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[Taichi Terashima](https://orcid.org/0000-0001-9239-0621), [Shinya Uji](https://orcid.org/0000-0001-9351-6388), Yuji Matsuda, Takasada Shibauchi, Shigeru Kasahara

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[Apparent nonreciprocal transport in FeSe bulk crystals](https://mdr.nims.go.jp/datasets/409d0077-b4b2-4763-8fbc-b2fc8f77ccfe)

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ver3.1Apparent nonreciprocal transport in FeSe bulk crystalsTaichi Terashima,1, ∗ Shinya Uji,1 Yuji Matsuda,2 Takasada Shibauchi,3 and Shigeru Kasahara4, †1Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science, Tsukuba 305-0003, Japan2Department of Physics, Kyoto University, Kyoto 606-8502, Japan3Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan4Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan(Dated: February 21, 2025)We performed low-frequency ac first- and second-harmonic resistance measurements and dc I−Vmeasurements on bulk FeSe crystals in a temperature range between 1.8 and 150 K and in mag-netic field up to 14 T. We observed considerable second-harmonic resistance, indicative of nonre-ciprocal charge transport, in some samples. By examining correlation between contact resistancesand second-harmonic signals, we concluded that the second-harmonic resistance was not due tothe genuine nonreciprocal transport effect but was caused by joule heating at a current contactthrough the thermoelectric effect. Our conclusion is consistent with a recent preprint (Nagata etal., arXiv:2409.01715), in which the authors reported a zero-field superconducting diode effect indevices fabricated with FeSe flakes and attributed it to the thermoelectric effect.I. INTRODUCTIONNonreciprocal charge transport in quantum materialslacking space-inversion symmetry attracts considerableattention in recent years, not only because of fundamen-tal interest but also because of potential technologicalimportance [1]. The nonreciprocal charge transport hererefers to a state where the electrical resistance differs be-tween a right-going (+I) and a left-going current (−I). Itusually requires that time-reversal symmetry is also bro-ken. This is because, as long as the time-reversal symme-try is present, Kramers degeneracy enforces the relationεσ(k) = ε−σ(−k) on the electronic band energy, which isusually considered to preclude the nonreciprocal chargetransport. Some recent theories however claim that thenonreciprocal transport can occur without time-reversalsymmetry breaking by considering skew scattering [2] orelectron correlation [3], for example.The most impressive manifestation of the nonrecip-rocal charge transport may be a superconducting diodewhere the resistance is zero for one current direction whilefinite for the other. Although the initial realization ofthe superconducting diode with an artificial superlattice[Nb/V/Ta]n required the application of a magnetic fieldto break time-reversal symmetry [4], there are recent re-ports that superconducting diodes may be achieved atzero magnetic field without breaking time-reversal sym-metry [5, 6]. Especially intriguing is the observationof the superconducting diode effect at zero field in de-vices fabricated with an FeSe flake [6]. FeSe is a uniqueparent compound of iron-based superconductors [7]: Itexhibits a tetragonal-to-orthorhombic structural phasetransition at Ts ∼ 90 K but no antiferromagnetic order.It becomes superconducting below about Tc ∼ 9 K. The∗ TERASHIMA.Taichi@nims.go.jp† kasa@okayama-u.ac.jpTABLE I. Contact resistance. The accuracy is estimated tobe about 0.1 Ω.sample (contact number) contact resistance in Ω#1 (1) 0.7 (2) 4.7 (3) 0.7 (4) 0.4#3 (9) 0.3 (10) 0.7 (11) 0.4 (12) 0.6#4 (13) 0.6 (14) 3.0 (15) 0.5 (16) 0.8space group of the low-temperature orthorhombic latticeis Cmme preserving inversion symmetry [8, 9]. Hencethe superconducting diode effect was observed despitethat both space-inversion and time-reversal symmetrieswere retained. The authors argued that the observed ef-fect was actually due to the large thermoelectric effectof FeSe rather than the intrinsic nonreciprocal transporteffect [6, 10]. Namely, because the shapes of the FeSeflakes were asymmetric, roughly triangular, the two cur-rent contacts attached to them differed in size and hencethe joule heating was larger at the narrower contact, re-sulting in the temperature gradient. This caused addi-tional unidirectional current flow via the large thermo-electric effect and broke superconductivity for one cur-rent direction. In this work, we show that similar ap-parent nonreciprocal charge transport is observed evenin bulk FeSe crystals.II. EXPERIMENTSHigh-quality single crystals of FeSe were grown bya chemical vapor transport method [11]. Figure 1shows the experimental setup. We chose approxi-mately rectangular-shaped single crystals, avoiding mul-tiple fused crystals. Their length directions were alongthe tetragonal [100] axis, and their dimensions are givenin the caption. Four 10-µm gold wires were spot-weldedto each sample and the contacts were reinforced by con-ducting silver paint. The contacts were numbered as2123 4 (5) (6)(7) (8)910111213 141516sample #1 (sample #2)sample #3 sample #43 mm B θφθφBFIG. 1. FeSe samples mounted on a rotation platform at φ= 0 and θ = −90◦. Contact numbers are indicated. Thepolar θ and azimuthal φ angles of the applied magnetic fieldwere defined with respect to the platform (right panel). θ = 0corresponds to B ‖ c. Data from sample #2 were not usedbecause the electrical contacts were unstable. The dimensionsof the samples are 1.0 × 0.26 × 0.12 mm3, 2.2 × 0.78 ×0.02 mm3, and 1.0 × 0.40 × 0.02 mm3 for #1, 3, and 4,respectively.shown in the figure. Although four samples were pre-pared, we do not mention data from sample #2 becausethe electrical contacts were unstable. The residual resis-tivity ratio RRR at T = 11 K and the midpoint Tc are(RRR, Tc) = (20, 8.5 K), (26, 9.1 K), and (29, 9.1 K)for sample #1, 3, and 4, respectively. The contact resis-tances (except those for sample #2) were estimated atroom temperature from two-wire measurements throughdifferent contacts and wire resistances estimated fromtheir lengths and resistivities with an accuracy of about0.1 Ω (Table I). Each sample except sample #3 was fixedon a polyether ether ketone (PEEK) or quartz substrateat one end with epoxy or GE (General Electric) varnish toavoid vibration due to Lorentz force acting on ac current.Sample #3 was fixed on an FRP (fiber reinforced plas-tic) substrate with vacuum grease. The substrate mate-rial and glue were different between the samples, but thiswas not intentional, simply because some of the sampleshad previously been used for different purposes and werereused in this experiment. The samples were mountedon a two-axis rotation platform and loaded into a He-4variable temperature insert equipped with a 17-T super-conducting magnet. The polar θ and azimuthal φ anglesof the applied magnetic field were defined with respectto the platform as shown in the figure. We performedlow-frequency (f = 7–19 Hz) ac lock-in measurements ofthe resistance at the fundamental and second-harmonicfrequencies and dc I − V measurements.III. NONRECIPROCAL TRANSPORT ANDNONLINEAR I − V CHARACTERISTICSBefore presenting experimental data, we briefly reviewthe nonreciprocal transport and nonlinear I − V charac-teristics.Rikken et al. provided a basic framework for the non-reciprocal transport under broken time-reversal symme-try on the basis of symmetry arguments [12, 13]. Theyshowed that the nonreciprocal transport appears in chiraland polar structures, and the resistance depends linearlyon both the electrical current and the magnetic field asfollows:R = Ro(1 + βB2 + γI ·B)(1)for chiral structures andR = Ro[1 + βB2 + γI · (P ×B)], (2)for polar structures. Here, Ro is the resistance withoutmagnetic field, βB2 the normal quadratic magnetoresis-tance, γ the magnitude of the nonreciprocal effect, andP the axis of the polar structure. Because γ depends onthe sample size, a normalized parameter γ′ = Aγ, whereA is the cross-sectional area of a sample, sometimes usedfor comparison between different materials [1].The nonreciprocal transport under time-reversal sym-metry is more elusive. Theoretical studies suggest thatit requires electron correlation or skew scattering [2, 3].We now derive the expressions that will be used inanalyzing experimental data. We assume the followingI − V relation:V = R1I +R2I2 +R3I3, (3)where Ri is a function of temperature T and magneticfield B.For dc I−V measurements, we perform measurementsat a positive (+B) and a negative (−B) field and calcu-late the symmetric (wrt B) Rs and the antisymmetriccomponent Ra of the resistance:Rs(B) =12V (B) + V (−B)I= Rs1(B) +Rs2(B)I +Rs3(B)I2, and (4)Ra(B) =12V (B)− V (−B)I= Ra1(B) +Ra2(B)I, (5)where we have neglected the antisymmetric part Ra3 .For ac lock-in measurements, we apply the currentI = Io sinωt where ω = 2πf and obtain the followingrelations:V1 = R1I = R1Io sinωt, and (6)V2 = R2I2 = R2I2o sin2 ωt=12R2I2o[1− sin(2ωt+π2)]. (7)3Because lock-in amplifiers output rms values, their out-puts are expressed as follows:V Lx1 =1√2R1Io = R1Irms, and (8)V Ly2 = − 12√2R2I2o = − 1√2R2I2rms, (9)where VLx(y)i is the in-phase (quadrature) component ofthe i-th harmonic lock-in voltage. The (anti)symmetricpart Rs(a)i (i = 1, 2) is obtained by performing the mea-surements at +B and −B and (anti)symmetrizing Ri.Rs1 is the usual longitudinal resistance. Rs2 and Ra2 canbe due to genuine nonreciprocal transport effects undertime-reversal symmetry and under broken time-reversalsymmetry, respectively. However, in addition to thoseintrinsic effects, we have to consider extrinsic origins (orexperimental artifacts) as well. Ra1 can arise from thecontamination of the Hall voltage. Rs2 and Ra2 can arisefrom the Seebeck and Nernst effect, respectively: a tem-perature gradient due to joule heating at contacts, whichis proportional to I2, gives rise to a voltage proportionalto I2 due to the Seebeck and Nernst effects. Notice thatthe Nernst effect is antisymmetric with respect to B. Rs3can arise from a resistance change ∆R due to sampleheating: a temperature change ∆T ∝ I2 due to sam-ple heating gives rise to ∆R = (dR/dT )∆T and hence avoltage proportional to I3.IV. RESULTSA. sample #4Figure 2 shows results of ac resistance versus temper-ature measurements on sample #4 at B = 0 and ±14T applied along the c axis (θ = 0). The contacts 13and 14 were used as current contacts (see Fig. 1, here-after referred to as contact configuration N, N indicating‘normal’), and the current flows in the ab plane. Thefrequency and magnitude of the current were f = 18.8Hz, and Irms = 3 mA. Note that the contact resistanceof 14 is relatively large, being 3.0 Ω (Table I). For thesecond-harmonic resistance R2, the symmetric part Rs2(= R2) at B = 0 as well as the symmetric Rs2 and theantisymmetric part Ra2 at B = 14 T are shown. Thesymmetric part Rs2 at B = 0 is already finite at 150 K,shows a relatively weak temperature dependence, and itsmagnitude starts decreasing near 80 K. The symmetricpart Rs2 at B = 14 T is indistinguishable from that at B= 0 above Ts, deviates downward below Ts, and startsapproaching zero below ∼9 K. The two curves exhibita very faint kink at Ts. The antisymmetric componentRa2 at B = 14 T appears from slightly above Ts and in-creases with decreasing temperature down to ∼6 K, thenstarts decreasing. No clear anomaly was observed in Ra2at Ts. The parameter γ′ is 2.6×10−9 m2A−1T−1 at 6 K(we re-defined γ as γ = Ra2(B)/[Rs1(B)B]). Although it0.150.100.050.00-0.05R2 (Ω/A)140120100806040200T (K)0.140.120.100.080.060.040.020.00R1 (Ω)B = +/-14 TB = 0R2a(14 T)R2s(14 T)R2s(0 T)Tssample #4B // c, Irms = 3 mAFIG. 2. Ac resistance versus temperature measurements onsample #4 at B = 0 and ±14 T along the c axis. The con-tacts 13 and 14 were used as current contacts (see Fig. 1,contact configuration N). For the second-harmonic resistance,the symmetric Rs2 and the antisymmetric part Ra2 at B = 14T and R2 = Rs2 at B = 0 are shown. The circles show val-ues deduced from dc I − V measurements (ex. Fig. 5) forcomparison (after corrected for the field-angle difference, ifnecessary).is smaller than γ′ ∼10−7 m2A−1T−1 in WTe2 and ZrTe5[14, 15], it is considerably larger than ∼10−12 m2A−1T−1in BiTeBr [16].Figure 3(a) shows the magnetic field dependence ofthe second-harmonic resistance R2 and its decomposi-tion into the symmetric Rs2 and the antisymmetric partRa2 . The magnetic field was applied at an angle θ of 18◦from the c axis (φ = 90◦) for a technical reason. Theantisymmetric part is linear in B. Figure 3(b) shows theantisymmetric part Ra2 measured with different currentstrengths. All the curves coincide within error, indicat-ing that the resistance linearly depends on the current.Thus the bilinear nature (wrt B and I) of the nonrecip-rocal transport is confirmed.Figure 4 shows the field-angle θ dependence of the anti-symmetric partRa2 atB = 14 T and T = 30 K. The curvesfor different φ’s coincide and follow the cos θ dependenceindicated by the thick broken curve within error. This isthe behavior expected from Eq. (2) when the vector Pis in the ab plane.We now turn to dc I − V measurements. For thosemeasurements, the current was swept as follows: 0 →10 mA → 0 → −10 mA → 0 → 10 mA → 0. Figures5(a) and (b) show results of dc I − V measurements on4-0.08-0.06-0.04-0.020.000.020.040.06R2 (Ω/A)-10 -5 0 5 10B (T)R2R2aR2ssample #4B: 18° off the c axisIrms = 3 mA0.060.040.020.00R2 (Ω/A)12840B (T)R2a  Irms =  1 mA  2 mA  3 mA  4 mA(a)(b)FIG. 3. Second-harmonic resistance versus magnetic field atT = 30 K for sample #4 with contact configuration N. Themagnetic field was applied at an angle θ of 18◦ from the c axis(φ= 90◦). (a) The raw data R2 and its decomposition into thesymmetric Rs2 and the antisymmetric part Ra2 are shown. (b)The antisymmetric part Ra2 for four different current values.0.080.060.040.020.00-0.02R2a  (Ω/A)-90 -60 -30 0 30 60 90θ (deg)sample #4B = 14 T, T = 30 K φ = 90 to -75° cosθ FIG. 4. Dependence of Ra2 in sample #4 with contact config-uration N at B = 14 T and T = 30 K on the polar angle θof the applied field measured at different φ’s. The azimuthalangle φ was varied from 90 to -75◦ in steps of 15◦. The thickbroken curve shows a cos θ dependence.sample #4 with contact configuration N at T = 30 K forB = 0 and ±14 T, respectively. The magnetic field direc-tion is the same as that in Fig. 3: i.e., (θ, φ) = (18◦, 90◦).The measured R(I) (= Rs) curve at B = 0 is asymmetricwith respect to I = 0, indicating a finite linear-in-I term[Fig. 5(a)]. A fit to Eq. (4) (broken line) gives a linearcoefficient Rs2 = -0.013 Ω/A. It is in good agreement withac measurements [see circles in Fig. 2 and Rs2 at B = 0 inFig. 3(a)]. The existence of the quadratic term suggestsjoule heating of the sample. Using the fit coefficient Rs3= 5.4 Ω/A2 and dR/dT = 0.81 mΩ/K from Fig. 2, wecan estimate the increase in the average sample temper-ature to be 0.06 K at I = 3 mA. Similarly, we fit Eq.(4) to the symmetric component Rs at B = 14 T [Fig.5(b)] and find that Rs2 = -0.024 Ω/A and that Rs3 = 1.3Ω/A2. The former is again in good agreement with acmeasurements [Fig. 2 and Fig. 3(a)], and the latter givesan estimate of the average sample temperature increaseof 0.08 K at I = 3 mA (dR/dT = 0.15 mΩ/K at B = 14T from Fig. 2). The antisymmetric component Ra at B24.023.823.6R (mΩ)-10 -5 0 5 10I (mA)sample #4T = 30 KB = 048.548.047.547.046.5R, Rs  (mΩ)-10 -5 0 5 10I (mA)-0.8-0.40.00.4Ra  (mΩ)B = 14 TB = -14 TRsRa(a)(b)FIG. 5. Dc I − V measurements on sample #4 with contactconfiguration N at T = 30 K and (a) at B = 0 and (b) ±14 Tplotted in the form of R versus I. The magnetic field directionis the same as that in Fig. 3: (θ, φ) = (18◦, 90◦). In (b), thesymmetric (wrt B) Rs and the antisymmetric component Raare also shown. Second-order polynomial fits to R(I) at B =0 (a) and Rs(I) at B = 14 T (b) and a linear fit to Ra(I) atB = 14 T (b) are shown by grey broken lines. At low currentvalues (i.e., |I| <∼ 2 mA), measured voltages were too smallto accurately determine R, which caused apparently divergingbehavior of R as I → 0.0.50.40.30.20.10.0abs(V) (μV)1086420abs(I) (mA)sample #4T = 8 K, B = 0 I > 0 I < 0FIG. 6. Absolute voltage versus absolute current in sample#4 measured at T = 8 K without magnetic field.= 14 T is nicely linear in I (except at very low currentvalues where voltage is too small to accurately determineR), suggesting the existence of the nonreciprocal trans-port under broken time-reversal symmetry [Fig. 5(b)].By fitting Eq. (5) to the Ra curve, we obtain Ra2 = 0.066Ω/A, which is in good agreement with ac measurements[Fig. 2 and Fig. 3(a)]. Notice that the fit line intersectsthe I = 0 axis at a finite value, i.e., Ra1 6= 0. This in-dicates that the measured voltage was contaminated bythe transverse voltage due to the Hall effect, and hencethat the transverse voltage due to the Nernst effect couldalso be detected with this contact arrangement.Finally, we show that a faint field-free superconductingdiode effect was observed in bulk FeSe. Figure 6 plots theabsolute voltage versus absolute current measured at T =8 K without a magnetic field. For a current range roughly5-0.2-0.10.00.1R2 (Ω/A)65432T (K)0.030.020.01R1 (Ω)B = 14 TB = -14 T4.2 Kλsample #4FIG. 7. First- and second-harmonic resistance (R1 and R2)measured on sample #4 with contact configuration N at B =±14 T as a function of temperature.from 7 to 9 mA, the voltage is almost zero for negativecurrent, while it is finite for positive current. The ob-served effect is much weaker than reported in [6]. Thegradual rise in the voltage for positive current before thesharp increase above I = 9.5 mA suggests the existenceof areas where Tc and the critical current density Jc aresmaller than in other areas. Probably, in those areas, thethermally-induced current can become a significant frac-tion of the critical current, as in the tiny FeSe flakes usedin [6], giving rise to the superconducting diode effect.The data presented above strongly suggested the ex-istence of the nonreciprocal transport coefficient Rs2 un-der time-reversal symmetry as well as Ra2 under brokentime-reversal symmetry that is compatible with Eq. (2).However, we were forced to reconsider the origin of thenonzero Rs2 and Ra2 by the following observation: Figure7 shows the first- and second-harmonic resistance mea-sured on sample #4 with contact configuration N at B= ±14 T as a function of temperature. Initially, thesample was immersed in superfluid 4He at T = 1.8 K,when the first-harmonic resistance R1 was finite, but thesecond-harmonic one R2 was almost zero. As the temper-ature was slowly raised, the second-harmonic resistanceR2 suddenly jumped at T = 2.2 K, i.e., the λ point,and became clearly finite. With further increasing tem-perature, it again jumped and became much larger atT = 4.2 K as the sample became no longer surroundedby liquid helium. The first-harmonic resistance R1 alsoshowed a slight kink at T = 4.2 K. These observationsstrongly suggested that joule heating at a current contact(or maybe both current contacts) and resulting temper-ature gradient in the sample caused the nonzero Rs2 andRa2 through the large thermoelectric effect [10]. Heatexchange between the sample and the environment andhence the temperature gradient inside the sample dependon whether the sample is surrounded by liquid heliumand also whether the liquid is superfluid or not. Su-perfluid helium-4 is expected to provide the best heatexchange, and hence the least temperature gradient isexpected, which is consistent with the vanishingly smallR2 below the λ point (Fig. 7). In the following, we sub-0.060.040.020.00-0.02V 2Ly, V2Lx (mV)1050-5-10B (T)sample #4, Irms = 3 mAf V2Ly V2Lx18.8 Hz56.4 Hz112.8 Hz188 HzFIG. 8. Quadrature and in-phase second-harmonic voltage(V Ly2 and V Lx2 ) measured on sample #4 with contact config-uration N at T = 30 K for different current frequencies.stantiate this conjecture.Figure 8 shows the second-harmonic voltage as a func-tion of field measured on sample #4 at T = 30 K withIrms = 3 mA. Four different frequencies were used, andboth the quadrature V Ly2 and the in-phase componentV Lx2 are shown. Note that the former V Ly2 correspondsto R2. As the frequency increases, the magnitude ofV Ly2 decreases, and V Lx2 develops, indicating increas-ing phase delay. This is expected behavior when thesecond-harmonic voltage is due to the thermoelectric ef-fect: When an ac current i sinωt is applied, the powerdissipation at a current contact with a resistance r is(1/2)i2r[1 − sin(2ωt + π/2)]. Accordingly, the tempera-ture difference between the contact and the environmentoscillates as sin(2ωt+π/2−ϕ), in addition to a constantincrease [17, 18]. The phase delay ϕ increases with thefrequency, resulting in the increase in the phase-shiftedcomponent V Lx2 .All the data presented in Figs. 2–8 were obtained byusing contacts 13 and 14 as current contacts, and 15 and16 as voltage ones (namely, contact configuration N). Wealso measured the resistance using contacts 15 and 16 ascurrent contacts, and 13 and 14 as voltage ones (contactconfiguration R, R indicating ‘reversed’, Fig. 9). Notethat the contact resistances were 0.5 and 0.8 Ω for con-tacts 15 and 16, respectively, which are smaller than 3.0Ω for contact 14. The first-harmonic resistance R1 mea-sured with the two contact configurations coincided withthe maximum difference of only ∼0.2% as expected forproper four-contact resistance measurements. Unexpect-edly, the antisymmetric part Ra2 of the second-harmonicresistance measured with contact configuration R hadthe opposite sign to that measured with configurationN [compare Fig. 9(a) and Fig. 3(a)]. Also, the magni-tude of R2 was smaller. The dc R versus I curve at B =0 is almost symmetric with respect to I = 0, indicatinga tiny linear-in-I component [Fig. 9(b)]. The resistanceincrease with current is smaller than that in Fig. 5(a),indicating smaller joule heating. A second-order polyno-mial fit (grey broken curve) gives Rs2 = -0.0004 Ω/A andthat Rs3 = 1.5 Ω/A2. The former is much smaller than6-0.04-0.020.000.020.04R2 (Ω/A)-10 -5 0 5 10B (T)R2R2aR2ssample #4I contacts 15 & 1623.6023.5523.5023.4523.40R (mΩ)-10 -5 0 5 10I (mA)sample #4I contacts 15 & 16T = 30 KB = 048.047.847.647.447.247.0R, Rs  (mΩ)1050-5-10I (mA)0.40.20.0-0.2Ra  (mΩ)B = 14 TB = -14 TRsRaT = 30 KB = +/-14 T(a)(b)(c)FIG. 9. Sample #4 with contact configuration R: contacts 15and 16 were used as current contacts. T = 30 K. (a) Second-harmonic resistance R2 as a function of magnetic field. Thedecomposition into Rs2 and Ra2 is shown. (b) Resistance versuscurrent at B = 0. (c) Resistance versus current at B = ±14T. The symmetric (wrt B) Rs and the antisymmetric part Raare also shown. The grey broken lines show fitting results.the value of -0.013 Ω/A obtained with contact configu-ration N [Fig. 5(a)]. From the latter, the increase in theaverage sample temperature at I = 3 mA is estimatedto be 0.02 K, three times smaller than that with contactconfiguration N. The R versus I curves at B = ±14 Tindicate the existence of a linear-in-I term, but the slopeof Ra is opposite to that obtained with contact config-uration N [compare Fig. 9(c) and Fig. 5(b)], which isconsistent with the ac measurements [Fig. 9(a)]. A lin-ear fit to Ra (grey broken line) gives Ra2 = -0.041 Ω/A,which is in good agreement with the ac measurements.The symmetric (wrt B) part Rs is almost symmetric withrespect to I = 0, and the resistance increase with currentis small, similar to that at B = 0. From Rs at B = 14 T,the average temperature increase is estimated to be 0.02K at I = 3 mA.The fact that the antisymmetric second-harmonic re-sistance Ra2 changed sign as the current and voltage con-tacts were exchanged is difficult to explain if the second-harmonic resistance was intrinsic to sample bulk. How-ever, it is understandable if the second-harmonic voltageis due to joule heating at a current contact (or contacts):the use of different contacts produces different tempera-ture gradient, resulting in different second-harmonic volt-age generation. The smaller magnitude of the second-harmonic resistance with current contacts 15 and 16 (con-tact configuration R) is also understandable because theircontact resistances were smaller than the contact resis-tance of 14.0.040.030.02R1 (Ω)-10 -5 0 5 10B (T)#3#4-0.008-0.006-0.004-0.0020.0000.0020.004R2 (Ω/A)-10 -5 0 5 10B (T)R2R2aR2s#4x (1 / 10)(a) (b)FIG. 10. (a) First- and (b) second-harmonic resistance as afunction of B at T = 30 K in sample #3. R2 is decomposedinto Rs2 and Ra2 . The magnetic field direction is the same asthat in Fig. 3. For comparison, R1 and R2 in sample #4 withconfiguration N are shown with broken lines. Note that R2for sample #4 is scaled down by 1/10.B. sample #3We now present results for sample #3, which show thatthe second-harmonic resistance can be negligibly smallwhen contacts resistances are small. Figure 10 shows thefirst- and second-harmonic resistances as a function of Bmeasured on sample #3 at T = 30 K. The contacts 9 and10 were used as current contacts (configuration N). Bothcontact resistances in sample #3 were small (0.3 and 0.7Ω) (Table I), in contrast to the fact that the contact re-sistance of contact 14 in sample #4 was as large as 3.0 Ω.Figure 10 also shows R1 and R2 in sample #4 with con-tact configuration N (broken lines), for comparison. Notethat R2 in sample #4 is scaled down by a factor of 1/10.Although R1 is of similar magnitude between samples#3 and 4, R2 is much smaller in sample #3 than in #4.The parameter γ′ at B = 14 T is 3.5×10−11 m2A−1T−1in sample #3, more than one order-of-magnitude smallerthan 8.8×10−10 m2A−1T−1 in sample #4 with contactconfiguration N.Figure 11 shows results of dc I − V measurements onsample #3. Compared to the corresponding curve in Fig.5(a), the R versus I curve at B = 0 in Fig. 11(a) showsmuch less resistance change with increasing current andalso is much less asymmetric with respect to I = 0. Thesecond-order polynomial fit (grey broken line) gives Rs2= -0.0019 Ω/A and Rs3 = 1.1 Ω/A2. The former is con-sistent with the ac measurements [Fig. 10(b)]. Using thelatter value combined with a separately measured dR/dT= 0.53 mΩ/K, the increase in the average sample tem-perature at I = 3 mA is estimated to be 0.02 K, muchsmaller than the value estimated for sample #4 with con-tact configuration N (0.06 K). Figure 11(b) shows the Rversus I curves at B = ±14 T, together with the sym-metric (wrt B) Rs and the antisymmetric part Ra. Asecond-order polynomial fit to Rs at B = 14 T [grey bro-ken line in Fig. 11(b)] yields Rs2 = -0.0044 Ω/A and Rs3= 0.54 Ω/A2. The former is in reasonable agreementwith the ac measurements at B = 14 T [Fig. 10(b)], andthe latter with dR/dT = 0.25 mΩ/K at B = 14 T gives717.1017.0517.0016.9516.9016.8516.80R (mΩ)-10 -5 0 5 10I (mA)sample #3T = 30 KB = 029.129.028.928.8R, Rs  (mΩ)-10 -5 0 5 10I (mA)-0.10-0.050.000.050.10Ra  (mΩ)B = 14 TB = -14 TRsRa(a)(b)FIG. 11. Dc I − V measurements on sample #3 at T = 30 Kand (a) at B = 0 and (b)±14 T plotted in the form of R versusI. The magnetic field direction is the same as that in Fig. 3.In (b), the symmetric (wrt. B) Rs and the antisymmetriccomponent Ra are also shown. Second-order polynomial fitsto R(I) at B = 0 (a) and Rs(I) at B = 14 T (b) and a linearfit to Ra(I) at B = 14 T (b) are shown by grey broken lines.the estimated increase in the average sample tempera-ture at I = 3 mA of 0.02 K, again much smaller than thecorresponding value for sample #4 (0.08 K). The anti-symmetric part Ra is almost flat. A linear fit to Ra at B= 14 T [grey broken line in Fig. 11(b)] gives a tiny valueof Ra2 = 0.00036 Ω/A, which is in reasonable agreementwith the ac measurements at B = 14 T [Fig. 10(b)].To summarize, the ac and dc measurements on sample#3 indicated that joule heating of sample #3 was muchless than that of sample #4 with contact configurationN and that the second-harmonic resistances Rs2 and Ra2were correspondingly smaller.C. sample #1Finally, we show results obtained for sample #1, whichwill confirm that the nonzero Rs2 and Ra2 that we ob-served in bulk FeSe crystals were due to joule heating atcurrent contacts and the thermoelectric effect. Figure 12shows results obtained with two contact configurations:In configuration N, contacts 1 and 2 were used as currentcontacts while 3 and 4 voltage ones. In configuration R,contacts 3 and 4 were used as current contacts while 1and 2 voltage ones. Note that the contact resistance 2was as large as 4.7 Ω, whereas the rest of the contactresistances were small (Table I).Figure 12(a) shows the first-harmonic resistances R1for the two configurations measured at T = 30 K as afunction of B. The resistances R1 for the two configura-tions were the same within experimental accuracy. Fig-ure 12(b) shows the second-harmonic resistances R2 forthe two configurations and the symmetric and antisym-metric parts for configuration N. The second-harmonicresistances R2 were clearly different between the two con-1514131211R1s  (mΩ)-10 -5 0 5 10B (T)sample #1, T = 30 K config. N(I contacts = 1 & 2) config. R(I contacts = 3 & 4)-0.04-0.03-0.02-0.010.000.01R2 (Ω/A)-10 -5 0 5 10B (T)R2R2aR2s config. N config. RR2(a) (b)11.611.411.211.010.810.6R (mΩ)-10 -5 0 5 10I (mA)T = 30 K,  B = 0 config. N config. R(c)FIG. 12. Sample #1 with two different contact configurationsN and R: The current contacts were 1 and 2 in configurationN, while 3 and 4 in R. (a) First- and (b) second-harmonicresistance as a function of B at T = 30 K for the two config-urations. The magnetic field direction is the same as that inFig. 3. In (b), only R2 in configuration N is decomposed intoRs2 and Ra2 . (c) dc R versus I curves at T = 30 K and B = 0for the two configurations.figurations: R2 for contact configuration N was finite,roughly of similar magnitude to that in sample #4 withcontact configuration N [Fig. 3(a)], whereas R2 for con-figuration R is practically zero, below the noise level.Figure 12(c) shows dc R versus I curves for the two con-figurations. The resistance increase with current is muchlarger for configuration N than R, indicating larger jouleheating for N. A second-order polynomial fit (grey bro-ken lines) gives Rs2 = -0.022 Ω/A and Rs3 = 8.0 Ω/A2for configuration N, and Rs2 = -0.00038 Ω/A and Rs3 =0.68 Ω/A2 for R. Rs2 is much larger for configuration Nthan for R, consistent with the ac results [Fig. 12(b)].Rs3 is also much larger for configuration N. Using dR/dT= 0.31 mΩ/K from a separate measurement, the increasein the average sample temperature at I = 3 mA is esti-mated to be 0.23 and 0.02 K for configuration N and R,respectively. Thus, it is confirmed that the observation ofthe second-harmonic resistance is correlated to the jouleheating at a current contact.V. DISCUSSIONWe observed a rough but clear correlation betweenjoule heating of samples and appearance of second-harmonic resistance: A substantial second-harmonic re-sistance was observed in sample #4 with configuration Nand sample #1 with configuration N (Figs. 3 and 12),for which the estimated average sample temperature in-crease at I = 3 mA was 0.06 and 0.23 K, respectively. In8contrast, the second-harmonic resistance was vanishinglysmall in sample #3 with configuration N and sample #1with configuration R (Figs. 10 and 12), for which thetemperature increase was smaller, 0.02 K. It is howeverto be noted that, because the temperature gradient andresulting thermoelectric voltage depends not only on themagnitude of joule heating but also on details of heat ex-change between the sample and environment, no simplerelation between the average temperature increase andthe magnitude of the second-harmonic resistance is ex-pected. This explains the observation that a significantsecond-harmonic resistance was observed in sample #4with configuration R despite the estimated temperatureincrease of 0.02 K (Fig. 9). It is also to be noted thatthe local temperature increase at a current contact is ex-pected to be much larger than the above mentioned av-erage temperature increases. It is this local temperatureincrease that governs the magnitude of the thermoelec-tric voltage.The thermoelectric coefficients in FeSe can be foundin the literature such as refs. [10, 19–21]. The Seebeckcoefficient S is positive at room temperature and changessign near 230 K. Although it shows a broad negative peaknear 100 K, the temperature dependence below 150 Kis weak, S being of the order of −10 µVK−1, and theanomaly associated with the structural transition at Tsis subtle [19]. The Nernst coefficient ν reported in [20]gradually increases from∼150 K peaks near 80 K, ν being∼0.9 µVK−1T−1. The anomaly at Ts is unclear. Thesereports are consistent with the present observations thatthe second-harmonic resistances Rs2 and Ra2 started toappear far above Ts and that almost no anomaly wasobserved at Ts (Fig. 2). The magnitudes of the thermo-electric coefficients show substantial sample dependence.One to two orders-of-magnitude larger values of S andν than those in [19, 20] were reported in [10]. This islikely related with the fact that the sample of [10] showsmuch larger magnetoresistance than that of [20]. Themagnetoresistance of the present samples is also muchlarger than that of [20], and hence the magnitudes oftheir thermoelectric coefficients are likely close to thosein [10].Let us make an order-of-magnitude estimate of ther-moelectric coefficients necessary to explain the observedRs2 and Ra2 . Let us assume a typical size of the second-harmonic resistance to be 0.02 Ω/A. This corresponds toa voltage of ∼0.2 µV at I = 3 mA. Let us assume thetemperature difference between the voltage contacts of 1K. Then, a Seebeck coefficient of 0.2 µVK−1 suffices. Forthe Nernst effect, the estimation is more difficult becausethe direction of the temperature gradient and the geome-try of contacts are involved. For simplicity, we consider atemperature difference of 1 K in some direction in the abplane. A Nernst coefficient of 2 µVK−1T−1 gives a volt-age of 20 µV in the perpendicular direction at B = 10 Tapplied along the c axis. Only 1% of this voltage is neces-sary to produce Ra2 = 0.02 Ω/A. Considering large ther-moelectric coefficients reported in [10], the above valuesof S and ν are reasonable assumptions. Thus, our claimthat the observed second-harmonic resistance is actuallydue to the thermoelectric effect is justified.The unexpectedly large influence of the thermoelectriceffect on the second-harmonic resistance is due to thelarge thermoelectric coefficients in FeSe [10]. Our resultsclearly support ref. [6], in which the authors observed thezero-field superconducting diode effect in FeSe devicesand ascribed it to the thermoelectric effect. The implica-tions of the present results are twofold: Firstly, the ther-moelectric effect can be used to design superconductingdiodes. By using superconductors with large thermo-electric effect like FeSe and intentionally making tem-perature gradient in a device, effective superconductingdiodes may be achieved as in [6]. Secondly, caution hasto be taken when using the second-harmonic resistanceto make a diagnosis of broken space-inversion symmetry.The appearance of the antisymmetric second-order har-monic resistance is sometimes taken as evidence for bro-ken space-inversion symmetry. However, such diagnosishas to be made with extreme caution to exclude possi-ble contamination by the thermoelectric effect. 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