Large anomalous Nernst and spin Nernst effects in noncollinear antiferromagnets Mn_3X (X = Sn, Ge, Ga)
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Large anomalous Nernst and spin Nernst effects in noncollinear antiferromagnetsMn X ( X = Sn, Ge, Ga) Guang-Yu Guo
1, 2, ∗ and Tzu-Cheng Wang Department of Physics and Center for Theoretical Physics,National Taiwan University, Taipei 10617, Taiwan Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan Department of Physics and Center for Theoretical Sciences,National Taiwan University, Taipei 10617, Taiwan (Dated: December 12, 2017)Noncollinear antiferromagnets have recently been attracting considerable interest partly due torecent surprising discoveries of the anomalous Hall effect (AHE) in them and partly because theyhave promising applications in antiferromagnetic spintronics. Here we study the anomalous Nernsteffect (ANE), a phenomenon having the same origin as the AHE, and also the spin Nernst effect(SNE) as well as AHE and the spin Hall effect (SHE) in noncollinear antiferromagnetic Mn X ( X =Sn, Ge, Ga) within the Berry phase formalism based on ab initio relativistic band structure calcula-tions. For comparison, we also calculate the anomalous Nernst conductivity (ANC) and anomalousHall conductivity (AHC) of ferromagnetic iron as well as the spin Nernst conductivity (SNC) ofplatinum metal. Remarkably, the calculated ANC at room temperature (300 K) for all three alloysis huge, being 10 ∼
40 times larger than that of iron. Moreover, the calculated SNC for Mn Snand Mn Ga is also larger, being about five times larger than that of platinum. This suggests thatthese antiferromagnets would be useful materials for thermoelectronic devices and spin caloritronicdevices. The calculated ANC of Mn Sn and iron are in reasonably good agreement with the veryrecent experiments. The calculated SNC of platinum also agrees with the very recent experimentsin both sign and magnitude. The calculated thermoelectric and thermomagnetic properties are an-alyzed in terms of the band structures as well as the energy-dependent AHC, ANC, SNC and spinHall conductivity via the Mott relations.
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I. INTRODUCTION
In recent two decades, spin transport electronics (spin-tronics) has attracted enormous interest because of itspromising applications in information storage and pro-cessing and other electronic technologies . Spin cur-rent generation, detection and manipulation are threekey issues in the spintronics. In this context, spin-relatedtransport phenomena in solids especially in those materi-als that can provide highly spin-polarized charge currentand large pure spin current, have been intensively investi-gated recently. The anomalous Hall effect (AHE), discov-ered in 1881 by Hall , and the spin Hall effect (SHE), pre-dicted in 1971 by Dyakonov and Perel , are two principalspin-related transports and thus have received renewedinterests. Intuitively, spin-up and spin-down electronsmoving through the relativistic band structure of a solidexperience opposite transverse velocities caused by an ap-plied electric field. In a ferromagnet where an unbalanceof spin-up and spin-down electrons exists, these oppositecurrents result in a spin-polarized transverse charge cur-rent and hence the (intrinsic) AHE. Therefore, the AHEis usually assumed to be proportional to the magneti-zation of the magnetic material. In a nonmagnetic ma-terial where spin-up and spin-down electrons are equalin numbers, this process gives rise to a pure transversespin current, and this is known as the (intrinsic) SHE.The SHE is particularly important for spintronics be- cause it enables us to generate, detect and control spincurrent without magnetic field or magnetic materials.
Furthermore, the pure spin current is dissipationless andis thus especially useful for the development of low power-consumption nanoscale spintronic devices .Interestingly, Chen et al. recently showed that largeAHE could occur in noncollinear antiferromagnets with-out net magnetization such as cubic Mn Ir. This surpris-ing result arises from the fact that in a three-sublatticekagome lattice with a noncollinear triangle antiferromag-netic structure, not only the time-reversal symmetry ( T )is broken but also there is no spatial symmetry opera-tion ( S ) which, in conjunction with T , is a good symme-try that preserves the Kramers theorem. Subsequently,large AHE was observed in hexagonal noncollinear an-tiferromagnets Mn Sn and Mn Ge . In the mean-time, large SHE was predicted in noncolinear antiferro-magnets Mn X ( X = Sn, Ge, Ir) and was also ob-served in Mn Ir . All these fascinating findings suggestthat these noncollinear antiferromagnets may find excit-ing applications in spintronics, an emergent field calledantiferromagnetic spintronics . Antiferromagnetic spin-tronics has been attracting increasing attention in recentyears because antiferromagnetic materials have severaladvantages over ferromagnetic materials. In particular,antiferromagnetic elements would not magnetically af-fact their neighbors and are insensitive to stray magneticfields. Moreover, antiferromagnets have faster spin dy-namics than ferromagnets, and this would lead to ultra-fast data processing.In a ferromagnet, the charge Hall current could alsoarise when a temperature gradient ( ∇ T ) instead of anelectric field, is applied. This phenomenon, due to thesimultaneous presence of the spin-orbit coupling (SOC)and net magnetization in the ferromagnet, is refered toas the anomalous Nernst effect (ANE) . Similarly, atemperature gradient could also generate the spin Hallcurrent in a nonmagnetic material, and this is known asthe spin Nernst effect (SNE) . Clearly, materials thatexhibit large ANE and SNE would have useful applica-tions for spin thermoelectronic devices driven by heat,a new field known as spin caloritronics . This offersexciting prospects of ’green’ spintronics powered by, e.g.,waste ohmic heat. Since the ANE and SNE, respectively,have the same physical origins as the AHE and SHE,one could expect significant ANE and SNE in the above-mentioned noncollinear antiferromagents as well. Inother words, noncollinear antiferromagnets Mn X ( X =Sn, Ge, Ga) could also be useful materials for developingantiferromagnetic spin caloritronics. Nevertheless, no in-vestigation of the SNE in noncollinear antiferromagnetshas been reported and only two reports on the measure-ment of the ANE in Mn Sn appeared very recently.
In this paper, therefore, we perform an ab initio studyon the ANE and SNE in hexagonal noncollinear antifer-romagnets Mn Ga, Mn Ge and Mn Sn (Fig. 1), basedon the density functional theory (DFT) with the general-ized gradient approximation (GGA) . For comparison,we also study the ANE in bcc Fe, a ferromagnetic transi-tion metal having large AHE , and the SNE in fcc Pt, aheavy nonmagnetic transition metal exhibiting giganticSHE . Indeed, we find that the anomalous Nernst con-ductivity at room temperature of all three alloys is large,being more than ten times larger than that of bcc Fe. Thespin Nernst conductivity of Mn Ga and Mn Sn is largerthan that of fcc Pt. The rest of this paper is organizedas follows. In the next section, we briefly describe theBerry phase formalism for calculating the intrinsic Halland Nernst conductivities as well as the computationaldetails. Section III consists of three subsections. We firstpresent the calculated total energy and magnetic prop-erties of two low-energy noncollinear antiferromagneticstructures [Figs. 1(c) and 1(d)] of Mn X and also com-pared our results with available previous experimentaland theoretical reports in Subsec. III A. We then reportthe calculated anomalous Nernst conductivity as well asanomalous Hall conductivity for these magnetic struc-tures in Subsec. III B. We finally present the calculatedspin Nernst conductivity and also spin Hall conductivityin Subsec. III C. Finally, the conclusions drawn from thiswork are summarized in Sec. IV. FIG. 1: (Color online) (a) Layered hexagonal ( D h ) structureof Mn X ( X = Sn, Ge, Ga) with (b) the associated hexagonalBrillouin zone (BZ). (c) Type A and (d) type B antiferromag-netic configurations considered in this paper. Both magneticstructures have an orthorhombic symmetry and thus theirirreducible BZ wedge (IBZW) [i.e., the trapzian prism indi-cated by the blue dashed lines in (b)] is three times largerthan the hexagonal IBZW [i.e., the triangle prism indicatedby the blue dashed lines in (b)]. The vertical [horizontal]black dashed line in (c) [(d)] denotes the mirror plane. II. THEORY AND COMPUTATIONALMETHOD
Here we consider ordered Mn Ga, Mn Ge and Mn Snalloys in the layered hexagonal DO ( P /mmc or D h )structure [see Fig. 1(a)] and use the experimental latticeconstants of a = 5 .
36 ˚A and c = 4 .
33 ˚A , a = 5 .
34 ˚Aand c = 4 .
31 ˚A , and a = 5 .
66 ˚A and c = 4 .
53 ˚A ,respectively. The primitive unit cell contains two lay-ers of Mn triangles stacked along the c -axis, and in eachlayer the three Mn atoms form a kagome lattice with the X atom located at the center of each hexagon (Fig. 1).The total energy and electronic structure are calculatedbased on the DFT with the GGA in the form of Perdew-Burke-Ernzerhof . The accurate projector-augmentedwave (PAW) method , as implemented in the Vienna ab initio simulation package ( vasp ) , is used. Thefully relativistic PAW potentials are adopted in orderto include the SOC. The valence configurations of Mn,Sn, Ge and Ga atoms taken into account in the calcula-tions are 3 d s , 4 d s p , 3 d s p and 3 d s p ,respectively. A large plane-wave cutoff energy of 350eV is used throughout. In the self-consistent electronicstructure calculations, a fine Γ-centered k -point mesh of20 × ×
20 [i.e., 2112 k -points over the orthorhombic ir-reducible Brillouin zone wedge (IBZW) (see Fig. 1)] isadopted for the Brillouin zone (BZ) integration using thetetrahedron method .The anomalous Hall conductivity (AHC) and anoma-lous Nernst conductivity (ANC) are calculated based onthe elegant Berry-phase formalism . Within this Berry-phase formalism, the AHC ( σ Aij = J ci /E j ) is simply givenas a BZ integration of the Berry curvature for all theoccupied bands, σ Aij = − e ~ X n Z BZ d k (2 π ) f k n Ω nij ( k ) , Ω nij ( k ) = − X n ′ = n h k n | v i | k n ′ ih k n ′ | v j | k n i ]( ǫ k n − ǫ k n ′ ) , (1)where f k n and Ω nij ( k ) are the Fermi distribution func-tion and the Berry curvature for the n th band at k , re-spectively. i and j ∈ ( x, y, z ), and i = j . J ci is the i -component of the charge current density J c and E j isthe j -component of the electric field E . Moreover, theANC ( α Aij = − J ci / ∇ j T ) can be written as α Aij = 1
T e ~ X n Z BZ d k (2 π ) Ω nij ( k ) × [( ǫ k n − µ ) f k n + k B T ln(1 + e − β ( ǫ k n − µ ) )] , (2)where µ is the chemical potential and k B is the Boltz-mann constant. The Berry curvature Ω nij ( k ) can be considered as apseudovector , just like the spin, and thus can be writ-ten as [Ω xn ( k ) , Ω yn ( k ) , Ω zn ( k )] = [Ω nyz ( k ) , Ω nzx ( k ) , Ω nxy ( k )].Thus, Ω n ( k ) = Ω n ( − k ) if the system has spatial in-version symmetry ( P ) and Ω n ( k ) = − Ω n ( − k ) if ithas T symmetry. Obviously, if the system hasboth P and T symmetries, Ω n ( k ) becomes identi-cally zero. The AHC and ANC are also pseudovectorsand can be written as [ σ xA , σ yA , σ zA ] = [ σ Ayz , σ
Azx , σ
Axy ] and[ α xA , α yA , α zA ] = [ α Ayz , α
Azx , α
Axy ], respectively.Similarly, the spin Hall conductivity ( σ sij = J si /E j ) isgiven by a BZ integration of the spin Berry curvature(Ω n,sij ( k )) for all the occupied bands, σ sij = − e X n Z BZ d k (2 π ) f k n Ω n,sij ( k ) , Ω n,sij ( k ) = − X n ′ = n h k n |{ τ s , v i } / | k n ′ ih k n ′ | v j | k n i ]( ǫ k n − ǫ k n ′ ) , (3)where s denotes the spin direction and τ s is a Paulimatrix. Then the spin Nernst conductivity ( α sij = − J si / ∇ j T ) can be written as α sij = 1 T X n Z BZ d k (2 π ) Ω sij ( k n ) × [( ǫ k n − µ ) f k n + k B T ln(1 + e − β ( ǫ k n − µ ) )] , (4)where J si denotes the i -component of the spin currentdensity j s with spin being along the s -axis.In the AHC, SHC, ANC and SNC calcula-tions, the velocity ( h k n | v i | k n ′ i ) and spin-velocity( h k n |{ τ s , v i } / | k n ′ i ) matrix elements are obtained from the self-consistent electronic structure within the PAWformalism. To obtain accurate AHC, SHC, ANC andSNC, a dense k -point mesh would be needed. There-fore, we use a very fine mesh of 97344 k-points on themagnetic IBZW (1/8 BZ), together with the tetrahedronmethod . This is equivalent to a large number of k -points of ∼ n d = 50 intervals. Furthercalculations using n d = 20, 30 and 40 (i.e., 7260, 22272,51597 k -points in the IBZW, respectively) indicate thatthe AHC, SHC, ANC and SNC obtained using n d = 50converge to within a few %. Indeed, the curves of AHC,SHC, ANC and SNC as a function of energy (see Figs.3-5 below) and also the curves of ANC and SNC as afunction of temperature (see Fig. 6 below) obtained with n d = 40 and 50 are indistinguishable. Moreover, the cal-culated AHC, SHC, ANC and SNC versus the inverse ofthe number ( N k ) of k -points in the IBZW are plottedand fitted to a straight line to get the converged theoret-ical values listed in Table II below (i.e., the extrapolatedvalues at N k = ∞ ) (see Refs. [36,37]) Note that the dif-ferences between the converged theoretical AHC, SHC,ANC and SNC values and the corresponding n d = 50values are within a few %. As mentioned before, wealso calculate the AHC and ANC of ferromagnetic bccFe and the SNC of nonmagnetic fcc Pt for comparison.In the calculation of the AHC and ANC of bcc Fe, wealso adopt a very fine mesh of 360396 k -points on themagnetic IBZW (1/16 BZ). In the SHC and SNC calcu-lations for fcc Pt, a very find grid of 253044 k -points onthe magnetic IBZW (1/16 BZ) is used. III. RESULTS AND DISCUSSION
The energetics of many possible magnetic configura-tions for Mn Sn in the hexagonal DO structure hasalready been investigated with the ab initio density func-tional calculations . Therefore, in this paper we con-sider only two low-energy noncollinear triangular antifer-romagnetic configurations for Mn X ( X = Sn, Ge, Ga)[see Fig. 1(c) and Fig. 1(d) in Ref. ], namely, type Aand type B configurations as illustrated in Fig. 1(c) andFig. 1(d), respectively. For comparison, the ferromag-netic state (FM) of Mn Ga with magnetic moments inthe ˆ x -direction is also investigated. A. Magnetic properties
The calculated total energies and spin magnetic mo-ments are listed in Table I, together with the reportedexperimental values. Table I shows that in all three al-loys, magnetic structure A has a lower energy than mag-netic structure B, although the total energy difference isin the order of ∼ Sn , Mn Ge and Mn Ga. . In TABLE I: Calculated total energy ( E t ) and total spin mag-netic moment ( m st ) as well as averaged Mn spin magnetic mo-ment ( m sMn ) for the A and B magnetic structures of Mn X ( X = Sn, Ge, Ga). Total magnetic moments are parallel tothe ˆ x -axis in configuration A but to the ˆ y -axis in configurationB. The X atoms have a nearly zero magnetic moment (beingless than 0.01 µ B ) and thus are not listed. Note that there aretwo formula units [i.e., 2(Mn X )] per unit cell. For compari-son, the results of the magnetic moment direction-constrainedcalculation for Mn Ga in configuration A (denoted A ∗ ) andthe ferromagnetic calculation for Mn Ga with the magneticmoments in the ˆ x -direction (denoted FM) are given as well.Some previously reported total and Mn spin moments are alsolisted for comparison. E t m s Mn m s t (meV/cell) ( µ B / atom) (10 − µ B / cell)Mn Sn A 0.0 3.13, 3.0 a b B 0.03 3.13 22Mn Ge A 0.0 2.70, 2.4 c d B 0.03 2.68 2.3, 30 e Mn Ga A 0.0 2.75, 2.4 f ∗ a Ref. 41 (experiment), b Ref. 11 (experiment), c Ref. 42(experiment), d Ref. 12 (experiment), e Ref. 13 (experiment), f Ref. 27 (experiment). an earlier DFT calculation for Mn Sn , configuration Awas also found to be slightly lower in total energy thanconfiguration B. Nevertheless, the total energy differenceis very small and such a small energy difference is perhapswithin the numerical uncertainty. This small energy dif-ference between the two configurations is consistent withthe experimental fact that the magnetic moments can beeasily rotated in the hexagonal plane by a small mag-netic field . In contrast, the total energy of theFM structure of Mn Ga is well above that of the twononcollinear antiferromagnetic structures (Table I).The calculated Mn spin magnetic moments in all threeMn X ( X = Sn, Ge, Ga) alloys are large, being ∼ . µ B , while the calculated spin magnetic moments of the X atoms are nearly zero, being less than ∼ . µ B . Table Iindicates that the calculated Mn spin magnetic momentsagree fairly well with previous experiments. Dueto rather strong exchange coupling between large spinmagnetic momonets on the Mn moments, the N´eel tem-peratures in these Mn-based alloys are as high as 420 Kin Mn Sn , 365 K in Mn Ge and 470 K in Mn Ga .Interestingly, in Mn Y ( Y = Ir, Rh, Pt), the calcu-lated total spin magnetic moment is zero and the twocoplanar noncollinear T1 and T2 antiferromagnetic struc-tures have the same total energy in the absence of theSOC. This suggests that the small total magneticmoment obtained with the SOC included in Mn Y ( Y =Ir, Rh, Pt), is induced by the spin-canting caused by theDzyaloshinskii-Moriya interaction (DMI) (i.e., the SOC).In contrast, the nonzero total spin magnetic moment al- ready exists in Mn X ( X = Sn, Ge, Ga) even without theSOC. For example, the total spin magnetic moment cal-culated without the SOC, is 4 × − µ B /cell along the ˆ x -axis in magnetic structure A of Mn Sn, being larger thanthat in the presence of the SOC (Table I). This is becauseboth magnetic structures A and B are orthorhombic withonly one mirror plane (Fig. 1). In the A magnetic struc-ture, the mirror plane M x is parallel to the yz plane [seeFig. 1(c)]. Since the total magnetic moment m t is apseudovector, m t,y and m t,z that are parallel to the yz plane transform, respectively, to − m t,y and − m t,z underthe M x reflection, while m t,x remains unchanged. Con-sequently, m t,y and m t,z must be zero and only m t,x canbe nonzero. In the B structure, the mirror plane M y is parallel to the zx plane [see Fig. 1(d)], and a mirrorreflection plus a translation τ = (0 , , c/
2) would bringthe magnetic structure back onto itself. In this case, only m t,y can be nonzero. The calculated magnetic momentsare consistent with these symmetry requirements (Ta-ble I). In contrast, the T1 and T2 magnetic structures ofMn Y are hexagonal and have three mirror planes, and thus all three components of the magnetic momentsmust be zero. Furthermore, the calculated total energiesof the T1 and T2 structures are the same. B. Anomalous Nernst effect
Table II lists the calculated anomalous Nernst conduc-tivity ( α Aij ), anomalous Hall conductivity ( σ Aij ) and den-sity of states at the Fermi level [ N ( E F )] of Mn X ( X =Sn, Ge, Ga) alloys. As discussed before, the AHC andANC are pseudovectors, just like the total magnetic mo-ment. Thus, in the A (B) magnetic structure, only α Ayz ( α Azx ) and σ Ayz ( σ Azx ) can be nonzero. This can also beseen from the k -space distribution of the Berry curva-ture Ω ( k ) =[Ω x ( k ) , Ω y ( k ) , Ω z ( k )], as displayed in Fig. 2for configuration A of Mn Sn. Figure 2(a) shows that inthe k x k y (i.e., k z = 0) plane, Ω y ( k ) is an odd function of k x while Ω x ( k ) is an even function of k x . In Fig. 2(b),Ω z ( k ) is found to be an odd function of k x while Ω x ( k )is again an even function of k x in the k x k z (i.e., k y = 0)plane. Consequently, Eqs. (1) and (2) would indicatethat σ Azx and σ Axy as well as α Azx and α Axy should be zero.The present results (Table II) are consistent with thesesymmetry properties. It is also clear from Table II thatthe AHC, ANC and N ( E F ) for both A and B configu-rations are very similar. This is consistent with the factthat the two configurations have nearly degenerate totalenergies and very similar magnetic properties (Table I).The calculated α Axy and σ Axy of iron metal are also listedthere for comparison. Table II shows that the AHC ofall the Mn X alloys is rather large, being in the sameorder of magnitude as that of ferromagnetic iron with alarge net magnetic moment of 2.27 µ B /atom. Remark-ably, all the Mn X alloys have a huge ANC, which is10 ∼
40 times larger than that of Fe (Table II). Thisstrongly suggests that these noncollinear antiferromag-
TABLE II: Calculated density of states at the Fermi level [ N ( E F )] (states/eV/spin/f.u.), anomalous Hall conductivity (AHC; σ AH ) ( σ Ayz , σ Azx ) and anomalous Nernst conductivity (ANC; α AN ) ( α Ayz , α Azx ) as well as spin Hall conductivity (SHC) ( σ zxy ) andspin Nernst conductivity (SNC) ( α zxy ) of Mn X ( X = Sn, Ge, Ga). For comparison, the calculated related properties of bcc Fe( σ Axy , α Axy ) and fcc Pt ( σ zxy , α zxy ) are also listed. Note that ANC and SNC listed here were calculated at temperature T = 300K. The ANC for Mn Sn in brackets were calculated at T = 210 K and the SNC for fcc Pt in brackets were calculated at T = 255 K. For comparison, the results of the magnetic moment direction-constrained calculation for Mn Ga in configurationA (denoted A ∗ ) and the ferromagnetic calculation for Mn Ga with the magnetic moments in the ˆ x -direction (denoted FM) aregiven as well. Some previous experimental results are also listed for comparison. N ( E F ) σ AH σ AH ( µ ) ′ α AN σ zxy σ zxy ( µ ) ′ α zxy (S/cm) (S/cm-eV) (A/m-K) ( ~ /e)(S/cm) ( ~ /e)(S/cm-eV) ( ~ /e)(A/m-K)Mn Sn A 1.96 -132, -68 a ,-90 b -456 -4.6 (-1.16),-0.39 b ,-0.28 c
72 -845 7.7B 1.96 -132, -126 a ,-80 b -444 -4.7 (-1.18),-0.32 b
74 -834 7.5Mn Ge A 2.37 -298, 310 d ,150 e -9020 -6.4 56 691 1.02B 2.38 -298, 380 d ,500 e -8289 -6.4 63 1000 0.63Mn Ga A 5.99 -104 -3722 17.1 -219 -5323 7.3A ∗ e -230 0.40,1.8 f - - -fcc Pt 1.75 - - - 2139 1214 -1.11 (-0.92),-1.57 ga Ref. 11 (experiment at 50 K), b Ref. 22 (experiment at 210 K), c Ref. 23 (experiment at 200 K), d Ref. 12 (experiment at 10K), e Ref. 13 (experiment at 2 K). f Ref. 22 (experiment at 300 K). g Extracted from the experiment at 255 K [ 46]. nets would find promising applications in thermoelectricdevices, heat nanosensors and also spin caloritronics.One may wonder whether the nonzero ANC and AHCare caused by the presence of the small net magne-tization in these noncollinear antiferromagnetic struc-tures, as in the case of ferromagnets where the ANCand AHC are proportional to the net magnetization.To address this issue, we perform the magnetic momentdirection-constrained GGA calculation for the A struc-ture of Mn Ga in order to make the total magnetic mo-ment vanished. The results of this calculation are listedin Tables I and II. Table II shows that the resultant ANCand AHC remain nearly unchanged, although the netmagnetic moment is reduced by a factor of ∼
20 (TableI). Moreover, we also carry out the GGA calculation forMn Ga in the ferromagnetic state (FM) with magneti-zation along the ˆ x -axis. Interestingly, although the totalmagnetic moment of the FM state is three orders of mag-nitude larger than that of the A structure (Table I), theANE gets reduced by 20 %, compared with that of theA structure.The calculated ANC ( α Ayz ) and AHC ( σ Ayz ) of mag-netic structure A as a function of the Fermi energy ( E F )as well as the relativistic band structure are plotted inFig. 3 for Mn Sn, in Fig. 4 for Mn Ge and in Fig. 5 forMn Ga. Figure 3 shows that for up to 0.33 eV above the E F , the σ Ayz of Mn Sn is negative and rather flat withsmall ripples. However, if the Fermi energy is loweredto -0.114 eV, one sees a very pronounced negative peakin σ Ayz . The peak σ Ayz value is -979 S/cm. To reach thisenergy level, the number of valence electrons should bereduced by 0.206 per formula unit (f.u.), indicating sub-stitution of ∼
20 % of Sn by In or Ga. Examination ofthe calculated band-resolved Berry curvatures suggests that this peak arises predominantly from the large Ω yz on the top valence band in the vicinity of the gap at M-point [see Fig. 3(a)]. The shape of the σ Ayz versus E F curve in Mn Ge [Fig. 4(b)] is similar to that of Mn Sn[Fig. 3(b)], and this is understandable because both al-loys are isoelectronic. Mn Ga has roughly the same σ Ayz versus E F curve [Fig. 5(b)] as that of Mn Ge and Mn Snexcept that the Fermi level is now about 0.25 eV lowermainly because Mn Ga has one less valence electron.To understand the features in the α Ayz versus E F curve,one should note that at low temperatures, Eq. (2) canbe simplified as the Mott relation, α Axy = − π k B Te σ
Axy ( µ ) ′ , (5)which relates the ANC to the energy derivative of theAHC. This Mott relation roughly explains why in Mn Sn[Fig. 3(c)] there is a prominant peak in α Ayz at -0.070eV, where σ Ayz has a steep slope [Fig. 3(b)]. The peak α Ayz value is as large as -19 A/m-K at 300 K. One couldreach this point by reducing the valence electrons by 0.13electron per Mn Sn, i.e., by merely substituting 13 % Snwith In or Ga. As mentioned before, σ Ayz is rather flatabove the Fermi level [Fig. 3(b)], and this explains why α Ayz becomes nearly zero slightly above the E F [Fig. 3(c)].We have also calculated the ANC of all the alloys as afunction of temperature ( T ) and the results are displayedin Fig. 6(a) together with the calculated T -dependent α Axy of bcc Fe. Figure 6(a) shows that at high tempera-ture (300 ∼
400 K) Mn Ga has a very large α Ayz , being upto ∼
20 A/m-K which is 50 times larger than that of bccFe. The α Ayz of Mn Ga decreases steadily with decreasing T and eventually approaches zero at ∼
50 K. The mag-nitude of the ANC of Mn Sn and Mn Ge is also large (a)(b)(a) ( Ω x , Ω y )( Ω x , Ω z ) ΓΓ K KM A H
FIG. 2: (Color online) Berry curvature [Ω x ( k ) , Ω y ( k ) , Ω z ( k )](in units of ˚A ) of configuration A Mn Sn. (a) (Ω x , Ω y ) onthe k x k y ( k z = 0) plane and (b) (Ω x , Ω z ) on the k x k z ( k y = 0)plane. at high temperatures (e.g., ∼
10 A/m-K at T = 400 K)but the sign of the ANC is negative, being opposite tothat of Mn Ga. The magnitudes of the ANC of Mn Snand Mn Ge decrease monotonically as T decreases andchange sign at 175 K and 200 K, respectively. As T further cools, the ANC of Mn Ge increases steadily andreaches to 5.2 A/m-K at 50 K, while that of Mn Sn staysaround 0.6 A/m-K with small fluctuations. Because oftheir large ANC at room temperature [being at least oneorder of magnitude larger than that of bcc Fe (see TableII)], all three Mn X alloys could serve as a thermoelectricmaterial for spin caloritronics.To examine the validity of the Mott relation [Eq. (5)],we calculated the energy derivative of the AHC for allthe alloys and bcc Fe, as listed in Table II. The ANC at100 K calculated using Eq. (5) and the energy deriva-tives of the AHC are shown in Fig. 6(a). Figure 6(a)indicates that the ANC values calculated this way agreein sign with those calculated directly using Eq. (2) forMn Sn, Mn Ge and bcc Fe. However, the magnitudes -101 E ne r g y ( e V ) -1200 0 1200 σ H (S/cm) AHCSHC -30 0 30 α N (A/m-K) -101 ANCSNC
A L K M ΓΓ (a) (b) Mn Sn (c)(A) FIG. 3: (Color online) Mn Sn. (a) Relativistic band structurein magnetic structure A. (b) Anomalous Hall conductivity(AHC) ( σ Ayz ) and spin Hall conductivity (SHC) ( σ zxy ) as wellas (c) anomalous Nernst conductivity (ANC) ( α Ayz ) and spinNernst conductivity (SNC) ( α zxy ) as a function of energy. TheNernst conductivities were calculated at T = 300 K. TheFermi level is at the zero energy. Note that in (b) [(c)], theunit for the SHC [SNC] should be ( ~ /e)S/cm [( ~ /e)A/m-K]. -101 E ne r g y ( e V ) -1200 0 1200 σ H (S/cm) AHCSHC -30 0 30 α N (A/m-K) -101 ANCSNC
A L K M ΓΓ (a) (b) Mn Ge (c)(A) FIG. 4: (Color online) Mn Ge. (a) Relativistic band structurein the A magnetic structure. (b) Anomalous Hall conductivity(AHC) ( σ Azx ) and spin Hall conductivity (SHC) ( σ zxy ) as wellas (c) anomalous Nernst conductivity (ANC) ( α Azx ) and spinNernst conductivity (SNC) ( α zxy ) as a function of energy. TheNernst conductivities were calculated at T = 300 K. TheFermi level is at the zero energy. Note that in (b) [(c)], theunit for the SHC [SNC] should be ( ~ /e)S/cm [( ~ /e)A/m-K]. differ significantly. At 300 K, the ANC for all Mn X al-loys estimated using Eq. (5) would differ in sign fromthose from Eq. (2) (listed in Table II). We note that inthe magnetized Pt and Pd, at 100 K the α Axy calculatedusing the Mott relation [Eq. (5)] and directly from Eq.(2) agree quantitatively, and even at 300 K they agreewith each other quite well. The band structures of magnetic structures A and B ofall three alloys are almost identical and thus their bandstructures for the B configuration are not presented inthis paper. Furthermore, the two magnetic configura- -101 E ne r g y ( e V ) -1200 0 1200 σ H (S/cm) AHCSHC -30 0 30 α N (A/m-K) -101 ANCSNC
A L K M ΓΓ (a) (b) Mn Ga (c)(A) FIG. 5: (Color online) Mn Ga. (a) Relativistic band structurein the A magnetic structure. (b) Anomalous Hall conductivity(AHC) ( σ Azx ) and spin Hall conductivity (SHC) ( σ zxy ) as wellas (c) anomalous Nernst conductivity (ANC) ( α Azx ) and spinNernst conductivity (SNC) ( α zxy ) as a function of energy. TheNernst conductivities were calculated at T = 300 K. TheFermi level is at the zero energy. Note that in (b) [(c)], theunit for the SHC [SNC] should be ( ~ /e)S/cm [( ~ /e)A/m-K]. tions for each alloy have similar AHC and ANC as a func-tion of energy and hence the AHC and ANC as a functionof energy of the B configuration are not displayed here ei-ther. The present band structures of Mn Sn (Fig. 3) andMn Ge (Fig. 4) are in good agreement with the previousGGA results . The present (Fig. 5) and previous GGA band structures for Mn Ga also agree quite wellalong all the high symmetry lines except the KM linewhere the two band structures differ quite significantly.As mentioned before, the AHE in Mn Sn and Mn Gein noncollinear antiferromagnetic states have been ex-perimentally investigated by several groups.
Thecalculated AHC (132 S/cm) for Mn Sn in configurationB agrees well with the measured value (126 S/cm at 50K) reported in Ref. [11], although the theoretical AHC(132 S/cm) in configuration A is nearly twice as large asthe measured value (68 S/sm) (see Table II). The calcu-lated AHC for Mn Ge in both configuration A and B isalso in good agreement with the experimental value at10 K reported in Ref. [12], although for configurationB it is about 30 % smaller than the measured one (500S/cm at 2 K) presented in Ref. [13] and for configurationA it is twice as large as the measured one . All thesesuggest that the anomalous Hall effect in these alloys isdominated by the intrinsic mechanism due to the nonzeroBerry curvatures in the momentum space. This is alsothe conclusion drawn in Ref. [22] based on the experi-mental examination on the validity of the Wiedemann-Franz law. The AHC of Mn Sn (Mn Ge) presented inTable II is in excellent agreement with the GGA resultof 133 (330) S/cm of Mn Sn (Mn Ge) reported in Refs.[14,47].However, unlike the AHE case, so far merely two pa-pers very recently reported on the experiments on the ANE in Mn Sn.
It was found that the ANE signalsare significant and easily detectable.
Furthermore,the thermal and Nernst conductivities was found to cor-relate according to the Wiedemann-Franz law, indicatingthe intrinsic origin of the ANE. Overall, this is consistentwith our finding of large intrinsic ANE in these alloys.Also the measured and calculated ANC at ∼
210 K agreein sign with respect to that of AHC, although the mea-sured ANC (0 .
39 and 0 .
28 A/m-K) for configuration A isa few times smaller than the calculated ANC (1.16 A/m-K) (Table II). Nevertheless, experimentally, the ANCand AHC were found to decrease steadily as the T isincreased from 200 K to 400 K, in contrast to themonotonical increase of the calculated ANC with T [Fig.6(a)]. This significant discrepancy could arise from sev-eral reasons. First of all, the temperature range of 200 ∼
400 K is close to the antiferromagnetic transition ( T N )and consequently the magnetism gets weaker as the T N is approached. In the theoretical calculation, however,the T = 0 magnetism is assumed and the T -dependenceenters only through the Fermi function [see Eq. (2)]. Sec-ondly, although the ANC is calculated directly from theband structure [see Eq. (2)], experimentally, the ANCcannot be measured directly and thus is estimated usingmeasurable longitudinal ( ρ ii ) and Hall ( ρ ij ) resistivitiesas well as Seebeck ( S ii ) and Seebeck-Nernst ( S ij ) coeffi-cients via α Ayz = − ρ zz S yz − ρ yz S zz ρ yy ρ zz − ρ yz ρ zy ≈ ρ zz S yz − ρ zz S yz ρ yy ρ zz . (6)Clearly, to obtain accurate estimated ANC, all thesequantities must be accurately measured on the same sam-ple, but this often is not the case. Given all these compli-cations, we believe that the level of agreement betweenthe experiment and calculation is quite good. Table IIshows that the experimental ANC of iron at 300 K alsoreported in [22] is ∼ of iron gave a value of 0.16 A/m-K, being more thantwo times smaller than the present GGA result. Furtherexperiments on the ANE and AHE on these alloys areclearly needed. C. Spin Nernst effect
The SNC ( α sxy ; s, i, j = x, y, z ) and SHC ( σ sxy )are third-order tensors. A recent symmetry analysis showed that only elements σ xyz ( σ xzy ), σ yxz ( σ yzx ) and σ zxy ( σ zyx ) can be nonzero. Furthermore, the ab initio cal-culations of the SHC of Mn X ( X = Sn, Ge, Ga) in-dicated that only σ zxy and σ zyx are significantly nonzero.Therefore, in this paper we consider only α zxy and σ zxy .The calculated α zxy and σ zxy of all Mn X alloys are listedin Table II. The α zxy and σ zxy of platinum metal arealso listed there for comparison. Table II shows that theSHC of the Mn X alloys is rather small, compared to α A ( A / m - K ) Mn SnMn GeMn GaFe α xy α z ( h / e )( A / m - K ) Pt xy - (a) ( b ) x10 y z A100 K100 K
FIG. 6: (color online) (a) Anomalous Nernst conductivity(ANC) ( α A ) and (b) spin Nernst conductivity (SNC) ( α Sxy )as a function of temperature. The values displayed on thevertical dashed line (a) and (b) are the ANC and SNC at 100K calculated using the Mott relations [Eq. (5) and Eq. (6)]and the energy derivatives of AHC and SHC listed in TableII, respectively. that of platinum, which has the largest intrinsic SHCamong transition metals.
Remarkably, the SNC ofMn Sn and Mn Ga is very large, being about five timeslarger than that of Pt (Table II). Mn Ge also has a largerSNC than platinum. This shows that noncollinear anti-ferromagnets Mn X ( X = Sn, Ge, Ga) would be veryuseful materials for spin thermoelectric devices and spincaloritronics, just like Pt metal for spintronics.The calculated SNC ( α zxy ) and SHC ( σ zxy ) as a func-tion of the Fermi energy ( E F ) of Mn Sn, Mn Ge andMn Ga are displayed in Fig. 3, Fig. 4 and Fig. 5,respectively. Figures 3(b) and 4(b) show that in bothMn Sn and Mn Ge the σ zxy in the vicinity of the E F israther small, thus resulting in a small value at the E F (Table II). Nevertheless, the σ zxy in Mn Ge has a broadprominant peak near -0.30 eV, and the peak value is aslarge as -750 ( ~ /e)S/cm [Fig. 4(b)]. This peak can bereached by a reduction of the valence electrons of ∼ Ga which has one less valence electron,the E F is lowered to just below this peak [Fig. 5(b)],thus resulting in a much larger σ zxy value (Table II).To understand the features in the α zyz versus E F curve,one should note again that Eq. (4) would be reduced to the simple Mott relation at low temperatures, α zxy = − π k B Te σ zxy ( µ ) ′ , (7)which relates the SNC to the energy derivative of theSHC. This Mott relation roughly explains why in Mn Sn[Fig. 3(c)] the α zyz has a broad plateau from -0.09 eVto 0.23 eV around the E F , where σ zyz has more or lessa constant negative slope [Fig. 3(b)]. The plateau α zyz value is about 9 ( ~ /e)A/m-K at 300 K. In Mn Ge, the α zyz is rather small in the vicinity of the E F because σ zyz israther flat (and small) (Fig. 4). Nevertheless, the α zyz hasa prominant negative peak at -0.21 eV [Fig. 4(c)] where σ zyz has a steep slope [Fig. 4(b)]. Within the rigid bandmodel, the α zyz peak could be reached by reducing thenumber of valence electrons by ∼ Ga,the E F sits on the upper side of the pronounced peakat -0.035 eV and thus α zyz is large, being as large as 14( ~ /e)A/m-K at 300 K. Again, this is because the σ zyz hasa steep slope at -0.035 eV [Fig. 5(b)].Very recently, the SNE in platinum was studied ex-perimentally and a large spin Nernst angle ( θ SN ) wasobserved. The spin Nernst angle is comparable in sizebut opposite in sign to the spin Hall angle ( θ SH ) with θ SH / θ SN = − . It can be shown that α zxy = − σ zxy S yy / ( θ SH /θ SN ). Using the theoretical σ zxy = 2139( ~ /e)(S/cm) (Table II) and measured Seebeck coefficient S yy = − . µ V/K at 255 K, we would obtain an esti-mated experimental α zxy = − .
57 ( ~ /e)(S/cm), agreeingquite well with the calculated value of -0.92 ( ~ /e)A/m-K(Table II).In Fig. 6(b), the calculated T -dependence of the SNCfor all three alloys as well as Pt metal are displayed. Fig.6(b) shows that the magnitude of the SNC of Mn Sn isvery large at high temperatures (e.g., ∼
10 ( ~ /e)A/m-Kat T = 400 K). Nevertheless, the SNC decreases mono-tonically as the T decreases down to 50 K. Interestingly,the SNC of Pt has a very similar T -dependence, albeitwith a much smaller magnitude and an opposite sign. Incontrast, Mn Ga has a smaller SNC at high tempera-tures (e.g., ∼ ~ /e)A/m-K at T = 400 K). However,the SNC of Mn Ga increases steadily as the T is low-ered, and it reaches its maximum of ∼
10 ( ~ /e)A/m-Kat T = 175 K. It then decreases monotonically as the T further decreases. Mn Ge has a small SNC at high tem-peratures (e.g., ∼ ~ /e)A/m-K at T = 400 K). TheSNC of Mn Ge decreases gradually as the T decreasesand changes sign at T = 225 K. After passing 225 K, itfurther decreases as the T is lowered to 125 K, and itthen increases slightly as the T decreases to 50 K.We calculated the energy derivative of the SHC for allthe alloys and fcc Pt, as listed in Table II, in order toexamine the validity of the Mott relation [Eq. (7)]. TheSNC at 100 K calculated using Eq. (7) and the energyderivatives of the SHC are shown in Fig. 6(b). Figure6(b) indicates that all the SNC values calculated this wayagree in sign with those calculated directly using Eq. (4).For fcc Pt, the SNC values [-0.30 and -0.31 ( ~ /e)S/cm]agree rather well. This level of agreement [-0.89 and -1.11 ( ~ /e)S/cm] is maintained even at 300 K. For Mn Ga(Mn Ge), the SNC value from Eq. (7) is 2.5 (five) timessmaller than that from Eq. (4). For Mn Sn, the SNCvalues from Eq. (7) and Eq. (4) differ by one order ofmagnitude [Fig. 6(b)].
IV. CONCLUSIONS
We have studied theoretically the ANE, a phenomenonhaving the same origin as the AHE, and also the SNE aswell as the AHE and SHE in noncollinear antiferromag-netic Mn X ( X = Sn, Ge, Ga) based on ab initio rel-ativistic band structure calculations. As references, wealso calculate the ANC and AHC of ferromagnetic ironas well as the SNC of platinum metal. Fascinatingly,the calculated ANC at room temperature (300 K) forall three alloys is huge, being 10 ∼
40 times larger thanthat of iron. Further, the calculated SNC for Mn Snand Mn Ga is also larger, being about five times largerthan that of platinum. This suggests that these anti- ferromagnets would be useful materials for thermoelec-tronic devices and spin caloritronic devices. The calcu-lated ANC of Mn Sn and iron are in reasonably goodagreement with the very recent experiments . The cal-culated SNC of platinum also agree well with the veryrecent experiments in both sign and magnitude. Thecalculated thermoelectric and thermomagnetic propertiesare analyzed in terms of the band structures as well asthe energy-dependent AHC, ANC, SNC and SHC via theMott relations. We hope that our interesting theoreticalresults would stimulate further experimental works onthese noncollinear antiferromagnets. Acknowledgments
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