finally it hasbeen established that type I superconductivity is robustunder pressure .Although the specific heat of PdTe was reportedbefore, the focus was on elucidating the symmetry ofthe gap structure . Heat capacity techniques can alsobe utilized to ascertain whether superconductors aretype I or type II. Unlike type II superconductors, typeI superconductors, when subjected to a magnetic field,will undergo a first order phase transition. This can beverified by measuring the heat capacity in a magneticfield, which involves the latent heat associated with thetransition. In this case the latent heat appears as anextra contribution to the jump in the specific heat at T c , such that the jump size exceeds the value in zeromagnetic field. Furthermore, for type I superconducting a r X i v : . [ c ond - m a t . s up r- c on ] F e b samples that have a shape resulting in a nonzerodemagnetization factor, the intermediate state emerges.The intermediate state contribution broadens the su-perconducting transition towards lower temperaturesdue to the gradual transformation of normal domains tosuperconducting domains. Hitherto, no thermodynamicevidence in favor of type I or type II superconductivityhas been reported. This warrants a second specific heatstudy focusing on these aspects.In this paper heat capacity measurements of PdTe inzero and applied magnetic fields are reported. The datain field show the presence of latent heat associated with afirst order transition and thus type I superconductivity.The temperature variation of the critical field, B c ( T ),follows the expected quadratic temperature variation upto 9.5 mT. The data at higher applied fields reveal thepresence of a second, minority superconducting phase inthe PdTe crystal. II. EXPERIMENTAL
PdTe crystallizes in the trigonal CdI structure (spacegroup P¯3m1) . The single crystal investigated in thisstudy is taken from a batch grown with the modifiedBridgman technique as reported in Ref. 6. The proper1:2 stoichiometry within the 0.5 % experimental reso-lution was inferred from scanning electron microscopy(SEM) with energy dispersive X-ray (EDX) spectroscopy.Magnetization measurements showed a bulk T c of 1.64 Kand B c = 13 . . The rectangular shaped single crystallinesample used in this study has sizes of 3 · · . along the a, a ∗ and c axes, respectively and a mass of39.66(2) mg.Heat capacity measurements were carried out in an Ox-ford Instruments Heliox He refrigerator down to 0.3 Kby use of the dual slope thermal relaxation calorimetrytechnique . In this technique the sample is kept at astable temperature T . Heat is then applied to heat thesample from T by ∆ T /T to T , which is recorded. Thedata recorded represents the heating curve. Subsequentlythe heat is removed, and the sample cools back to a sta-ble temperature T , which is recorded as well. This rep-resents the cooling curve. The increase in temperature∆ T /T ≈ . , sufficiently large to probe theintermediate state. All measurements in a magnetic fieldhave been carried out with the sample first cooled down in field from the normal state to the base temperature.The data points are collected by step-wise heating to thedesired temperature T . III. RESULTS
The as-measured total specific heat, consisting of theelectronic and phononic contributions, is reported in theSupplementary Material file . At low temperatures, thespecific heat of a simple metal in the normal state isgiven by C = γT + βT , where γ is the Sommerfeldcoefficient and β is the phononic coefficient. We havedetermined γ and β by the usual procedure and ob-tained values of 4.4 mJ/molK and 0.70 mJ/molK , re-spectively. This γ value compares reasonably well to the6.0 mJ/molK derived in previous work . The value β = 0.70 mJ/molK compares well to 0.66 mJ/molK of the previous heat capacity study . The Debye tem-perature Θ D can be calculated using Θ D = (cid:18) S π R β (cid:19) ,where S is the number of atoms per formula unit and R is the gas constant. We obtain Θ D = 202 K, whichagrees well with the previously reported value of 207 K and the calculated value of 211 K . After subtractingthe phonon contribution the electronic specific heat, C el ,results.The overal temperature variation of the electronicspecific heat is presented in figure 1 in reduced temper- (cid:1) C / (cid:1) T c = 1 . 4 2 Cel (mJ/molK)
T / T c d a t a p o i n t s M u h l s c h l e g e l + g r e s T g T P d T e d a t a B C S + g r e s T Cel (mJ/molK)
T / T c (cid:1) / k B T c = 1 . 7 6 FIG. 1. Reduced temperature (
T /T c ) dependence of the elec-tronic specific heat C el of PdTe in zero field. Red dots andline: experimental data; green solid line: BCS temperaturedependence according to M¨uhlschlegel with a small residualterm γ res T added; black dashed line: extrapolation to zeroof the linear electronic specific heat in the normal state. Thejump in the specific heat quantified with the BCS relation∆ C/γT c is equal to 1.42. Inset: Specific heat at low tempera-tures compared with the low temperature BCS behavior witha small residual term γ res T (see text). ature ( T /T c ) with T c = 1.54 K. Here T c is taken as thetemperature where C el has its maximum value.The jump at T c quantified with the BCS relation∆ C/γT c , where ∆ C is the jump in the specific heat,equals 1.42, which is close to the textbook value of1.43, confirming the weak-coupling BCS nature ofsuperconductivity in PdTe . The full range temperaturedependence of a weak coupling BCS superconductor astabulated by M¨uhlschlegel is given by the green line infigure 1. In order to better match the experimental data,a small residual linear term with γ res = 0 .
10 mJ/molK is added. This accounts for 2.2 % of the sample thatapparently remains in the normal state. At low temper-atures the superconducting specific heat is described bythe relation C = C n . T − . e − . /T (Ref. 29), where C n is the specific heat of the electronic normal stateat T = T c . Here the BCS gap relation ∆ k B T c = 1 . crystal.Figure 2 shows the temperature dependence of theelectronic specific heat, C el ( T ), in zero field and mag-netic fields ranging up to 18.5 mT. The same dataplotted as C el /T versus T are presented in Figure S2of the Supplemental Material file . An increase in theheight of the transition peak for fields up to 4.5 mTcompared to the peak at 0 mT is observed. This impliesextra energy is necessary to complete the transformationinto the normal phase in small fields. At higher fields,especially at 6.5 mT and 8.5 mT, a broadening of thetransition temperature towards lower temperatures isvisible. In the experimental configuration used, thecrystal has a demagnetization factor of 0.14 causingthe intermediate state to form. It is likely that thesuperconducting transition is considerably broadened athigher fields due to the intermediate state. The region inthe B − T phase diagram occupied by the intermediatephase is shown in figure 3. At even higher magneticfields, up to 16.5 mT, the transition broadens furtherand is no longer observed above this field. Remark-ably, for B ≥ . C abruptly reduces.In figure 3 the B − T phase diagram is mapped out bytracing the onset temperatures of superconductivity inapplied magnetic fields, indicated by the arrows in figure2. In previous research the phase diagram for bulk su-perconductivity probed by different techniques was foundto follow the textbook relation B c ( T ) = B c (0) (cid:2) − ( T /T c ) (cid:3) , (1)where B c (0) = 13 . T c = 1 .
64 K. The newdata are in good agreement with the previous resultwith T c = 1 .
60 K (solid blue line in figure 3). Forfields B ≥ . T c than expected, which presents the onset Cel (mJ/molK)
T e m p e r a t u r e ( K ) B a = 0 m T 0 . 5 m T 1 . 5 m T 2 . 5 m T 4 . 5 m T 6 . 5 m T 8 . 5 m T 1 0 . 5 m T 1 2 . 5 m T 1 4 . 5 m T 1 6 . 5 m T 1 8 . 5 m T P d T e Cel (mJ/molK) T ( K ) FIG. 2. Temperature dependence of the electronic specificheat C el of PdTe in zero field and in magnetic fields up to18.5 mT as indicated. An increase in the size of the specificheat jump at T c is observed in field. T c ’s are indicated byarrows.For B ≥ . ≤ B ≤ . temperature of the transition with reduced specific heatstep (see the inset in figure 2). We attribute the reduced∆ C to a second, minority superconducting phase (seeDiscussion). The Meissner-to-intermediate phase line isgiven by the thin blue line. Its position is calculatedby assuming that a type I superconductor is in theintermediate state for B c (1 − N ) < B app < B c where B app is the applied magnetic field and N = 0 .
14 is thedemagnetization factor.Figure 4 depicts the zero field cooled (ZFC) and fieldcooled (FC) specific heat data as a function of tempera-ture at 4.5 mT and 8.5 mT. All measurements here havebeen carried out by cooling down to base temperatureeither in field or without field. At the base temperaturethe field is applied (ZFC) or kept constant (FC). Nextthe sample was heated to different temperatures whilekeeping the field constant. The measurements carriedout at 4.5 mT are given in black and red symbols, re-spectively, and no difference between the FC and ZFCdata is found. This shows the phase transformation isthe same FC and ZFC at this particular field strength.In the case of 8.5 mT, however, an odd feature is observedin the ZFC data in the temperature range 0.75-0.87 K.The heating curve of the first thermal relaxation mea-surement results in a much larger specific heat (see theSupplemental Material file ). This is shown by the bluesymbols. All subsequent data points (cyan symbols), in-cluding those derived from the cooling curve of the firstrelaxation measurement, fall on top of the FC data set(green symbols). This effect is only observed in the tem- Magnetic field (mT)
T e m p e r a t u r e ( K ) N = 0 . 1 4 M e i s s n e rp h a s e n o r m a lp h a s ei n t e r m e d i a t e p h a s e
P d T e I M Pa t 8 . 5 m T
FIG. 3. The B − T phase diagram of PdTe obtainedby plotting the onset superconducting transition temper-ature for different magnetic fields. Blue symbols: datapoints; thick solid blue line: B c ( T ) = B c (0) (cid:2) − ( T /T c ) (cid:3) with B c (0) = 13 . T c = 1 .
60 K; thin solid blueline: Meissner-to-intermediate phase (IMP) transition line B IMP ( T ) = B c ( T )(1 − N ) with N = 0 .
14; green symbols: T c of a second, minority phase; dashed green line: guide tothe eye; red solid bar: temperature range of the intermediatestate at 8.5 mT (see text). perature range of the intermediate phase. IV. DISCUSSION
The overall temperature variation of the superconduct-ing contribution to the specific heat is in very good agree-ment with the tabulated M¨uhlschlegel values. At thesame time the low temperature data
T /T c < . ∆ k B T c = 1 . . The jump size ∆ C/γT c = 1.42 is con-form with the weak-coupling BCS expectation of 1.43.These results compare well to previous work where aweak- to moderately coupled superconducting state and aconventional isotropic gap are reported . Comparedto the previous specific heat study , the γ value of 4.4mJ/molK is nearly 20 % lower. This is possibly relatedto a different carrier density n considering the semimetal- Cel (mJ/molK)
T e m p e r a t u r e ( K )
P d T e I M P
FIG. 4. Temperature dependence of the electronic specificheat C el of PdTe measured FC and ZFC at B = 4.5 and 8.5mT. The square symbols depict FC data, whereas the roundsymbols depict ZFC data. No difference between FC dataand ZFC data is observed at B = 4.5 mT. In the ZFC datataken at 8.5 mT a large specific heat is observed, but onlywhen derived from the heating part of the first relaxationcurve (see text). No difference is observed with respect to FCdata for subsequent measurements. The red bar depicts thetemperature interval where the intermediate phase in 8.5 mTis expected according to figure 3. lic properties of PdTe . Differences in carrier density arealso inferred from penetration depth measurements. In aprevious study using single crystals from the same batch,values for the penetration depth λ (0) were obtained thatranged from 377 nm to 482 nm . There λ (0) was di-rectly related to n in an extended London model usedto analyze the data where the assumption n s = n wasmade, with n s the superfluid density. The difference inthe value for ∆ C/γT c between the previous heat capac-ity study (1.52 ) and this work (1.42) is understood asa difference in coupling strength. This is in line withthe results of penetration depth studies where simi-lar differences in ∆ /k B T c , ranging from 1.77 to 1.83, werefound. The γ value can be related to the critical field :∆ C = 4 B c (0) µ T c = 1 . γT c , (2)where ∆ C and γ are per unit volume. With the values γ = 4 .
46 mJ/molK , T c = 1 .
62 K, ∆
C/γT c = 1 . . · − m /mol, we cal-culate B c (0) = 10 . B c (0) = 13 . , where γ = 6 .
01 mJ/molK, T c = 1 . C/γT c = 1.52 were reported, eq. 2gives B c (0) = 14 . B c (0) = 19 . .The temperature dependence of the electronic specificheat C el in magnetic fields shown in figure 2 is consistentwith that of a first order phase transition. The latentheat appearing with a first order phase transition isvisible as the increased peak height in the specific heatin small fields relative to zero field. Consequently, weconclude that the type I nature of PdTe is successfullyprobed via the presence of latent heat near the super-conducting transition in field. Further evidence for theexistence of type I superconductivity can be obtained byprobing the intermediate state. In this study the sampleand field geometry results in a demagnetization factor N = 0 .
14. From figure 3 it is clear that for B ≥ . B ≥ . B c ( T ) data points traced in figure 3 closely followthe results probed by dc and ac magnetization measure-ments in previous work up to B = 8 . B c (0) = 13.6 mT and T c = 1.60 K. However, above 8.5 mT, where ∆ C is sud-denly reduced, superconductivity is observed above theexpected B c ( T )-curve. It is of importance to investigatewhether this can be caused by the intermediate phase.The temperature dependence of the normal state volumefraction F N in the intermediate state in fixed fields for B c (1 − N ) < B app < B c is given by F n = 1 − t t c − − NN + 1 N , (3)where t is the reduced temperature T /T c , t c the reducedcritical temperature T c ( B app ) /T c (0) and N = 0 .
14 is thedemagnetization factor . Eq. 3 shows F n has a smoothtemperature variation and cannot suddenly collapse.The reduced critical temperature t c in eq. 3 can berewritten using eq. 1: t c = (cid:113) − B c ( T ) B c (0) . From this, nosudden decrease in F n is possible as well. We thereforeexclude the intermediate phase as a possible cause for theelevated T c and reduced specific heat step. A more likelyexplanation is that superconductivity survives in a smallvolume fraction ( ∼
10 %) of the crystal with a slightlydifferent PdTe x stoichiometry . We remark a similaradditional phase line was obtained in a previous study by analyzing the screening signal in the ac-susceptibilityfor small driving fields. Since the screening signalpersisted above B c it was attributed to superconductiv-ity of the surface sheath with a critical field B sc ≈
35 mT.In the heating curves of the first relaxation measure-ments of the ZFC data detailed in figure 4 an increase ofthe specific heat at 8.5 mT appears, whereas no such in-crease was found at 4.5 mT. Given the temperature rangein which it appears, 0.75-0.87 K, it can be attributed tothe intermediate phase. We remark this range is a littlelower than the expected range 0.83-0.98 K (red bar) cal-culated from the phase diagram in figure 3. The spatialarrangement and size of the normal and superconductingdomains will depend on the field and temperature historybecause of pinning effects. This may cause hystereticbehavior. Such a history dependence was also reportedby probing the intermediate phase in PdTe by scanningsquid magnetometry . The absence of irreversibility inthe relaxation curves at 4.5 mT shows the phenomenonis much weaker at this field. Moreover, the increase inspecific heat in the first measurement point is more dif-ficult to observe at 4.5 mT due to the smaller tempera-ture range in which the intermediate phase is present. Acloser examination of the irreversibility in ZFC calorime-try should be possible with ac calorimetry, as long as thechange in the specific heat does not exceed the amplitudeof the ac heat pulse. V. CONCLUSION
The temperature dependence of the specific heat ofPdTe in zero field and magnetic fields was measured inorder to produce thermodynamic evidence of the typeI nature of superconductivity. From the zero field dataa weak-coupling BCS superconducting state is inferredconform with the literature. The data in small magneticfields show the presence of latent heat at the supercon-ducting transition, where the step in the specific heat∆ C exceeds the zero field value. The intermediate statewas probed by (i) a significant broadening of the transi-tion onto lower temperatures for B > . B c = 13 . T c = 1 .
60 K for B ≤ . B ≥ . ∼
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1. Measured total specific heat in zero field. C (mJ/molK) T e m p e r a t u r e ( K )
P d T e z e r o f i e l d C n = g T + b T C/T (mJ/molK2) T ( K ) C n / T = g + b T g = 4 . 4 6 ( 2 ) m J / m o l K b = 0 . 7 1 ( 3 ) m J / m o l K FIG. S1. As-measured specific heat of PdTe in zero field in a plot of C versus T . The dashed line shows the normal statecontribution C n = γT + βT , with the fitted values of γ and β listed in the graph. Inset: Same data in a plot of C/T versus T .
2. Electronic specific heat in a plot of C el /T versus T in zero and in applied magneticfields. C el/ T (mJ/molK2) T e m p e r a t u r e ( K )
0 m T 0 . 5 m T 1 . 5 m T 2 . 5 m T 4 . 5 m T 6 . 5 m T 8 . 5 m T 1 0 . 5 m T 1 2 . 5 m T 1 4 . 5 m T 1 6 . 5 m T 1 8 . 5 m T
P d T e a p p l i e d f i e l d FIG. S2. Temperature variation of C el /T of PdTe in zero field and in = magnetic fields up to 18.5 mT as indicated. Theincrease of the step size of the specific heat at the superconducting transition temperature measured in magnetic field is clearlyobserved. This is due to the first order nature of the transition which involves latent heat. For B ≥ .
3. Relaxation curves at T = 0 . K and B = 8 . mT. Temperature (K)