Non-linear transport without spin-orbit coupling or warping in two-dimensional Diracsemimetals

Sai Satyam Samal,1 S. Nandy,2 and Kush Saha1, 3

1National Institute of Science Education and Research, Jatni, Odisha 752050, India2Department of Physics, University of Virginia, Charlottesville, VA 22904, USA

3Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India

It has been recently realized that the first-order moment of the Berry curvature, namely theBerry curvature dipole (BCD) can give rise to non-linear current in a wide variety of time-reversalinvariant and non-centrosymmetric materials. While the BCD in two-dimensional Dirac systems isknown to be finite only in the presence of either substantial spin-orbit coupling where low-energyDirac quasiparticles form tilted cones or higher order warping of the Fermi surface, we argue thatthe low-energy Dirac quasiparticles arising from the merging of a pair of Dirac points without anytilt or warping of the Fermi surface can lead to a non-zero BCD. Remarkably, in such systems, theBCD is found to be independent of Dirac velocity as opposed to the Dirac dispersion with a tilt orwarping effects. We further show that the proposed systems can naturally host helicity-dependentphotocurrent due to their linear momentum-dependent Berry curvatures. Finally, we discuss animportant byproduct of this work, i.e., nonlinear anomalous Nernst effect as a second-order thermalresponse.

Introduction: The Berry phases of electronic wavefunc-tions can substantially modify the transport propertiesand give rise to a plethora of anomalous transport phe-nomena in the linear response regime such as anoma-lous Hall effect (AHE), anomalous Nernst effect (ANE),quantum charge pumping, etc [1] in systems with bro-ken time-reversal symmetry. Recently, it has also beenproposed that the first-order moment of the Berry cur-vature, namely the Berry curvature dipole (BCD) cangive rise to non-linear current in a time-reversal invariantand non-centrosymmetric material [2–4]. Subsequently,the discovery [5, 6] of non-linear anomalous Hall ef-fect (NLAHE) in layered transition metal dichalcogenides(TMDCs) makes the non-linear transport as one of theprime topics of interest to both theorists and experimen-talists [2–4, 7–22] in recent times.

Unlike the linear transport properties induced by theBerry curvature, the BCD-induced non-linear transportphenomena such as non-linear anomalous Hall effect,non-linear anomalous Nernst effect (NLANE) and non-linear anomalous thermal Hall effect (NLATHE) is foundto be Fermi surface quantities [2, 13, 23, 24]. To find finitevalues of these non-linear phenomena, one requires eitherlow-symmetry crystals or Dirac Hamitonian with higherorder corrections. For example, it has been shown thatthe presence of substantial spin-orbit coupling (SOC)where low-energy Dirac quasiparticles form tilted conesis necessary to find a finite NLAHE in two-dimensional(2D) Dirac systems[2]. On the other hand, a recentstudy [25] showed that the NLAHE can survive in 2DDirac semimetals even in the absence of complete SOCbut with higher order correction to the linearly disper-sive Dirac Hamiltonian, particularly with warping of theFermi surface.

This raises an important question to address, perti-nent to several recent and upcoming experiments on 2D

δ0 μ 0 δ0=μ>0 δ0=0<μ δ0<0<μ

FIG. 1. Evolution of energy spectrum (Eq. 1) (top panel) andcorresponding Fermi surface topologies[26] (bottom panel) fordifferent values of parameter δ0. For finite chemical potential(µ), the saddle point between the two Dirac nodes evolveswith δ0 and crosses the chemical potential at δ0 = µ, leadingto topological Lifshitz transition. Note that, for fixed δ0 > 0,the similar topological transition is obtained by varying µ.

materials: is it possible to realize a finite BCD in a sim-ple low-energy 2D Dirac Hamiltonian without any tiltingand warping effect. Remarkbly, we find that the tilt-ing of the Dirac cone or warping of the Fermi surface isnot necessary to observe a non-zero BCD in 2D DSMs.Rather, a simple low-energy Hamiltonian with a pair ofDirac nodes close to each other with a saddle point inbetween the nodes, or a Dirac system where two Diracnodes merge with each other at a point turn out to be asimple platform for realizing sizeable BCD-induced non-linear Hall effect. Further, we find that the BCD is inde-pendent of Dirac velocity but predominantly depends onthe effective mass parameter for a fixed band gap. More-over, these systems can naturally give rise to helicity-dependent photocurrent due to their linear momentum-dependent Berry curvatures, as opposed to the typical

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gapped Dirac and semiconducting systems[3]. We finallydiscuss the contribution of the Berry curvature to thenon-linear anomalous Nernst effect.

Deformed graphene with uniaxial stress turns outto be an ideal platform for realizing such modelHamiltonian[27–29]. It has also been argued thatTiO2/VO2 heterostructures [30, 31] under quantum con-finement, (BEDT-TTF)2I3 organic salts under pressure[32], photonic metamaterials [33] can be ideal candidatematerials to host these types of low energy dispersions.Recent experimental realization of such model Hamilto-nian in optical lattices [34] has renewed the quest formaterials with tunable Dirac nodes.

Model Hamiltonian: The merging of a pair of Diracnodes can be modeled by the low-energy Hamiltonian[29,35]

H = d(k) · σ, (1)

where σ’s are the Pauli matrices in pseudospin space andd(k) = (αk2x − δ0, v ky, 0). Here k = (kx, ky) is the crys-tal momentum, α=~2/2m is the inverse of quasiparticlemass along x, v is the Dirac velocity along y, and δ0is the parameter which drives the transition between ametallic and insulating phase. For δ0 > 0, two gaplessDirac nodes are found at (±

√δ0/α, 0). At δ0 = 0, the

two Dirac nodes merge, leading to a special dispersionwhere electron disperses quadratically along the y direc-tion and linearly along the orthogonal direction. This istypically called semi-Dirac point. For δ0 < 0, a gappedsystem with trivial insulating phase is obtained. The cor-responding spectrum is shown in Fig. (1). Evidently, thesaddle point between the two Dirac nodes evolves with δ0,which in turn leads to the topological Lifshitz transitionfor a fixed chemical potential (µ) (cf. Fig. (1)). Conse-quently, the area of the Fermi surface (SF ) is found tovary nonmonotonically with δ0. This is expected to bereflected in the BCD since it is a Fermi surface quantity.We note that for fixed δ0, the Liftshiz transition can alsobe obtained with the variation of µ.

Note that, Eq. 1 obeys effective time-reversal (Θ = K),particle-hole (P = σy) and chiral symmetries (C = σyK),where K is the complex conjugation operator. Fur-ther, we can define two effective mirror symmetries Mx :(x, y)→ (−x, y) and My(x, y) : (x, y)→ (x,−y). In mo-mentum space, Mx = σ0 and My = σx[36]. It is worthmentioning that Eq. 1 can be effectively obtained from afour band isotropic Dirac Hamiltonian with an anisotropyalong x direction[37] without any tilt.

Berry curvature dipole in 2D Dirac semimetals: Thesimple form of Eq. 1 in terms of fictitious magnetic fieldd(k) allows us to write Berry curvature as

Ωa(k) = εabc1

2|d(k)|3d(k) ·

(∂d(k)

∂kb× ∂d(k)

∂kc

), (2)

where εabc is the usual Levi-Civita and (abc) ∈ (xyz).With this, the Berry curvature dipole as characterized

-π 0 π-π

0

π

kx

k y

a)

-π 0 π-π

0

π

kx

k y

b)

-0.15

-0.10

-0.05

0

0.05

0.10

0.15

-π 0 π-π

0

π

kx

k y

c)

-π 0 π-π

0

π

kx

k y

d)

-0.15

-0.10

-0.05

0

0.05

0.10

0.15

FIG. 2. Top Panel: (a) Contour plot of Berry curvature forisotropic gapped Dirac Hamiltonian. (b) The same plot forthe gapped semi-Dirac Hamiltonian discussed in the text. Ev-idently, Ω(−kx,−ky) = −Ω(kx, ky) as a manifestation of time-reversal symmetry. Lower Panel: (c) and (d) are the contourplots of the derivatives of Ω with respect to ky and kx, re-spectively. Since ∂yΩ(kx, ky) = −∂yΩ(kx,−ky), the integralover ky at zero temperature is found to be zero, leading tozero Berry curvature dipole along y direction. In contrast,∂xΩ(kx, ky) = ∂xΩ(−kx, ky) which in turn leads to finiteBerry dipole. The white spaces in all plots represent verylarge values of Berry curvatures (peaks or dips in 3D plots).

by the first-order moment of the Berry curvature overthe occupied states is defined as

Dab =

∫k

fo(∂aΩb), (3)

where∫k

=∫ddk/(2π)d and fo is the Fermi-Dirac dis-

tribution function. In three-dimension (3D), the Berrycurvature is a pseudovector and consequently, the BCD(Dab) becomes a pseudotensor. On the other hand, in thecase of a two-dimensional system, the only nonzero com-ponent of the Ω is Ωz(k) (a = z), indicating the fact thatthe Berry curvature behaves as a pseudoscalar. Thusin 2D, the pseudotensorial quantity Dab is reduced toa pseudovector quantity (Da) confined in the 2D planewith only two independent (x and y) components. Wepoint out that for finite Berry curvature dipole to exist,the system must have at most one mirror symmetry leftwhere the single mirror symmetry forces BCD to be or-thogonal to the mirror line (i.e. a mirror plane that isperpendicular to the 2D system).

It is easy to see from Eq. (2) that Ω(k) = 0 for boththe bands in Eq. (1) since dz = 0. This is attributed tothe fact that Eq. (1) preserves all relevant symmetriesprotecting the gapless point. To generate finite Berrycurvature, we introduce a perturbation δh = m0 σz toH, which breaks both P and My symmetry but preserves

3

Material m/me m0 (eV) µ (eV) Dx(nm)

(TiO2)5/(VO2)3 13.6 0.2 0.25 0.27

α-(BEDT-TTF)2I3 3.1 0.1 0.15 0.86

Photonic crystals 1.2× 10−3 1.0 1.5 13.0

TABLE I. Microscopic parameters for semi-Dirac materials,with representative gaps (m0), chemical potentials, effectivemasses (m) and corresponding Berry curvature dipoles[30, 33,38]. Here me represents the free electron mass. Evidently,the Dx increases with the decrease in effective band massesas explained in the main text.

Mx and time-reversal symmetry. This gives

Ω(k) =β kxE3

k

, (4)

where Ek =√|d(k)| and β = 2α vm0. Since Ω(k) =

−Ω(−k), the integral of the Berry curvature over the en-tire Brillouin zone, namely the Chern number turns outto be zero, hence we obtain zero linear anomalous Hallconductivity. It is also apparent from the right top panelof Fig. (2) . For comparison, we have also shown Berrycurvature for a isotropic Dirac Hamiltonian (σ ·k) in theleft top panel of Fig. (2). Since Ω(k) = Ω(−k), the inte-gral of Ω over the entire Brillouin zone is expected to befinite. However, whether the total Chern number finite orzero depends on the contribution coming from all inequiv-alent Dirac nodes of the respective lattice model. Forexample, in graphene, the perturbation δh, namely ”Se-menoff mass” arising from the staggered on-site potentialbetween two sublattices gives rise to insulting phase asthe Chern numbers for the two inequivalent gapped Diracnodes are equal but opposite in sign. We note that suchmass term in deformed graphene and other possible can-didate materials can be induced by light with differentpolarization.

At zero temperature, the BCD can be computed us-ing Eq. 3, where the momentum integral is restricted tothe region Ek < µ. For the present model, the Fermisurface topology adds complexities in finding the ana-lytical results using Eq. 3. Thus we rewrite Eq. (3) asDi = −

∫kvi Ωz f ′0, where vi = ∇kiEk with i ∈ (x, y) and

′ denotes the derivative with respect to the energy. Withthis, we obtain

Dx = 2m0

√α

√µ2 −m2

0

µ3I(µ, δ0),

Dy = 0, (5)

where I(µ, δ0) =∫ 2π

0dθ(F+ − F−Θ(F−

2))| cos θ| with

F s =(s√

(µ2 −m20)| cos θ|+ δ0

)1/2, Θ(x) is the usual

Heaviside function. For δ0 = 0, I(µ, δ0) can be further

0.2 0.3 0.4 0.5 0.6µ (eV)

0

0.5

1

1.5

2

Dx (n

m)

δ0=0.0δ0=0.1δ0=0.2δ0=0.3δ0=0.4

0 0.1 0.2 0.3 0.4 0.5δ0 (eV)

10

15

20

25

30

S F (nm

-2)

µ=0.2

FIG. 3. Berry curvature dipole (BCD) as a function of chemi-cal potential for different values of δ0. Clearly for low doping,the BCD increases as the Dirac nodes move close to each other(δ0 → 0). This is attributed to the enhancement of the areaof the Fermi surface (SF ) as evident from Fig. (1) and theinset of Fig. 3. However, the Berry dipole suddenly reducesas the two Dirac nodes merge (δ0 = 0.0) into a single one,corroborating the nonmonotonic nature of the SF (see inset).For high doping, the SF does not change substantially withδ0, leading to similar Dx. Here, we have used m0 = 0.1 eVand m/me = 13.6 for (TiO2)5/(VO2)3.

simplified and Eq. 5 reads off

Dx = 2m0

√α I0

(µ2 −m20)3/4

µ3, (6)

where I0 ' 3.5. To obtain Eq. (5), we have used

the parametrization kx = sign[cos θ](r | cos θ|+δ0

α

)1/2and

ky = r sin θv [39]. Note that the Berry curvature dipole

along x survives due to the surviving mirror symmetryMx while Dy = 0 as a consequence of broken My. More-over, Dx vanishes if µ lies inside the bulk gap i.e. µ ≤ m0.Notice that Dx is independent of Dirac velocity v, whichis in sharp contrast to the BCD for typical Dirac dis-persion with a tilt or warping terms[2, 25]. Also, it isindeed apparent that for materials with small band gapand small effective mass, the BCD is expected to be verylarge. In Table I, we have presented typical mass param-eters with representative gaps and corresponding Berrycurvature dipoles for proposed semi-Dirac materials. Ev-idently, the BCD in a semi-Dirac material is comparableor larger than the BCD in materials with tilting such asSnTe (D ∼ 3nm), TMDCs (D ∼ 10−2 nm )[2] or mate-rials with warping such as graphene (D ∼ 10−3 nm)[25].

Figure 3 demonstrates the Berry curvature dipole withthe variation of δ0. For low doping, Dx increases as δ0decreases. However, at δ0 = 0, the Dx suddenly reducescompared to the Dx at δ0 > 0 as apparent from the

4

0.2 0.4 0.6µ (eV)

0

2

4

6S x (n

m-K

B2 )

δ0=0.0

δ0=0.1

δ0=0.2

δ0=0.3

δ0=0.4

0.2 0.4 0.6µ (eV)

-0.2

0

0.2

0.4

0.6

0.8δ0=0.0

δ0=0.1

δ0=0.2

δ0=0.3

δ0=0.4

T=5 K T=200 K

a) b)

FIG. 4. a) Nonlinear Nernst coefficient (Sx) as a function ofchemical potential (µ) for different values of δ0. It is apparentthat Sx mimics the Dx even in the presence of additional term(Ek −µ)2 at low temperature (T=5 K). b) The same plot forT=200 K. All parameters are used same as Fig. 3.

Fig. (3). The is because the area of the Fermi surface,SF changes nonmonotonically with δ0 as illustrated inthe inset of Fig. (3). This behavior can further be under-stood from the Fig. (1). For finite µ in the conductionband, we have a single Fermi surface at δ0 = 0 since theband has only one minima (refers to semi-Dirac node).As we move away from δ0 = 0, the single minima splitsinto two minimas (refers to Dirac nodes) and a saddlepoint appears between them. However, the single Fermisurface retains as long as δ0 <

√µ2 −m2

0. Accordingly,the area increases due to the additional curvature arisingfrom the spliting of the single minima. As we further in-crease δ0, the saddle point crosses µ at δ0 =

√µ2 −m2

0,leading to two Fermi surfaces. Consequently, the areastarts to decrease. If we increase δ0 even more, the bandsnear the Dirac nodes become narrower, which results inthe reduction of SF . For high doping, the Fermi surfacetopology almost remains same irrespective of the valuesof δ0, leading to similar Dx. Note that this feature ofBCD differs from the case for fixed δ0 but varying chem-ical potential where SF monotonically increases with µ(not shown here). We also note that the BCD is foundto have a little kink at µ =

√δ20 +m2

0 as a manifestationof topological Lifshitz transition.

Non-linear photo current: We next move to the Berrycurvature contribution to the dc photocurrent. In thepresence of an external electric field (Eeiωt) with E ∈ C,the non-linear current is produced due to the Berry phaseof Bloch electrons. In particular, the anomalous velocityof the Bloch electrons gives rise to a net current:

j = e

∫k

[k×Ω(k)

]fneq(k), (7)

where fneq(k) is the non-equilibrium electron distribu-

tion function and k = Re ( e~Eeiωt). In the relaxation

time approximation with energy-independent scatteringtime τ , the non-equilibrium electron distribution is ob-

tained to be[3]

fneq(k) = 2 τ e f ′0(Re E − ω τ Im E) · v

1 + ω2 τ2(8)

Assuming low temperature and low frequency, Eq. (7)together with Eq. (8) leads to

j =χ

1 + ω2 τ2

[1

τ[EyE∗x ]+x+ i ω [EyE∗x ]−x−

2

τ|Ex|2 y

],

(9)

where [AB∗]± = AB∗ ±A∗B and χ is obtained to be

χ =e3

~

∫d2k

2πvx Ωz(k) δ(µ− Ek) =

e3

~Dx. (10)

The first two terms in Eq. (9) are known as linear(LPGE) and circular (CPGE) photogalvanic currents, re-spectively. The last term in Eq. (9) denotes typical pho-tovoltaic effect. It is clear that the CPGE changes signwith the helicity of the light wave and it is maximum forcircularly polarized light. In contrast, the LPGE is dom-inant and maximum for linearly polarized light. We fur-ther see that the non-linear conductivity is nothing butthe Berry curvature dipole, corroborating the relationbetween photocurrent and Berry dipole as expected[2].For a laser power of 1 Watt, ω τ = 1, m = 13.6me

((TiO2)5/(VO2)3), the current density is found to beroughly of the order of 300nA/mm, which can be eas-ily measured in standard experiments.

To this end, we note that the anomalous velocity as-sociated with the Berry curvature gives rise to helicity-dependent photocurrent if Ω(k) ∝ k as pointed outin Ref. [3]. It has been shown that while the Berryphase contribution vanishes in the bulk of a semicon-ductor quantum well, the surface confinement leads tothe helicity-dependent photocurrent due to the particularnature of the Berry curvature. Interesting, our presentmodel naturally gives rise to the helicity-dependent pho-tocurrent without any external perturbation as Ω =β kx/E

3k (cf. Eq. 4), in contrast to the typical Dirac

dispersion.Non-linear Anomalous Nernst Effect: Next, we turn to

non-linear anomalous Nernst effect in this system. Usingthe semiclassical Boltzmann theory within the relaxationtime approximation, the non-linear anomalous Nernst co-efficient at temperature (T ) can be defined as [13, 23]

Si =

∫k

(Ek − µ)2

T 2vi Ω(k)f ′0. (11)

The NLANE (second-order response function to the ap-plied temperature gradient), which refers to the nonlinearcurrent, flowing perpendicular to the temperature gradi-ent even in the absence of a magnetic field, is induced bythe BCD and therefore, only x-component of it i.e., Sxis finite in this system. In Fig. (4), we show the behavior

5

of Nernst coefficient at two different temperatures T = 5K and T = 200 K. The qualitative feature turns to besame as BCD at low temperature even in the presence ofthe additional term (EK − µ)2 in Eq. (11). However, athigh temperature it deviates substantially from the lowtemperature behavior as evident from Fig. (4). Noticethat in both cases, the non-monotonicity of the area ofthe Fermi surface as discussed before is reflected.

Conclusion: To conclude, we have identified a simpleplatform to observe substantial non-linear transport phe-nomena arising from the Berry curvature dipole (BCD).Specifically, we have shown that the sizable BCD canbe obtained in a low-energy Dirac Hamiltonian with twoDirac nodes close to each other or their merging at asingle node, namely the semi-Dirac node. Remarkably,the BCD for the present model is found to be indepen-dent of the Dirac velocity and predominantly depends onthe inverse of effective mass of the Dirac quasiparticles ascompared to the typical isotropic Dirac systems with wellseparated Dirac nodes. Indeed, this is one of the interest-ing findings of this study. For typical Dirac materials, vis of the order of 105m/s and does not vary significantlyfrom one material to another. However, the effectivemasses may vary significantly as evident from the Ta-ble I of possible semi-Dirac systems. This fact may guideus to identify materials with low effective masses respon-sible for a sizable BCD than the only velocity dependentBCD. For example, the candidate semi-Dirac materialshave very low effective masses (Tabel. I), hence they arepotential platforms to observe BCD-induced large non-linear transport phenomena. We further show that thepresent model naturally host the helicity-dependent pho-tocurrent due to the linear momentum-dependent Berrycurvature in contrast to the isotropic Dirac dispersion.Finally, we present non-linear Nernst effect, arising solelydue to the Berry curvature effect. Since the NLAHE cantransform ac electric fields into dc currents, a processknown as rectification[40], the proposed Dirac materialsmay have great potential applications for next-generationwireless and energy-harvesting devices.

Acknowledgement: KS is thankful to A. Jaiswal foruseful discussion.

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