Magnetic susceptibility of Dirac electrons in single-component molecular

conductor [Pd(dddt)2] under pressure

Yoshikazu Suzumura1* and Reizo Kato2

1Department of Physics, Nagoya University, Nagoya 464-8602, Japan2RIKEN, Wako, Saitama 351-0198, Japan

*E-mail: suzumura@s.phys.nagoya-u.ac.jp

Received November 14, 2016; accepted February 9, 2017; published online April 4, 2017

Using a tight-binding model with four lattice sites per unit cell, we examine a three-dimensional Dirac electron in a single-component molecularconductor [Pd(dddt)2], which consists of HOMO and LUMO orbitals. The Dirac cone, which originates from the interplay of the intralayer andinterlayer transfer energies, gives a semimetallic state owing to a slight variation in energy along the line of the Dirac point. Electronic states of theDirac electron are examined by calculating the temperature (T ) dependence of magnetic (spin) susceptibility. It is shown that magneticsusceptibility remains finite at zero temperature and the variation with increasing temperature exhibits a T-linear dependence. The role of theHOMO and LUMO orbitals is discussed in terms of local susceptibility. © 2017 The Japan Society of Applied Physics

1. Introduction

Since the discovery of the quantum Hall effect in graphene,two-dimensional massless Dirac fermions have been one ofthe recent topics.1) In addition to graphene with a monolayer,Dirac electrons in molecular conductors provide a novelaspect as the bulk system. For the two-dimensional organicconductor α-(BEDT-TTF)2I32) [BEDT-TTF = bis(ethylenedi-thio)tetrathiafulvalene], the Dirac electron with a tilted Diraccone3) followed by the vanishing of the density of states atthe Fermi surface4) was found in the tight-binding modelwith the transfer energy estimated by the extended Hückelmethod.5) The existence of the Dirac cone was verified byfirst-principles calculation.6) Since the Dirac electron explainswell the transport experiment under pressure,7,8) the Diracelectron in the organic conductor has been studied exten-sively.9) Regarding the Dirac point that occurs as theaccidental degeneracy10) of the conduction and valencebands, the mechanism in α-(BEDT-TTF)2I3 was clarified onthe basis of the inversion symmetry of the crystal and theresultant effective Hamiltonian.11,12)

Recently, it has been shown that a single-componentmolecular conductor [Pd(dddt)2] (dddt = 5,6-dihydro-1,4-dithiin-2,3-dithiolate) exhibits Dirac electrons under pres-sure.13–15) The crystal consists of four molecules per unit cellwith the HOMO and LUMO orbitals being odd and evenaround the inversion center of the Pd atom, respectively.Compared with the case of the organic conductor, thismaterial provides the Dirac point in three-dimensionalmomentum space k ¼ ðkx; ky; kzÞ, corresponding to the a-,b-, and c-axes where there are transfer energies along theinterlayer direction (kz) in addition to the intralayer planeðkx; kyÞ. The Dirac cone located between the conduction andvalence bands was found by first-principles calculation15) andwas examined using a tight-binding model under pressure,16)

which is described by an 8 × 8 matrix Hamiltonian with theenergy band Eγ (k), [γ = 1 (top),…, 8 (bottom)]. At ambientpressure, the insulating state is obtained since the chemicalpotential, due to the half-filled band, is located between theLUMO and HOMO bands. Under high pressures, the Diracpoint emerges between E4(k) and E5(k) when the HOMOband becomes larger than the LUMO band at the Γ point(k = 0). The Dirac point found at the apex of a pair of Dirac

cones gives a line in a three-dimensional Brillouin zone.17)

The contour of the gap function of E4(k) − E5(k) with afixed magnitude is elliptical on a two-dimensional plane, andforms a cylindrical shape along the line, suggesting thethree-dimensional electronic states. The Dirac electron in[Pd(dddt)2] is exotic since the Dirac point originates from thecombined effect of the intralayer and interlayer transferenergies.13,18)

In the present study, the property of such a Dirac electronis examined by calculating the magnetic susceptibility, whichis useful for understanding the local electronic state of NMRas shown in α-(BEDT-TTF)2I3.19,20) In Sect. 2, the model andformulation are given. In Sect. 3, the three-dimensional Diracpoint located between the conduction and valence bands isshown for typical cases. The temperature dependence of themagnetic susceptibility is examined in Sect. 4. Section 5 isdevoted to summary and discussion.

2. Model and formulation

The crystal structure of [Pd(dddt)2], which consists of fourkinds of molecules (1, 2, 3, and 4) with HOMO and LUMOorbitals in the unit cell, is shown in Fig. 1. There are twokinds of layers; layer 1 includes molecules 1 and 3, andlayer 2 includes molecules 2 and 4. The interlayer transferenergies are given by bonds a (1 and 2 molecules, and 3 and

Fig. 1. (Color online) Crystal structure of [Pd(dddt)2] viewed along theb-axis, with four molecules 1, 2, 3, and 4 per unit cell where the locationðx; y; zÞ of the central Pd atom is given by ð0; 0; 0Þ, ð1=2; 1=2; 1=2Þ,ð1=2; 1=2; 0Þ, and ð0; 0; 1=2Þ in the unit of the lattice constant, respectively.

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4 molecules) and c (1 and 4 molecules, and 2 and 3molecules). The intralayer transfer energies are given bybonds p (1 and 3 molecules), q (2 and 4 molecules), and b(perpendicular to the a–c plane) being the largest transferenergy. For bonds p and q, there are two kinds of transferenergies owing to the tilting of the molecules.

On the basis of the crystal structure, the tight bindingmodel Hamiltonian is given by

H ¼Xi;j

ti; j;�;�ji; �ih j; �j; ð1Þ

where ∣i, α⟩ is the state vector and ti; j;�;� is the transfer energyin the unit of eV. The quantities i and j are the sites of the unitcell, and α and β denote the 8 molecular orbitals given by

HOMO ðH1; H2; H3; H4Þ and LUMO ðL1; L2; L3; L4Þ. Thelattice constant is taken as unity. The transfer energy ti; j;�;�consists of the HOMO–HOMO (H–H), LUMO–LUMO(L–L), and HOMO–LUMO (H–L) interactions (i.e., transferenergies). Since the symmetry of the molecular orbital is odd(even) with respect to the Pd atom for HOMO (LUMO), thetransfer energy between HOMO–LUMO orbitals shows analternation of the sign.

Using the Fourier transform jk; �i ¼Pj expð�ikrjÞjj; �i

with the wave vector k ¼ ðkx; ky; kzÞ, Eq. (1) is rewritten as

H ¼Xk

�ðkÞy ~HðkÞ�ðkÞ; ð2Þ

where ~HðkÞ is the Hermite matrix Hamiltonian given by13,16)

~HðkÞ ¼

tH1;H1 tH1;H2 tH1;H3 tH1;H4 tH1;L1 tH1;L2 tH1;L3 tH1;L4

tH2;H1 tH2;H2 tH2;H3 tH2;H4 tH2;L1 tH2;L2 tH2;L3 tH2;L4

tH3;H1 tH3;H2 tH3;H3 tH3;H4 tH3;L1 tH3;L2 tH3;L3 tH3;L4

tH4;H1 tH4;H2 tH4;H3 tH4;H4 tH4;L1 tH4;L2 tH4;L3 tH4;L4

tL1;H1 tL1;H2 tL1;H3 tL1;H4 tL1;L1 tL1;L2 tL1;L3 tL1;L4

tL2;H1 tL2;H2 tL2;H3 tL2;H4 tL2;L1 tL2;L2 tL2;L3 tL2;L4

tL3;H1 tL3;H2 tL3;H3 tL3;H4 tL3;L1 tL3;L2 tL3;L3 tL3;L4

tL4;H1 tL4;H2 tL4;H3 tL4;H4 tL4;L1 tL4;L2 tL4;L3 tL4;L4

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

ð3Þ

and Φ(k) is expressed as

�ðkÞ ¼ ðhH1j; hH2j; hH3j; hH4j; hL1j; hL2j; hL3j; hL4jÞ: ð4ÞThe matrix element of ~HðkÞ is given in Appendix. Thetransfer energy ti;j;�;� in the unit of eV is estimated to beaH = −0.0345, aL ≃ 0 aHL = 0.0260, b1H = 0.2040, b1L =0.0648, b1HL = 0.0219, b2H = 0.0762, b2L = −0.0413,b2HL = −0.0531, cH = 0.0118, cL = −0.0167, cHL = 0.0218,pH = 0.0398, pL = 0.0205, p1HL = −0.0275, p2HL = −0.0293,qH = 0.0247, qL = 0.0148, q1HL = −0.0186, and q2HL =−0.0191 by using the extended Hückel method based onthe structural data which is obtained by the first-principlesdensity functional theory (DFT) calculation.15)

The gap between the energies of HOMO and LUMOorbitals is taken as ΔE = 0.696 eV to reproduce the energyband of the first-principles calculation.15)

The energy band Eγ(k) is calculated fromX�

~H��ðkÞd��ðkÞ ¼ E�ðkÞd��ðkÞ; ð5Þ

where �; � ¼ H1; H2; . . . ; L4 and � ¼ 1; . . . ; 8. E1 >E2 > ⋯ > E8. Since the band is half-filled owing to the samenumber of HOMO and LUMO orbitals, we examine the gapfunction defined by

EgðkÞ ¼ minðE4ðkÞ � E5ðkÞÞ: ð6ÞThe insulating state is obtained when Eg(k) ≠ 0 for all kvalues. The Dirac point kD is obtained from Eg(kD) = 0 forthe fixed kz or ky as shown later.

The magnetic (spin) susceptibility caused by the Zeemaneffect is calculated as follows. Using the component of thewave function dαγ in Eq. (5), the spin response function perspin for the fixed kz is calculated as19)

���ðkzÞ ¼ � 1

NxNy

Xkx;ky

X8�¼1

X8�0¼1

fðE�ðkÞÞ � fðE�0 ðkÞÞE�ðkÞ � E�0 ðkÞ

� d��ðkÞ�d��ðkÞd��0 ðkÞ�d��0 ðkÞ; ð7Þwhere f(E) = 1=(exp[(E − μ)=T ] + 1) and the Boltzmannconstant kB and the Bohr magneton μB are taken as unity.The total number of lattice sites is given by N = NxNyNz.μ is the chemical potential and T is the temperature in the unitof eV. The local magnetic susceptibility at the α site, χα, isobtained as

�3D� ðTÞ ¼ 1

Nz

Xkz

X�

���ðkzÞ

¼ � 1

N

Xk

X�

@fðE�ðkÞÞ@E�ðkÞ

� �d��ðkÞd��ðkÞ�

¼ 1

Nz

Xkz

�2D� ðkzÞ; ð8Þ

where �2D� ðkzÞ ¼P

� ���ðkzÞ. For the total magnetic suscepti-bility χ total(T ), we obtain the simple formulas

�totalðTÞ ¼X�

�3D� ðTÞ ¼ �Z 1

�1d!

@fð!Þ@!

Dð!Þ; ð9Þ

Dð!Þ ¼ 1

N

Xk

X�

�ð! � E�ðkÞÞ; ð10Þ

where D(ω) denotes the density of states per spin and perunit cell, i.e., s dωD(ω) = 8. The chemical potential μ isdetermined self-consistently from the half-filled condition,which is given by

1

N

Xk

X�

fðE�ðkÞÞ ¼Z 1

�1d!Dð!Þfð!Þ ¼ 4: ð11Þ

3. Dirac electrons in three dimensions

We examine the Dirac point located between E4(k)

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(conduction band) and E5(k) (valence band) for the fixed kzor ky. At ambient pressure, the insulating state is found wherethere exists a minimum gap at the Γ point (k = 0), and theLUMO bands (E1; . . . ; E4) are separated from the HOMObands (E5; . . . ; E8). With increasing pressure, the gap at theΓ point decreases and the minimum of the LUMO band andthe maximum of the HOMO band are reversed at the Γ point.In this case, a surface given by E4(k) = E5(k) exists in theabsence of the H–L interaction, while a gap [E4(k) −E5(k) ≠ 0] appears on the surface in the presence of theH–L interaction. However, there is another surface on whichthe H–L interaction vanishes owing to the orthogonality ofthe HOMO and LUMO wave functions. The Dirac pointemerges along the line given by the intersection of these twosurfaces.13) We show this Dirac point on a two-dimensionalplane by taking two typical cases of kz = 0 (I) and ky = 0 (II).

Figure 2(a) shows case (I) with the Dirac point given bykD=� ¼ ðkx=�; ky=�; kz=�Þ ¼ ð0;�0:0875; 0Þ. The energybands of E4(k) and E5(k), which are shown on the plane ofkx and ky with the fixed kz = 0, correspond to the conductionand valence bands, respectively. The band E4(k) (E5(k)),which consists of HOMO (LUMO) orbitals at kx = ky = 0[i.e., Γ point], is mainly determined by the H1 and H3 (L2 andL4) orbitals. The Dirac point with an energy �D exists betweenthese two bands [i.e., E4ðkÞ > �D > E5ðkÞ], which touch at theDirac point forming a pair of Dirac cones. One-dimensional

band is seen along the ky-axis since the ky-axis correspondsto the most conducting direction. Figure 2(b) shows thecorresponding gap function Eg(k) [= E4(k) − E5(k)], wherethe Dirac point with Eg = 0 is found at the center of the brightregion. The valley that denotes the line of the minimum Eg(k)reduces to a line with E4(k) = E5(k) in the absence of theH–L interaction. There is another line of kx ≃ 0 (not shown)at which the H–L interaction vanishes. The Dirac point isobtained at the intersection of these two lines. The Dirac coneis tilted along the ky-axis, and the velocity is also anisotropicwith respect to the kx- and ky-axes, e.g., the velocity for ky isabout ten times as high as that for ky.

The Dirac point kD ¼ ðkx; kyÞ depends on kz. Withincreasing kz, ∣ky∣ decreases to zero leading to case (II).Figure 2(c) shows the Dirac cone for case (II) on the plane ofkx and kz with the Dirac point given by ðkx=�; ky=�; kz=�Þ ¼ð�0:155; 0;�1:09Þ. The Dirac cone is symmetric withrespect to kx = kz = 0 (Z point), at which the band E4(k)takes a saddle point. Two Dirac points are also symmetricwith respect to the Z point. One-dimensional band is seenalong the kx-axis and the velocity of the Dirac cone for thekx-axis is higher than that along the kz-axis. The correspond-ing gap function Eg(k) is shown in Fig. 2(d). The brightregion surrounding the Dirac point (Eg = 0) is much largerthan that of (b), suggesting that the velocity of the Dirac coneof (d) is lower than that of (b). The valley (minimum line of

(a) (b)

(c) (d)

Fig. 2. (Color online) Energy band of Ej ( j = 4 and 5) for kz=π = 0 (a) and ky=π = 0 (c). The apex of the Dirac cone is located between E4(k) and E5(k). Thecorresponding gap function of Eg [= E4(k) − E5(k)] is shown for kz=π = 0 (b) and ky=π = 0 (d). The bright region denotes Eg < 0.005, where the center denotesthe Dirac point given by ð0;�0:0875; 0Þ and ð�0:155; 0;�1:09Þ for (b) and (d), respectively.

Jpn. J. Appl. Phys. 56, 05FB02 (2017) Y. Suzumura and R. Kato

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Eg), which is almost parallel to the kz-axis, reduces to a linewith E4(k) = E5(k) in the absence of the H–L interaction. TheH–L interaction vanishes on the line with kz ∝ −kx (notshown), which connects the Z point and the Dirac point.Thus the Dirac point is obtained at the intersection of thesetwo lines.

4. Magnetic susceptibility

We note that �D slightly depends on kz where �D[= E4(kD) = E5(kD)], and E4ðkÞ > �D > E5ðkÞ for the fixedkz. For example, �D ¼ 0:5540, 0.5553, and 0.5567 for kz=π =0, 0.5, and 1.0, respectively. From Eq. (11), the chemicalpotential μ is estimated as μ = 0.5561 at T = 0, whichcorresponds to �D at kz=π ≃ 0.65. The dispersion of �DðkzÞwith a width of 0.0027 suggests a semimetallic state withthe electron and hole pockets along the line of the Diracpoint. The property of the Dirac electron close to thechemical potential is examined by calculating the magneticsusceptibility.

To examine the typical behavior associated with the Diraccone, first we show the two-dimensional susceptibility �2D� bytaking ¼ �D [i.e., E4(k) > μ > E5(k)]. For the fixed kz, �2D�is given in Eq. (8). We also examine �2D� for the fixed ky = 0.In Fig. 3(a), �2D� (� ¼ 1; 2; . . . ; 4) is shown as the function ofT, where �2D1 ¼ �2D3 and �2D2 ¼ �2D4 . The solid (dashed) linecorresponds to kz = 0 (ky = 0). The T-linear dependence forsmall T, which originates from the dispersion of the Diraccone, resembles that of the zero-gap state of the two-dimensional Dirac electron in the organic conductor.19) Theinequality of �2D� ðkz ¼ 0Þ < �2D� ðky ¼ 0Þ comes from thefollowing fact. The average velocity for the former is muchhigher than that for the latter as also seen from a fact that thebright region of the gap function of Fig. 2(b) is smaller thanthat of Fig. 2(d). The relation �2D1 ¼ �2D3 ( �2D2 ¼ �2D4 ) impliesthat molecules 1 and 3 (2 and 4) are equivalent. The localsusceptibility �2D2 (�2D1 ) is mainly determined by L2 (H1),and the contribution of H2 and L1 is negligibly small. Therelation �2D2 > �2D1 comes from the difference in dαγ inEq. (5). The contribution of the L2 component is larger than

that of the H1 component for E4(k) and E5(k). Thus, thesusceptibility of layer 2, which is determined by LUMOorbitals, is larger than that of layer 1, which is determined byHOMO orbitals.

The T-linear dependence of �2D� is corrected when ≠ �D,i.e., the effect of carrier doping caused by the electron andhole pockets along the Dirac line (as a function of kz). InFig. 3(b), �2D� ðkz ¼ 0Þ is calculated for μ = 0.5540 (solidline) and 0.5553 (dotted line), where the former (latter)corresponds to �D at kz=π = 0 (kz=π = 0.5). As shown by thedotted line, �2D� ðkz ¼ 0Þ remains finite even at T = 0 since thechemical potential is located above the energy of the Diracpoint. Noting that the energy at the Dirac point �D increaseswith increasing kz, the summation of �2D� with respect to kzgives a finite �3D� even at T = 0.

In Fig. 4, the temperature dependence of the localsusceptibility �3D� and μ(T ) − μ(0) is shown, where �3D2 takesa maximum in the interval region of 0 < T < 0.005. The

0 0.002 0.0040

0.2

0.4

0.6

T(eV)

Xα2D

α=1,3

α=2,4

Xα(kz=0)

(a)

Xα(ky=0)/10

0 0.002 0.0040

0.2

0.4

0.6

T (eV)

Xα2D

α=1,3

α=2,4

μ=0.55400.5553

kz=0

(b)

Fig. 3. (Color online) Magnetic susceptibility �2D� in the unit of (eV)−1 for kz = 0 (solid line) and ky = 0 (dashed line) (a). For �2D� , the chemical potential istaken as the energy of the respective Dirac point. i.e., μ = 0.5540 and 0.5569 eV, respectively. In (b) �2D� for kz = 0 is shown for μ = 0.5540 and 0.5553, whichcorrespond to �D at kz=π = 0 (solid line) and 0.5 (dotted line), respectively.

0 0.002 0.0040

0.2

0.4

0.6

−2

0

2

4

T (eV)

Xα3D

(μ(T)−μ(0))x1000

α=2,4

α=1,3

Fig. 4. Temperature dependence of the local susceptibility �3D� in the unitof (eV)−1 (α = 2 and 4 in layer 2, and α = 1 and 3 in layer 1) and thechemical potential μ(T ) − μ(0) with μ(0) = 0.5560. The dotted line denotesμ(T ) = μ(0).

Jpn. J. Appl. Phys. 56, 05FB02 (2017) Y. Suzumura and R. Kato

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chemical potential μ(T ) is calculated self-consistently fromEq. (11). With increasing temperature, μ decreases from0.5560 (T = 0) to 0.5546 (T = 0.005). The temperaturevariation of �3D� ðTÞ is compared with that in Fig. 3(b). Fora choice of kz with �D ≠ , there are two contributions for�3D� . One is the band energy closed to μ, which gives the partbeing independent of T owing to the finite density of states.Another is the energy far from μ, where the effect of ≠ �Dbecomes small, leading to the T-linear dependence found inthe Dirac cone. We also note that the decrease of μ(T ) withincreasing T suppresses �3D� ðTÞ, i.e., �3D� with μ(T ) is slightlysmaller than that with μ(0).

Figure 5 shows the temperature dependence of the totalsusceptibility χ total [Eq. (9)], which is expressed in terms ofthe density of states. The temperature variation of χ total

mainly comes from �3D2 and �3D4 with the L2 and L4components. The maximum χ total originates from the LUMOorbitals of layer 2. Note that �3D1 and �3D3 with the H1 and H3components also contribute since �total� ð0Þ is determined byboth layers 2 and 1. The inset shows the ω dependence ofthe density of states, D(ω), given by Eq. (10). The originω = 0 is taken as the chemical potential at T = 0, i.e., μ(0)(= 0.5561). It is found that ω corresponding to the minimumD(ω) is slightly larger than zero, and the width of the dip issmaller than that of �DðkzÞ. The comparison of the inset withthe main figure in Fig. 5 shows χ total(0) = D(0), which is alsoverified analytically from Eqs. (9) and (10).

Here, we explain the origin of χ total(0) ≠ 0 in Fig. 5 evenfor the half-filled band. It is reasonable to obtain �2D� ð0Þ ¼ 0

in Fig. 3(a) since the chemical potential is fixed at the energyof the Dirac point. However, in the present three-dimensionalcase, the energy of the Dirac point (�D) varies as a function ofkz, and then the chemical potential is taken somewhere alongthe line of the Dirac point. Thus, χ total(0) ≠ 0 comes from afact that the energy of the Dirac point deviates negatively orpositively from the chemical potential.

5. Summary and discussion

We examined an exotic Dirac electron in the single-component molecular conductor [Pd(dddt)2], which consists

of four molecules with HOMO and LUMO orbitals in theunit cell. The Dirac point forms a line in the three-dimensional Brillouin zone where the Dirac cone isrepresented on the kx–ky and kx–kz planes with increasing kzfrom zero. For the fixed kz, the valence and conduction bandsare separated by the Dirac point while the corresponding�DðkzÞ shows a dispersion with a width ∼0.0027 and thechemical potential at T = 0 exists in the region of the width.Thus, the system exhibits a semimetallic state followed by analternation of hole and electron pockets along the line.

Magnetic spin susceptibility is calculated with the chemi-cal potential determined self-consistently. For the fixed kz(or ky), the susceptibility �2D� with ¼ �D shows the typicalbehavior of the zero-gap state with the Dirac cone, i.e., theT-linear dependence. Since molecule 1 (and 3) is differentfrom molecule 2 (and 4), such a difference appears in themagnitude of magnetic susceptibility. The contribution fromthe LUMO orbital is larger than that from the HOMO orbital,since the �3D� of layer 2 is larger than that of layer 1. The totalmagnetic susceptibility remains finite even at T = 0, andshows a maximum at T ≃ 0.003 while χ total(T ) − χ total(0) ∝ Tat the interval temperature. The behavior at low temperaturesis understood from the density of states shown in the inset ofFig. 5, e.g., χ total(0) = D(0).

In the present study, we examined the Dirac electron usinga tight binding model only with transfer energies. Since theapplication of pressure expands the bandwidth and thepresent system does not exhibit the charge order shown inα-(BEDT-TTF)2I3,9) the effect of correlation on the Diracelectron in [Pd(dddt)2] is expected to be small.

Finally, we note a method to verify the Dirac point shownin Fig. 2. In a way similar to α-(BEDT-TTF)2I3,11) theproperties of the HOMO and LUMO orbitals, which are oddand even around the inversion center of the Pd atom, give riseto a parity at the time reversal invariant momentum (TRIM).In fact, the product of such a parity at the TRIMs gives thecondition for the Dirac point. The details will be reportedelsewhere.

Acknowledgement

This work was supported by JSPS KAKENHI GrantNumbers JP15H02108, JP26400355, and JP16H06346.

Appendix: Tight binding Hamiltonian

The 8 × 8 Hamiltonian, ~HðkÞ of Eq. (3) consists of HOMO(H) and LUMO (L) elements expressed as

~HðkÞ ¼ HðkÞH­H HðkÞH­LHðkÞL­H HðkÞL­L

!; ðA:1Þ

where HðkÞH­H, HðkÞL­L, and HðkÞH­L denote the 4 × 4matrix Hamiltonian. By taking x, y, and z in place of a, b, andc in Fig. 1, the matrix elements of Eq. (3) are given by

tH1;H1 ¼ tH3;H3 ¼ 2b1H cos ky;

tH2;H2 ¼ tH4;H4 ¼ 2b2H cos ky;

tH1;H2 ¼ aHð1 þ e�iðkxþkyþkzÞÞ;tH1;H3 ¼ pHð1 þ e�iky þ e�ikx þ e�iðkxþkyÞÞ;tH1;H4 ¼ cHð1 þ eikzÞ;tH2;H3 ¼ cHð1 þ eikzÞ;

0 0.002 0.0040

1

2

−0.005 0 0.0050

1

2

3

4

ω (eV)

D(ω )

T (eV)

Xtotal

Fig. 5. Temperature dependence of the total susceptibility χ total in the unitof (eV)−1. The inset denotes the density of states, D(ω), in the unit of (eV)−1,where ω = 0 corresponds to 0.5561 eV.

Jpn. J. Appl. Phys. 56, 05FB02 (2017) Y. Suzumura and R. Kato

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tH2;H4 ¼ qHðeiðkxþkzÞ þ eiðkxþkyþkzÞ þ eikz þ eiðkyþkzÞÞ;tH3;H4 ¼ aHðeiky þ eiðkxþkzÞÞ; ðA:2Þ

for HðkÞH­H,tH1;L1 ¼ tH3;L3 ¼ b1HLðeiky � e�ikyÞ;tH2;L2 ¼ tH4;L4 ¼ b2HLðeiky � e�ikyÞ;tH1;L2 ¼ tH1;L4 ¼ tH3;L2 ¼ tH3;L4 ¼ 0;

tH1;L3 ¼ p1HL þ p2HLe�iky � p2HLe

�ikx � p1HLe�iðkxþkyÞ;

tH2;L1 ¼ aHLð1 � eiðkxþkyþkzÞÞ;tH2;L3 ¼ cHLðe�iky � eiðkyþkzÞÞ;tH2;L4 ¼ q1HLe

iðkxþkzÞ þ q2HLeiðkxþkyþkzÞ � q2HLe

ikz

� q1HLeiðkyþkzÞ;

tH3;L1 ¼ p2HL þ p1HLeiky � p1HLe

ikx � p2HLeiðkxþkyÞ;

tH4;L1 ¼ cHLðe�iky � eiðky�kzÞÞ;tH4;L2 ¼ q2HLe

�iðkxþkzÞ þ q1HLe�iðkxþkyþkzÞ � q1HLe

�ikz

� q2HLe�iðkyþkzÞ;

tH4;L3 ¼ aHLðe�iky � e�iðkxþkzÞÞ; ðA:3Þfor HðkÞH­L, and

tL1;L1 ¼ tL3;L3 ¼ �E þ 2b1L cos ky;

tL2;L2 ¼ tL4;L4 ¼ �E þ 2b2L cos ky;

tL1;L2 ¼ tL3;L4 ¼ 0;

tL1;L3 ¼ pLð1 þ e�iky þ e�ikx þ e�iðkxþkyÞÞ;tL1;L4 ¼ cLðeiky þ eið�kyþkzÞÞ;tL2;L3 ¼ cLðe�iky þ eiðkyþkzÞÞ;tL2;L4 ¼ qLðeiðkxþkzÞ þ eiðkxþkyþkzÞ þ eikz þ eiðkyþkzÞÞ; ðA:4Þ

for HðkÞL­L, where t�;� ¼ t��;�. At the Γ point, the HOMO–LUMO interaction is absent, i.e., HðkÞH­L ! 0.

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