Magnetic Properties of Dirac Fermions in a Buckled Honeycomb Lattice
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Magnetic Properties of Dirac Fermions in a Buckled Honeycomb Lattice
C. J. Tabert , , , J. P. Carbotte , , and E. J. Nicol , , Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1 Canada Guelph-Waterloo Physics Institute, University of Guelph, Guelph, Ontario N1G 2W1 Canada Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 USA Department of Physics, McMaster University, Hamilton, Ontario L8S 4M1 Canada and Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8 Canada (Dated: June 20, 2018)We calculate the magnetic response of a buckled honeycomb lattice with intrinsic spin-orbit cou-pling (such as silicene) which supports valley-spin polarized energy bands when subjected to aperpendicular electric field E z . By changing the magnitude of the external electric field, the sizeof the two band gaps involved can be tuned, and a transition from a topological insulator (TI) toa trivial band insulator (BI) is induced as one of the gaps becomes zero, and the system enters avalley-spin polarized metallic state (VSPM). In an external magnetic field ( B ), a distinct signatureof the transition is seen in the derivative of the magnetization with respect to chemical potential( µ ) which gives the quantization of the Hall plateaus through the Streda relation. When plottedas a function of the external electric field, the magnetization has an abrupt change in slope at itsminimum which signals the VSPM state. The magnetic susceptibility ( χ ) shows jumps as a functionof µ when a band gap is crossed which provides a measure of the gaps’ variation as a function ofexternal electric field. Alternatively, at fixed µ , the susceptibility displays an increasingly largediamagnetic response as the electric field approaches the critical value of the VSPM phase. In theVSPM state, magnetic oscillations exist for any value of chemical potential while for the TI, andBI state, µ must be larger than the minimum gap in the system. When µ is larger than both gaps,there are two fundamental cyclotron frequencies (which can also be tuned by E z ) involved in thede-Haas van-Alphen oscillations which are close in magnitude. This causes a prominent beatingpattern to emerge. PACS numbers: 72.80.Vp, 75.60.Ej, 71.70.Di, 73.43.-f
I. INTRODUCTION
With the successful isolation of graphene in 2004, two-dimensional (2D) systems began to attract considerableattention. While graphene provides a platform for inves-tigating the physics of massless relativistic fermions ,other 2D crystals are increasing in popularity as theypromise to exhibit exciting phenomenon beyond thosefound in the single layer of graphite. One of the moreprominent extensions of graphene research has been the2D topological insulators (TIs) introduced in the semi-nal paper by Kane and Mele . In a 2D TI, the nontrivialtopology of the band structure allows helical edge chan-nels to exist at the boundary of the sample. These chan-nels are topologically protected, and are robust againsttime-reversal-symmetry preserving impurities. Recently,Hg x Cd − x Te quantum wells have shown signatures of thehelical edge-conduction .Another promising class of 2D TIs is the low-buckledhoneycomb lattice with intrinsic spin-orbit coupling. Atlow energy, these systems are expected to map onto theKane-Mele Hamiltonian for the quantum spin-Hall in-sulator (QSHI) . Two candidate materials are silicene(with a spin-orbit band gap of ∆ so ≈ . − . ),and germanene (∆ so ≈ −
93 meV ). The low-buckling of the honeycomb lattice causes the A, and Bsublattices to sit in vertical planes with a separation dis-tance d ( ≈ . ). In the presence ofan external electric field oriented perpendicular to the system, the A-B asymmetry causes a potential differ-ence ∆ z = E z d to arise between the sublattices. Thisspin splits the energy bands, and allows the resultingband gaps to be tuned. It has been argued that,as ∆ z becomes greater than ∆ so , the system transitionsfrom a TI to a trivial band insulator (BI). At the crit-ical value ∆ z = ∆ so , the lowest band gap closes into aDirac point, and the system is referred to as a valley-spinpolarized metal (VSPM) . For ∆ so =8 meV, the criticalelectric field associated with the VSPM is E z ≈ . ).Recent theoretical studies have examined the effect ofvarying ∆ z on the AC conductivity , magneto-opticalconductivity , spin, and valley Hall effects , po-larization function , anomalous spin Nernst effect ,and quantum oscillations .The magnetization of Dirac-like materials has longbeen known to be anomalous . A recent study showed how the orbital susceptibility of massless Diracfermions on a 2D T lattice evolves from diamag-netic to paramagnetic as a function of the couplingstrength between the honeycomb lattice, and an addi-tional carbon atom placed at the center of each hexagon(i.e. going from the graphene lattice to the dice lat-tice). For undoped graphene, the susceptibility showsa divergence as a function of the inverse magneticfield. This is lifted by finite temperature or other com-plications; these include the formation of a gap whichleads to massive Dirac fermions, effects of disorder , orby finite doping away from the charge neutral point .In the low-buckled honeycomb lattice discussed herein,we work with massive Dirac fermions except for the spe-cial case of the VSPM for which one of the gaps closes.In this paper, we study the magnetic properties of sucha system with particular emphasis on the changes inmagnetization as one goes from the nontrivial to trivialtopological phase.Our paper is organized as follows: In Sec. II, we presentthe theoretical background for the low-energy model of abuckled honeycomb lattice with intrinsic spin-orbit cou-pling. Section III contains a discussion of the grandthermodynamic potential, and results for the magneti-zation as a function of µ . We compute the µ derivativeof the magnetization which is related to the quantizedHall conductivity through the Streda formula. We alsoexplore the plateaus which emerge in the integrated den-sity of states, and optical spectral weight for finite mag-netic field. In Sec. IV, we extend the results of gappedgraphene to our system, and examine the magnetization,and magnetic susceptibility. Finite temperature, and im-purity effects are also considered. Section V contains ourresults for the magnetic oscillations at low field values.Our conclusions follow in Sec. VI. II. LOW-ENERGY BAND STRUCTURE
In the presence of an external electric field, the physicsof a low-buckled honeycomb lattice with intrinsic spin-orbit coupling is well described by a simple nearest-neighbour-tight-binding Hamiltonian . At lowenergy, an effective Hamiltonian can be written to de-scribe the physics in the vicinity of the two K pointsof the hexagonal first Brillouin zone. This Hamiltonianis of the well known Kane-Mele type for describing theQSHI. Written for a single spin, and valley,ˆ H ξσ = ¯ hv ( ξk x ˆ τ x + k y ˆ τ y ) − ξσ
12 ∆ so ˆ τ z + 12 ∆ z ˆ τ z , (1)where ˆ τ i are Pauli matrices associated with the pseu-dospin of the system. The two valleys K , and K ′ are in-dexed by ξ = ±
1, respectively, with ¯ hk x , and ¯ hk y beingthe momentum components measured relative to the K points. The real spin of the electrons is given by σ = ± v ( ≈ × m/s for silicene ). Includingthe first two terms, one obtains the Kane-Mele Hamilto-nian for intrinsic spin-orbit coupling leading to a spin-orbit band gap of ∆ so . The last term is associated withthe on-site potential difference ∆ z which arises betweenthe two sublattices when the system is subjected to anexternal electric field applied perpendicular to the planeof the lattice . As a matrix, Eqn. (1) is,ˆ H ξσ = (cid:18) ∆ ξσ ¯ hv ( ξk x − ik y )¯ hv ( ξk x + ik y ) − ∆ ξσ (cid:19) , (2) where ∆ ξσ = (1 / − ξσ ∆ so + ∆ z ), with | ∆ ++ | ≡ ∆ min ,and | ∆ + − | ≡ ∆ max . The eigenvalues of Eqn. (2) are ε ξσ = ± q ¯ h v k + ∆ ξσ . (3)A schematic representation of the band structure at the K point is given in Fig. 1 for varying ∆ z . Figure 1(a) FIG. 1. (Color online) Schematic representation of the low-energy band structure of silicene about the K point for vary-ing electric field strength. The purple bands represent spindegeneracy; the blue bands correspond to spin-up; and thered bands describe spin-down. Finite ∆ z spin splits the en-ergy bands. For ∆ z < ∆ so , the system is a TI. For ∆ z > ∆ so ,the system is a trivial BI. At the critical value ∆ z = ∆ so , thelowest gap closes, and the system is a VSPM. The spin labelsare reversed at the K ′ point. shows the low-energy dispersion for the Kane-Mele QSHI(∆ z = 0). The two purple bands are spin degenerate, andseparated by a gap of ∆ so . For finite ∆ z < ∆ so , the sys-tem remains a TI; however, the bands become spin split.The gap of the spin-up band (blue) decreases while thatof the spin-down band (red) increases [see Fig. 1(b)]. Thecritical value ∆ z = ∆ so is shown in Fig. 1(c). Here, thelowest band gap closes forming a Dirac point while theupper gap has continued to increase. For all ∆ z > ∆ so ,the system is a trivial BI [Fig. 1(d)]. With the reopen-ing of the lowest gap, a band inversion occurs associatedwith a change in the pseudospin label of the two lowestgapped bands. Now both energy gaps increase with ∆ z ,and the energy separation between the two spin bands at k = 0 remains at ∆ so . Throughout this paper, we willtake ∆ so = 8 meV. At the K ′ point, identical behaviouris observed; however, the spin labels are interchanged.Using a tight-binding hopping amplitude of t = 1 . ,the low-energy approximation is quite accurate for ener-gies between ±
800 meV .To discuss the magnetic properties of silicene, we mustfirst consider the effect of an external magnetic field B on the band structure. By choice, we orient B in the z direction, and work in the Landau gauge. Thus, themagnetic vector potential, given by B = ∇ × A , is writ-ten as A = ( − By, , hk i → ¯ hk i + eA i , (4)where we work in units of c = 1. The low-energy Hamil-tonian becomesˆ H ξσ = (cid:18) ∆ ξσ − g s µ B Bσ ¯ hv (cid:0) ξ (cid:2) k x − eB ¯ h y (cid:3) − ik y (cid:1) ¯ hv ( ξ (cid:2) k x − eB ¯ h y (cid:3) + ik y ) − ∆ ξσ − g s µ B Bσ (cid:19) , (5)where we include the Zeeman interaction − (1 / g s µ B Bσ z in Eqn. (1); g s is the Zeeman cou-pling strength which we take to be
23 to elucidatethe interesting features; µ B = e ¯ h/ (2 m e ) ≈ . × − meV/T is the Bohr magneton with m e , the mass of anelectron. For silicene, the Zeeman energy is small, andusually ignored . Retaining the Zeemaninteraction, Eqn. (5) can be solved to give the Landaulevel dispersion E ξσN,s = ( − g s µ B Bσ + s q ∆ ξσ + 2 N E , N = 1 , , , ... − ξ ∆ ξσ − g s µ B Bσ, N = 0 , (6)where s = ± is a band index, and E ≡ √ ¯ hev B ≈ . B = 1T. Note: in a sum over s = ± , there isonly a single contribution from the N = 0 levels. III. MAGNETIZATION AND THE HALLEFFECT
Our examination of the magnetic response of silicenebegins with the grand thermodynamic potential Ω( T, µ ) = − T Z ∞−∞ N ( ω )ln (cid:16) e ( µ − ω ) /T (cid:17) dω, (7)where T is the temperature, and N ( ω ) is the density ofstates. Note that we have taken k B = 1. In the absenceof impurity scattering, the density of states in a magneticfield is given by a series of Dirac-delta functions locatedat the Landau level energies. For a single ξ , and σ , N ξσ ( ω ) = eBh δ (cid:16) ω − E ξσ (cid:17) + ∞ X N =1 s = ± δ (cid:16) ω − E ξσN,s (cid:17) . (8)As will be discussed further on, the quantities of inter-est depend on a derivative of the grand potential withrespect to B ; thus, for convenience, we choose to add( µ/ R ∞−∞ N ( ω ) dω to Eqn. (7). Since the integral of thedensity of states over all energies gives the total num-ber of states (which is independent of B ), this term will not contribute to the magnetization ( − ∂ Ω /∂B ). At zerotemperature, Eqn. (7) then becomesΩ( µ ) = Z −∞ (cid:16) ω − µ (cid:17) N ( ω ) dω + Z µ ( ω − µ ) N ( ω ) dω + µ Z ∞ N ( ω ) dω. (9)For silicene, Z ∞ N ξσ ( ω ) dω = Z −∞ N ξσ ( ω ) dω − eBh Υ , (10)where (for realistic values of Zeeman splitting)Υ = − σ ∆ z < ∆ so ξ − σ z = ∆ so , g s = 0 ξ ∆ z = ∆ so , g s > ξ ∆ z > ∆ so . (11)Therefore,Ω ξσ ( µ ) = Z µ ( ω − µ ) N ξσ ( ω ) dω − eBµ h Υ + Z −∞ ωN ξσ ( ω ) dω. (12)The final term (which does not depend on µ ) gives thevacuum contribution, and will simply provide a constantbackground to the µ dependence of the magnetization. Inthe absence of impurity scattering, when summed over ξ and σ , the first two terms of Eqn. (12) give˜Ω( µ ) = eBh X ξ,σ = ± (cid:20) (cid:16) E ξσ − µ (cid:17) Θ (cid:16) µ − E ξσ (cid:17) Θ (cid:16) E ξσ (cid:17) − µ ∞ X N =1 (cid:16) E ξσN, + − µ (cid:17) Θ (cid:16) µ − E ξσN, + (cid:17) , (13)where we have used Eqn. (8) for the density of states,and have assumed that all the s = − states are negative.It is important to note that Θ(0) ≡ / ω = 0 is integrated. A plotof the magnetization derived from Eqn. (13) is shown inFig. 2 for ∆ z = 4 (solid black), and 8 meV (dashed red).In both cases, a jagged saw-tooth oscillation is seen. For∆ z = 4 meV, the system is in the TI phase and, thus,two N = 0 Landau levels are at positive energy. For lowchemical potential, two kinks are seen in M ( µ ) at µ = 2,and 6 meV associated with the spin-up N = 0 levelsat K , and K ′ , respectively. For the VSPM, only oneLandau level exists for ω > µ = 8 meV. When µ is greater than the N = 0 levels, the usual saw-tooth behaviour is present.The jumps occur at the Landau level energies of all the N >
FIG. 2. (Color online) Magnetization as a function of µ inthe TI phase (solid back curve), and VSPM regime (dashedred curve) for g s = 0. Inset: the slope of the magnetizationas a function of µ which is related to the Hall conductivitythrough ∂M/∂µ = (1 /e ) σ H . Note: in the TI regime, the Hallconductivity is zero for µ less than ∆ min . In the VSPM, afinite σ H persists to µ →
0. Plateaus have been offset fromtheir integer values for clarity. to that of the TI; however, the two low-energy kinks areassociated with the spin-down, and -up N = 0 levels at K ′ .The slope of the magnetization is of interest as it isrelated to the robust quantization of the Hall conductiv-ity through the Streda formula ∂M ( µ ) /∂µ = (1 /e ) σ H .The slope is shown in the inset of Fig. 2. For the TI, noLandau levels exist at zero energy, and thus the Hall con-ductivity is zero until µ reaches the energy of the lowestlevel. At this energy, it steps up by one unit of e /h un-til the next level at which point it increments by another e /h . For the VSPM, there does exist a level at chargeneutrality, and thus the slope of the magnetization is fi-nite for zero chemical potential. The BI behaves similarto the TI.The quantization of ∂M/∂µ can be seen by taking the µ derivative of Eqn. (13). For µ >
0, we find ∂ ˜Ω ∂µ = eBh X ξ,σ = ± (cid:20) − Θ (cid:16) µ − E ξσ (cid:17) Θ (cid:16) E ξσ (cid:17) − Υ2 − ∞ X N =1 Θ (cid:16) µ − E ξσN, + (cid:17) , (14)and, the slope of the magnetization is then − ∂∂B ∂ ˜Ω ∂µ ≡ ∂M∂µ = eh X ξ,σ = ± (cid:20) Θ (cid:16) µ − E ξσ (cid:17) Θ (cid:16) E ξσ (cid:17) + Υ2+ ∞ X N =1 Θ (cid:16) µ − E ξσN, + (cid:17) . (15)For the TI, and BI regimes in the absence of Zeemansplitting, the Hall conductivity σ H = ( e /h ) ν has fillingfactors ν = 0 , ± , ± , ± , ± , ... ; while, the VSPM hasfilling factors ν = ± , ± , ± , ± , ... . If ∆ z = 0, the fa-miliar values of gapped graphene ( ν = 0 , ± , ± , ± ... )are retained. For ∆ so = 0, and ∆ z = 0, the famousgraphene values ν = ± , ± , ± , ... are observed.Zeeman splitting breaks the valley degeneracy of the N > e /h . Figure 3(a) shows theslope of M ( µ ) in the presence of Zeeman interactions.The steps associated with the N = 0 levels are shifted[higher(lower) in µ for spin-up(down)]. The steps athigher µ are split into four integer plateaus which areclose in energy. For g s = 0, the four steps would re-duce to two steps of height 2 e /h . The inset shows aschematic of the Landau level energies for a TI at thetwo valleys K , and K ′ . The solid blue lines representspin-up while the dashed red lines correspond to thespin-down levels. µ = 0 is given by the dotted blackline. The effect of Zeeman splitting in gapped graphenehas previously been examined . For graphene with gap∆, there are two spin-degenerate N = 0 Landau levelsat − ξ ∆. Zeeman interactions split the degeneracy, andfour N = 0 plateaus are seen . The four-fold valley-spin-degenerate N > . This is equivalentto the limit ∆ so = 0, and ∆ z / T = 0. A finite temperature will smearthe edges of the steps on an energy scale associated with k B T , but will leave the quantization unaffected. There-fore, a very low temperature is required to see the finestructure, due to the Zeeman interaction, which is seenabove the central quartet of plateaus.Figure 3(b) shows the spin contributions to the Hallconductivity [solid blue(dashed red) for spin-up(down)].A negative Hall conductivity corresponds to a net edgecurrent in the opposite direction to that of the positiveHall effect. Thus, while at µ = 0 the charge Hall conduc-tivity is zero, there is a net spin Hall effect with a spin-up conductance of 2 e /h in the negative direction. For µ between the two spin-up N = 0 Landau levels, only the FIG. 3. (Color online) The slope of the magnetization whichgives the same integer values as the Hall conductivity. (a)The charge Hall conductivity as a function of µ in the TIregime. Four steps are seen near µ = 0 which are associatedwith the N = 0 Landau levels. A Zeeman interaction splitsthe two step feature of each of N = 1 , , , .. into a quartetof four plateaus. The Landau levels at K , and K ′ are shownin the inset for finite g s . Solid blue corresponds to spin-up,and dashed red to spin-down. (b) Spin contributions to thecharge Hall conductivity. (c) Valley contributions to the Halleffect. spin-down electrons contribute to the Hall effect. When µ is between the two spin-down N = 0 levels, only thespin-up electrons contribute. We note that for finite Zee-man splitting, the quartet of steps associated with eachof the N = 1 , , , ... levels also provides a spin imbal- ance. While the ν = ± N = 1 levels provide a spin-polarized response. This is lost when µ becomes greaterthan the remaining N = 1 levels. The pattern continuesfor higher values of N . Without a Zeeman interaction,all the N = 0 steps are spin degenerate.Figure 3(c) shows the valley contributions to the Halleffect (solid green for K , and dashed orange for K ′ ). Atcharge neutrality, there is no net valley conductivity. For µ between the two positive N = 0 Landau levels, onlythe K point contributes; while, for µ between the twonegative N = 0 levels, only K ′ contributes. Combiningthis result with those of Fig. 3(b), we note that for µ between the two spin-up N = 0 Landau levels, the finitecharge Hall effect is accompanied by finite spin-down,and momentum K spin, and valley Hall effects, respec-tively. Likewise, for µ between the two spin-down N = 0levels, there is a finite negative spin-up, and momentum K ′ spin, and valley Hall effect in addition to the chargeHall conductivity. Again, we note that Zeeman splittingallows for a valley polarized response from the N > g s = 0 are given byFigs. 4(c), (d) and (e), (f), respectively. Schematic rep-resentations of the Landau levels in these regimes can beseen in Figs. 4(a), and (b), respectively. Again, solid bluelines represent spin-up, and dashed red lines correspondto spin-down. The charge Hall effect, is given by the sumof the individual spin/valley conductivities. Therefore, inthe VSPM phase, a finite charge Hall effect is present at µ = 0; like the TI, there is no charge Hall conductivityat µ = 0 for the BI. The spin, and valley contributionsto the VSPM Hall effect [see Figs. 4(c), and (d), respec-tively] reveal that for 0 < µ < E K ′ ↑ , the finite charge Hallconductivity is comprised of spin-down, and momentum K fermions. For E K ↓ < µ <
0, the charge Hall conduc-tivity is made of spin-up, and momentum K ′ fermions.In the BI regime [see Figs. 4(e), and (f)], while there isno charge Hall effect for µ = 0, there is a finite valleyHall effect for E K ↑ < µ < E K ′ ↓ . For E K ′ ↓ < µ < E K ′ ↑ ,the charge Hall conductivity is made of spin-down, andmomentum K electrons. For negative µ between E K ↓ ,and E K ↑ , the Hall effect is comprised of spin-up, andmomentum K ′ electrons.To summarize, the total Hall conductivity at small µ is zero for the TI, and BI; however, a finite spin Hall ef-fect is present in the TI regime while an analogous valleyHall effect is found in the BI regime. The finite chargeHall conductivity is also accompanied by spin-valley Halleffects for various values of µ . Including Zeeman interac-tions results in a four step feature associated with each ofthe N > | µ | is small in the VSPM phase,the total Hall conductivity is finite, and carries the signof the chemical potential. Spin, and valley Hall effectsare also seen. Realistically, the Zeeman interaction will FIG. 4. (Color online) The valley separated Landau levelspectrum for the (a) VSPM, and (b) BI regimes for g s = 0.Solid blue lines represent spin-up, and dashed red correspondto spin-down. (c) Spin, and (d) valley contributions to theHall conductivity for ∆ z = ∆ so . Note that a valley-spin-polarized Hall effect persists to µ →
0. (e) Spin, and (f)valley contributions to the Hall conductivity for the BI. Notethat a valley-spin-polarized Hall effect is possible; however,there is no charge Hall response for µ → shift the Landau levels by much less than the inter-level spacing . Therefore, in what follows, we ignore the Zee-man term as its effects are negligible.As the filling factors, and valley contributions to theHall effect calculated from the magnetization are in con-trast to what is predicted by Tahir and Schwingenschl¨oglfor the DC Hall conductivity , we have verified ourresults by direct calculation of σ xy (Ω →
0) given byEqn. (14) in Ref. . Reference used the Kubo formulaand, thus, represents an independent check. In the DClimit Ω → µ >
0, we find σ xy ( µ ) = − e h E X ξ,σ = ± ∞ X n,m =0 s = ± ξ Θ (cid:0) µ − E ξσm,s (cid:1) − Θ (cid:16) µ − E ξσn, + (cid:17)(cid:16) E ξσn, + − E ξσm,s (cid:17) × h ( A m,s B n, + ) δ n,m − ξ − ( B m,s A n, + ) δ n,m + ξ i , (16)where A n,s = s q |E ξσn,s | + s ∆ ξσ q |E ξσn,s | , n = 0 , − ξ , n = 0 , (17)and B n,s = q |E ξσn,s | − s ∆ ξσ q |E ξσn,s | , n = 0 , ξ , n = 0 , (18)which gives the same quantized plateaus as Eqn. (15)after multiplication by a factor of e . A. Integrated Density of States and OpticalSpectral Weight
A similar quantization of plateaus is obtained for theintegrated density of states. We define the total inte-grated density of states up to energy ω max as I ( ω max ) = X ξ,σ = ± Z ω max N ξσ ( ω ) dω, (19)where N ξσ ( ω ) is given by Eqn. (8). In Fig. 5, I ( ω max )is plotted as a function of the cutoff energy ω max .Figure 5(a) applies to the TI phase for which the twogaps ∆ min and ∆ max are both finite, and equal to 2 and6 meV, respectively. Three values of magnetic field arepresented: B = 1T (solid red curve), B = 1 .
5T (dashedblue curve), and B = 0T (dash-dotted black curve). Webegin our discussion with B = 0T. Since all the electronicstates are gapped, I ( ω max ) = 0 for ω max < ∆ min =2meV. There is a change in slope at ω max = ∆ max = 6 FIG. 5. (Color online) (a) Integrated density of states forfinite ∆ z < ∆ so and B =0 (dash-dotted black curve), B = 1T(solid red), and B = 1 .
5T (dashed blue). For B = 0, theintegrated density of states has kinks at the values of the gaps(emphasized in the inset). For ω max < ∆ min , I ( ω max ) = 0.(b) For ∆ so = ∆ z , a finite I ( ω max ) exists for all ω max due toLandau levels at zero energy. meV which is highlighted in the inset. At higher energy, I ( ω max ) is proportional to ω which is traced back tothe linear energy dispersion of the Dirac fermions. Whenthe magnetic field is finite, the density of states formsa series of Landau level peaks but I ( ω max ) remains zerofor ω max < ∆ min = 2 meV after which it has a verticalstep associated with the occupation of the lowest Lan-dau level. The height of the curve reflects the spectralweight of the delta function. The first two steps have thesame height, and the other steps are twice as large dueto the spin, and valley degeneracy of the N = 0 levels.Note that the energy range between the steps is alter-nately small, and large, and that I ( ω max ) tends towardits B = 0 value as ω max increases. For B = 1 . B dependenceof the Landau level degeneracy [see Eqn. (8)]. The en-ergy range over which a given step persists is also changedas it reflects the energy between adjacent Landau levels.Similar results are found for the BI regime. In Fig. 5(b),we show equivalent results for the VSPM case which cor-responds to ∆ z = ∆ so . Here, ∆ min = 0, and ∆ max = 8meV. When B = 0T (dash-dotted black curve), a changein slope is seen for I ( ω max ) at ω max = ∆ max . However,a finite integrated density of states persists to the low-est energy ω max →
0. In fact, in a finite magnetic field, I ( ω max → = 0, and is constant in the entire rangeof ω max < ∆ max . This strikingly different behaviour be-tween the VSPM, and TI/BI regimes, seen in both theHall conductivity, and integrated density of states, canbe employed to monitor the phase transition between thetwo states of matter.Plateaus similar to those described in the integrateddensity of states also exist for the absorptive part of theAC longitudinal conductivity integrated to Ω c . The realpart of σ xx (Ω) is given by Eqn. (12) of Ref. ; the op-tical spectral weight up to Ω c (denoted by W (Ω c ) = R Ω c σ xx (Ω) d Ω) is W (Ω c ) = e h πE X ξ,σ = ± ∞ X n,m =0 s = ± Θ (cid:0) µ − E ξσm,s (cid:1) − Θ (cid:16) µ − E ξσn, + (cid:17) E ξσn, + − E ξσm,s × h ( A m,s B n, + ) δ n,m − ξ + ( B m,s A n, + ) δ n,m + ξ i × Θ (cid:16) ¯ h Ω c + E ξσm,s − E ξσn, + (cid:17) Θ (cid:16) E ξσn, + − E ξσm,s (cid:17) . (20)Equation (20) [like Eqn. (16)], is a simple algebraic sumthat depends only on the Landau level energies E ξσn,s , andgaps ∆ ξσ . It is readily evaluated. Results for the steps inthe integrated optical conductivity are presented in thethree frames of Fig. 6 for the TI, VSPM, and BI. Theseresults were obtained in two independent ways: by directintegration of the AC conductivity, and by employingEqn. (20). Figure 6(a), in which ∆ z = 4 meV, shows re-sults for the TI side of the VSPM transition point. Thedashed green curve is for a magnetic field of 1T withchemical potential µ = 15 meV. The dash-dotted ma-genta curve corresponds to µ = 25 meV, and B = 1T.For µ = 15 meV, and finite B , the chemical potentialis above all the N = 0 Landau levels but below all the N = 1 levels. The first four plateaus come from the fourlowest optical transitions that are possible; these involveonly N = 0 to 1 transitions (see Refs. ). For largervalues of Ω c , additional optical excitations are possible.These involve higher Landau levels within the constraintof the optical selection rules ( | N | → | N | ±
1) which arebuilt into Eqn. (20) through the Kronecker deltas. WhileEqn. (20) is compact, it is worth while writing down amore explicit form for the particular case of the dashed
FIG. 6. (Color online) Optical spectral weight in the (a) TI,(b) VSPM, and (c) BI phases. (a) For µ between the N = 0,and 1 levels (dashed green curve), there are four low-energysteps. (b) For the VSPM, only three steps result from the N =0 to 1 optical transitions (dashed blue curve). (c) The fourlow-energy steps reappear in the BI regime (dashed purplecurve). green curve of Fig. 6. It is W (Ω c ) = e h πE X ξ,σ = ± ∞ X n =0 Θ (cid:16) Ω c − E ξσn +1 , + + E ξσn, − (cid:17) × ξσ q n + 1) + ¯∆ ξσ q n + ¯∆ ξσ × Θ (cid:16) E ξσn +1 , + − µ (cid:17) Θ (cid:16) µ − E ξσn, − (cid:17)q n + ¯∆ ξσ + q n + 1) + ¯∆ ξσ . (21)Again, all plateaus in W (Ω c ) depend only on the en-ergies E ξσn,s , and the gaps ∆ ξσ which appear normal-ized ( ¯∆ ξσ ≡ ∆ ξσ /E ) in Eqn. (21). The solid blackcurve of Fig. 6(a) is for no magnetic field . This curveshows no quantized plateaus but provides an unstruc-tured background on which the discreteness of the Lan-dau levels superimpose a step structure. Importantly,as Ω c is increased, the signature of the Landau levelquantization becomes smaller, and fades into the back-ground. This is irrespective of B or µ as illustrated bythe dash-dotted magenta curve. Returning to the B = 0case, the first step in the solid black curve gives the op-tical spectral weight of the Drude contribution to theconductivity W D = X ξ,σ = ± e π h µ − ∆ ξσ | µ | Θ ( | µ | − | ∆ ξσ | ) . (22)For Ω c > µ > | ∆ ξσ | , the nearly linear slope above theDrude plateau is due to the universal background in the optical conductivity. This provides a spectralweight of W BG = X ξ,σ = ± e π h (cid:20) Ω c − µ + 4∆ ξσ (cid:18) µ − c (cid:19)(cid:21) (23)which is a linear function of Ω c when Ω c ≫ µ and | ∆ ξσ | .The dotted blue curve of Fig. 6(a) is also for B = 0 buthere the chemical potential is set at µ = 0. Therefore, W D = 0 , and W BG = X ξ,σ = ± e π h " Ω c − ξσ Ω c Θ (Ω c − | ∆ ξσ | ) , (24)which has the same slope as the solid black curve forΩ c → ∞ , and merges with it above 30 meV. However,for small Ω c , the dotted blue curve goes to zero since µ is below both gaps, and there is no Drude response.Figure 6(b) is for the VSPM. This is the special casewhich exists at the boundary between the TI [Fig. 6(a)],and the BI [Fig. 6(c)]. In this phase, the minimum gapis zero, and there are only three steps (as opposed tofour) associated with the lowest energy optical transi-tions which involve the N = 0 levels (dashed blue curve).This is a distinct feature of this particular case. As seenin Fig. 6(c), the four steps are restored in the BI regime.When comparing the optical spectral weight with the in-tegrated density of states in Fig. 5, it is important torealize that the energies at which a new step appearsis set by the difference between two Landau levels forwhich an optical transition is allowed. For the density ofstates, and the Hall plateaus, the energies of the steps areset by the Landau level energies themselves. A detaileddiscussion of the magneto-optical, and optical conductiv-ity of silicene is found in Refs. , and Ref. , respec-tively. Closely related work includes a quadratic-in-momentum (Schr¨odinger) contribution in addition to thelinear-in-momentum Dirac term of Eqn. (1) in the ab-sence of buckling. Including impurity effects will causethe edge of the steps to broaden and smear over an en-ergy range of order Γ . The predicted step structure willtherefore be resolved for Γ much less than the Landaulevel spacing (for the Hall conductivity and integrateddensity of states), or the allowed optical transition ener-gies (for the optical spectral weight). While these energyscales depend on B , for the moderate magnetic fieldsused herein, the dominant steps should be visible forΓ ∼ O (1meV). IV. MAGNETIZATION AND MAGNETICSUSCEPTIBILITYA. Zero Temperature, Clean Limit
For convenience, we now choose to work with the rel-ativistic form of the grand potential. That is ,Ω( T, µ ) = − T Z ∞−∞ N ( ω )ln (cid:18) ω − µ T (cid:19) dω. (25)For our purposes, this form is equivalent to Eqn. (7) .Equation (25) has been worked out analytically forgapped graphene at T = 0 , and may be extended tosilicene. After applying the Poisson summation formula,the grand potential may be written as the sum of a reg-ular, vacuum, and oscillating piece. For µ ≥
0, the singlespin, and valley contributions areΩ ξσ reg ( µ ) = − eB h Θ ( µ − | ∆ ξσ | ) " µ ( µ − ξσ )3 E − | ∆ ξσ |− / E ζ − , ξσ E ! , (26)Ω ξσ vac ( µ = 0) = − eB h " Λ∆ ξσ √ πE + | ∆ ξσ | +2 / E ζ − , ξσ E ! + O (cid:18) (cid:19) , (27) andΩ ξσ osc ( µ ) = eB h E µ ∞ X k =1 πk ) cos πk ( µ − ∆ ξσ ) E ! , (28)respectively [see Eqns. (6.3), (A4), and (8.7) in Ref. ],where ζ ( − / , x ) is the Hurwitz zeta function, and Λ isan ultraviolet cutoff associated with the band width. Themagnetization is the sum of M reg ( µ ) = − eh X ξ,σ = ± Θ ( µ − | ∆ ξσ | ) C (∆ ξσ , B ) , (29) M vac ( µ = 0) = eh X ξ,σ = ± C (∆ ξσ , B ) , (30)and M osc ( µ ) = − eh E µ X ξ,σ = ± ∞ X k =1 πk ) cos πk ( µ − ∆ ξσ ) E ! − eh X ξ,σ = ± µ − ∆ ξσ µ ∞ X k =1 πk sin πk ( µ − ∆ ξσ ) E ! , (31)where C (∆ ξσ , B ) = | ∆ ξσ | E √ ζ − , ξσ E ! − ∆ ξσ √ E ζ , ξσ E ! , (32)and we have used the relation ddx ζ ( s, x ) = − sζ ( s + 1 , x ) . (33)Comparing Eqns. (29), and (30), it is clear that for µ > | ∆ ξσ | , the regular, and vacuum contributions to themagnetization cancel with each other, and only the os-cillating piece remains. For small B fields, we use theexpansion ζ ( s, a ) ≈ a − s s − − a s + s a s (34)for large a to obtain M vac ( µ = 0) ≈ − X ξ,σ = ± eh M (∆ ξσ , B ) , (35)where M (∆ ξσ , B ) = E | ∆ ξσ | , ∆ ξσ = 03 E √ π ζ (cid:18) (cid:19) , ∆ ξσ = 0 , (36)0 FIG. 7. (Color online) Vacuum contribution to the magneti-zation as a function of (a) B , and (b) ∆ z . (a) The agreementbetween Eqn. (30) (solid curves), and Eqn. (35) is good forsmall B or ∆ so = ∆ z = 0. (b) Here, agreement is foundbetween the full result (solid curves), and approximate ex-pression when | ∆ z | ≫ ∆ so or ≪ ∆ so , and B is small. Aminimum is seen in M vac in the VSPM phase. and ζ ( x ) is the Riemann zeta function. Again, the regu-lar part is the same as the vacuum contribution up to afactor of − Θ( µ − | ∆ ξσ | ). Plots of the vacuum magneti-zation as a function of B , and ∆ z are given in Figs. 7(a),and (b), respectively. For ∆ so = ∆ z = 0, we recoverthe graphene limit. In this case, the full solution for M vac , given by Eqn. (30), and shown as a solid blackcurve in Fig. 7(a), agrees perfectly with the form givenin Eqn. (35) (dotted yellow) as it should since no ap-proximations were made when deriving Eqn. (35) fromEqn. (30) for ∆ ξσ = 0. As a function of B , M vac ∝ −√ B via the definition of E . This dependence is evident inthe figure. Since the VSPM case has a single graphene-like band, and one massive band at each valley, for low B ,there is mainly a −√ B dependence, and good agreementwith Eqn. (35) over the range of B shown. This arisesfrom the fact that the energy scale associated with thesesmall values of B is largely sampling the graphene band,and not the massive band. So, as B →
0, the dashed redcurve [Eqn. (35)] is in good agreement with the full resultof Eqn. (30), and is reduced by approximately a factorof two reflecting that there is only one spin-polarized lin-ear band at each K -point. With increasing field, themassive bands begin to contribute, and Eqn. (35) be-gins to deviate from the full result. The energy scale forsuch deviations will be controlled by ∆ ξσ ≈ E . Fi-nally, consider the Kane-Mele QSHI given by ∆ z = 0but finite ∆ so [solid green curve in Fig. 7(a)]. In thiscase, the result of Eqn. (35) depends on the approxima-tion a ≡ ∆ ξσ / (2 E ) = ∆ / (2 E ) ≫
1, and predicts M vac ∝ − B . This will only be a good approximation forsmall B ≪ ∆ / (2¯ hev ) as seen in Fig. 7(a) by comparingthe solid green (exact), and dash-double-dotted purple[approximate Eqn. (35)] curves. Thus, we expect that as B →
0, the simple formulae will be robust, and useful forcalculating the magnetic susceptibility ( ∂M/∂B | B → ). Ifone is at large B , the full expression must be applied.Continuing to Fig. 7(b), the vacuum magnetization isshown as a function of ∆ z for three values of magneticfield. Recall that the TI regime is for ∆ z < ∆ so = 8 meV,and the BI case corresponds to ∆ z > ∆ so . Hence, thedivergent features seen at ∆ z = ∆ so are occurring as theVSPM phase is approached. In the full calculation for thevacuum contribution (solid curves), no divergence occursbut rather there is a cusp in the magnetization that indi-cates the critical point between the TI, and BI; this marksthe VSPM. As already understood from the discussionsurrounding Fig. 7(a), for very low B , and small ∆ z , theapproximate Eqn. (35) works reasonably well even as ∆ z is varied away from zero. As ∆ z → ∆ so , however, the1 / | ∆ ξσ | factor in Eqn. (35) diverges. This is due to abreakdown in the approximation ∆ ξσ ≫ E that wasused in Eqn. (34). While Eqn. (35) works well only forvery small magnetic fields, and ∆ z away from the VSPMcritical point, it also is seen to work well for very large ∆ z where ∆ ξσ ≫ E is again satisfied. Finally, in order toreconcile the good agreement between Eqn. (35), and theVSPM curve in Fig. 7(a) with the divergence in Fig. 7(b),note that the data point ∆ z = ∆ so in Fig. 7(b) evaluatedby Eqn. (35) is not shown; it would be a discontinuouspoint that sits near the cusp position in good agreementwith the full solution. In summary, we caution againstusing the simple formula of Eqn. (35) beyond its intendedregion of validity determined by (∆ so − ∆ z ) ≫ hev B .We also note that the VSPM state is identified by a cusp(not a divergence) in the vacuum magnetization as theelectric field E z is varied.One can also calculate the magnetic susceptibility( − ∂ Ω /∂B | B → = χ ). Differentiating Eqn. (35) with1respect to B , and taking the limit B →
0, we obtain thevacuum susceptibility χ vac = − eh X ξ,σ = ± X (∆ ξσ ) , (37)where X (∆ ξσ ) = ¯ hev | ∆ ξσ | , ∆ ξσ = 0lim B → √ π r ¯ hev B ζ (cid:18) (cid:19) , ∆ ξσ = 0 . (38)For ∆ ξσ = 0, Eqn. (37) diverges as 1 / √ B for B → . While for finite ∆ ξσ → χ vac diverges as 1 / | ∆ ξσ | . The vacuum plus regular part of the suscep-tibility is given by χ = − eh X ξ,σ = ± X (∆ ξσ ) [1 − Θ( µ − | ∆ ξσ | )] . (39)A plot of the vacuum susceptibility as a function of ∆ z isgiven in Fig. 8(a). The result of differentiating Eqn. (30)[solid black curve] is compared with Eqn. (37) [dashedgreen curve]. Here, agreement is found between the fullresult, and that obtained from the approximate equationfor M vac . Unlike the magnetization, there is a singu-larity at ∆ z = ∆ so . This is due to the massless Diracfermions of the lowest band. Since the susceptibility isinversely proportional to the effective mass , a diver-gent diamagnetic response occurs when the VSPM phaseis approached . As the two gaps become larger (and theeffective mass of the electrons increases), the susceptibil-ity decreases in magnitude. The slope of the susceptibil-ity has a different sign in the two insulating regimes whichmay be used to distinguish the two gapped phases. Thecontribution to the susceptibility from both the vacuum,and regular parts of Ω( µ ) as a function of µ is shown inFig. 8(b). As in gapped graphene , the susceptibilityis not zero when µ is in the gap since the filled valenceband provides a constant contribution . For ∆ z = 0(solid black curve), there is only one gap, and the suscep-tibility is given by its vacuum value until µ = ∆ so /
2. Asthe chemical potential enters the conduction band, a stepoccurs, and the susceptibility is zero. This agrees withthe result of doped graphene . For finite ∆ z = ∆ so (dashed blue, and dash-double-dotted green curves), thesystem is characterized by two gaps. For µ less thanthe minimum gap, the magnitude of the susceptibility ismaximized. Two steps are seen at the values of the gaps∆ min and ∆ max . Again, for µ greater than the maxi-mum gap, the susceptibility is zero. In the VSPM phase(dash-dotted red curve), there is only one gap and, thus,a single step.The behaviour of the susceptibility should provide aprobe of the important energy scales of the system. Byvarying ∆ z , one can identify the VSPM transitions and,thus, ∆ so . In addition, by changing µ , the two gaps ∆ min ,and ∆ max can be determined. FIG. 8. (Color online) (a) Vacuum contribution to the mag-netic susceptibility as a function of ∆ z . Agreement is foundbetween the full result (solid black), and Eqn. (37) (dashedgreen). Unlike the magnetization, there is a singularity at∆ z = ∆ so . (b) Vacuum plus regular parts of the susceptibil-ity as a function of µ for various ∆ z . Steps are seen at thegap values ∆ min , and ∆ max . B. Impurity and Finite Temperature Effects
The Landau level broadening that results from impu-rity scattering (Γ), and finite temperature can be in-cluded by convolving the magnetization with the scat-tering, and temperature functions P Γ ( ω − µ ) = Γ π [( ω − µ ) + Γ ] , (40)and P T ( ω − µ ) = − ∂n F ∂ω = 14 T cosh (cid:0) ω − µ T (cid:1) , (41)2respectively, where n F is the Fermi distribution function.For simplicity, we have assumed (as done in Ref. andother works) that Γ is the same for all Landau levels.More complicated descriptions of impurity scattering ex-ist; in particular, see Ref. which deals explicitly withthe diamagnetism of disordered graphene. Therefore, M ( µ, Γ , T ) = Z ∞−∞ Z ∞−∞ P Γ ( ω − µ ) P T ( ω ′ − µ ) M ( ω ′ ) dω ′ dω, (42)where M ( ω ) is obtained by setting µ = ω in the Γ = 0,and T = 0 result [Eqns. (29)-(31)].Let us begin by examining the effect of impurity scat-tering on the vacuum plus regular piece of the magneti-zation. Combining Eqns. (29), and (30), Eqn. (42) gives M ( µ, Γ) = eh X ξ,σ = ± C (∆ ξσ , B ) Z | ∆ ξσ |−| ∆ ξσ | Γ π [( ω − µ ) + Γ ] dω = eh X ξ,σ = ± C (∆ ξσ , B ) D Γ , (43)where D Γ = 1 π " arctan | ∆ ξσ | Γ µ − ∆ ξσ + Γ ! , (44)and we have taken T = 0. For our low B expres-sion [see Eqn. (35)], C (∆ ξσ , B ) should be replaced with M (∆ ξσ , B ). Similarly, the effect of impurities on thevacuum plus regular part of the susceptibility is [seeEqn. (39)] χ (Γ) = − eh X ξ,σ = ± X (∆ ξσ ) D Γ . (45)This agrees with Eqn. (6) of Ref. in the gappedgraphene-limit | ∆ ξσ | → ∆. For both the magnetization,and susceptibility, the damping factor D Γ has a depen-dence on chemical potential. As µ increases, the mag-netic response decreases.The same procedure is applied when considering theeffect of finite temperature. Equation (42), with the com-bined result of Eqns. (29), and (30), gives M ( µ, T ) = eh X ξ,σ = ± C (∆ ξσ , B ) Z | ∆ ξσ |−| ∆ ξσ | (cid:18) − ∂n F ∂ω (cid:19) dω = eh X ξ,σ = ± C (∆ ξσ , B ) D T , (46)where D T = sinh( β | ∆ ξσ | )cosh( β | ∆ ξσ | ) + cosh( βµ ) , (47) β = T − , and Γ has been set to zero. While we haveprovided the general result here, there are limiting cases which are known, and discussed below. For ∆ ξσ → β | ∆ ξσ | ) ≈ β | ∆ ξσ | and D T ≈ β | ∆ ξσ | βµ ) . (48)The magnetic susceptibility is χ ( T ) = − eh X ξ,σ = ± X (∆ ξσ ) D T . (49)Using Eqn. (48), the susceptibility for ∆ ξσ → χ ( T ) = − eh X ξ,σ = ± ¯ hev β cosh ( βµ/ , (50)which is in agreement with the results of Refs. .Returning to Eqn. (47), for zero chemical potential, D T = tanh ( β | ∆ ξσ | ) , (51)and χ ( T ) = − eh X ξ,σ = ± X (∆ ξσ )tanh ( β | ∆ ξσ | ) . (52)This is in agreement with Eqn. (31) in Ref. .The µ = 0 susceptibility at finite temperature, andimpurity scattering is shown in Fig. 9(a) as a functionof ∆ z . At T = Γ = 0 (solid black curve), a singular-ity exists at the critical value ∆ so = ∆ z . The singu-larity is removed by the inclusion of finite temperatureor impurity effects . Note that the two curves for ei-ther temperature (dash-dotted blue) or impurity scatter-ing (dotted magenta) equal to 0.5 meV are very sim-ilar. While the singularity is washed out, there is aclear minimum at the position of the phase transition.This is independent of magnetic field since we are deal-ing with χ . Of course, the impurity scattering rate it-self depends on the type of impurities . Returning toEqn. (39), we note that finite µ will also remove thesingularity. This is shown in Fig. 9(b). There is anabrupt vertical drop in the absolute value of χ as themagnitude of ∆ ξσ is reduced below the value of µ . For∆ max > µ > ∆ min , the susceptibility remains finite, andis given by χ = − [ e/h ][¯ hev / (3∆ max )]. This value origi-nates entirely from the spin-valley branches of the Diracdispersion which have the maximum gap ∆ max . Thisabrupt change in χ can be used to determine the size of µ (i.e. the doping away from charge neutrality). The de-termining structures involved in such a measurement aremost prominent for small values of chemical potential. V. MAGNETIC OSCILLATIONSA. T = Γ = 0 We now return to Eqn. (31) to discuss the magnetic os-cillations in a doubly gapped Dirac system like silicene.3
FIG. 9. (Color online) (a) The effect of finite temperature,and impurity scattering on the susceptibility as a function of∆ z . (b) The effect of finite chemical potential on the suscep-tibility. The de-Haas van-Alphen (dHvA) effect is a low-field phe-nomenon which provides an experimental probe of theFermi surface. Semiclassically, M osc is found to oscillateaccording to M k osc ∝ sin (cid:18) πk (cid:20) ¯ hA ( µ )2 πeB − γ (cid:21)(cid:19) , (53)where A ( µ ) is the area of a cyclotron orbit at the Fermienergy ( A ( µ ) = πk F ), and γ is a phase offset related tothe Berry’s phase ( γ = 1 / π ). For silicene, γ = 0, and there are two possibleorbit areas. In silicene, the Fermi momentum for theband labelled by ξ and σ is k ξσF = s µ − ∆ ξσ ¯ h v , (54) where only real solutions are considered. Therefore, A ξσ ( µ ) = π ¯ h v " µ − (∆ so − ξσ ∆ z ) . (55)Indeed, taking only the leading term of Eqn. (31), wenote M osc ( µ ) = − eh X ξ,σ = ± µ − ∆ ξσ µ ∞ X k =1 πk sin (cid:18) πk (cid:20) ¯ hA ξσ ( µ )2 πeB − γ (cid:21)(cid:19) , (56)where γ = 0, and A ξσ ( µ ) is given by Eqn. (55). Equa-tion (55) is plotted in Fig. 10(a) for ¯ µ ≡ µ/ ∆ so ≤ / z ≡ ∆ z / ∆ so . The results are shownfor the lowest gapped bands (spin-up at K , and spin-down at K ′ ) since, for ¯ µ ≤ /
2, only one band is occu-pied. For ¯ µ < / − µ < ¯∆ z < µ . As µ is increased, the range of ∆ z grows for which A ( µ ) is nonzero. The maximum orbitarea is fixed at ¯ µ , and occurs at ∆ z = ∆ so (the VSPMphase). ¯ µ = 1 / z for which A ( µ ) > z = 0, and 2. In all cases, there is perfect symme-try between the TI, and BI regimes. Schematically, theregions of ∆ z which give a finite area are those whichensure µ sits in the µ position of Fig. 10(c). µ = µ cor-responds to no cyclotron orbits. Next, consider ¯ µ > / µ = µ in Fig. 10(c). Now bothbands are occupied. A ( µ ) as a function of ¯∆ z is shownin Fig. 10(b). The maximum area (solid curve) comesfrom the spin-up electrons at K , and the spin-down elec-trons at K ′ . The other spin-valley combinations makeup the minimum area (dashed curve). For A max ( µ ), themaximum value of ¯∆ z remains at 1 + 2¯ µ . 1 − µ nowoccurs for negative ¯∆ z ; thus, for ∆ z = 0, A max ( µ ) is fi-nite at a value of ¯ µ − /
4. The maximum still occursin the VSPM phase, and retains its value of ¯ µ . A min ( µ )persists until ¯∆ z = 2¯ µ −
1, and is equal to A max ( µ ) at∆ z = 0. There is perfect symmetry between the max-imum, and minimum areas if negative values of ¯∆ z areconsidered. In that region, the spin, and valley labelsof the two areas are interchanged. It is clear that for¯∆ z < µ −
1, there are two orbits which will contributeto the magnetic oscillations. For ∆ z = 0, the areas areequal, and will contribute an overall degeneracy to M osc .For finite ∆ z , the difference in the two orbits increaseswith the electric field strength. The interaction betweenthe two areas is clearly seen by looking at the total oscil-lating magnetization [Eqn. (56)]. We are only interestedin the lowest harmonic k = 1. Therefore, M osc ( µ ) = − eh (cid:26) µ − ∆ πµ sin (cid:18) π (cid:20) ¯ hA min ( µ )2 πeB (cid:21)(cid:19) + µ − ∆ πµ sin (cid:18) π (cid:20) ¯ hA max ( µ )2 πeB (cid:21)(cid:19)(cid:27) , (57)4 FIG. 10. (Color online) (a) The cyclotron orbit area as afunction of µ for µ between the two gaps. In this regime, asingle cyclotron orbit exists for 1 − µ < ¯∆ z < µ . Note:¯ µ ≡ µ/ ∆ so , and ¯∆ z ≡ ∆ z / ∆ so . (b) The two cyclotron orbitareas for µ above both gaps. (c) Schematic representationof the positive energy band structure at the K point. For µ = µ , no cyclotron orbits are present. For µ = µ [see (a)],only one orbit exists. For µ = µ [see (b)], two orbits exist. where A min ≡ A + − , and A max ≡ A ++ . Equation (57)shows that M osc is constructed from two sine waves with1 /B frequencies of ω min = ¯ hA min ( µ ) e = π ¯ hev " µ − (∆ so + ∆ z ) , (58)and ω max = ¯ hA max ( µ ) e = π ¯ hev " µ − (∆ so − ∆ z ) . (59)Therefore, for µ between the two gaps [ µ in Fig. 10(c)],there is only one characteristic dHvA frequency. For µ above both gaps [ µ in Fig. 10(c)], there are two fre-quencies, and interference will be observed . Whenthe difference between A min ( µ ), and A max ( µ ) is small,a strong beating is present. Looking at Fig. 10(b), thiswill occur when ∆ z is very small (but nonzero) or µ isvery large ( µ ≫ ∆ max ). The beating of the quantum os-cillations is shown in Fig. 11. Figure 11(a) shows M osc as a function of 1 /B for µ = 15 meV, and ∆ z = 0,2, and 4 meV (green, purple, and black curves, respec-tively). Here µ is above both gaps. For ∆ z = 0 meV,only one gap is present, and strong dHvA oscillations areobserved. For ∆ z = 2 meV, the bands have become spinsplit, and a beating phenomenon emerges in additionto the usual oscillations. For ∆ z = 4 meV, the beatingpersists; however, the beat frequency has increased. Thisphenomenon is a signature of a small difference betweenthe two orbit areas which we control here with the mag-nitude of the external electric field. For smaller valuesof µ , increasing ∆ z can lead from a single fundamentalfrequency to a beating regime, and back to a single fre-quency as ∆ z becomes large enough that µ is only abovethe smaller gap.The magnetic oscillations for fixed ∆ z = 4 meV, andvarying µ are shown in Fig. 11(b). For µ = 4 meV (dash-dotted black curve) only a single frequency is present.When µ = 7 meV (dashed blue curve), both bands con-tribute, and two characteristic frequencies become visi-ble. They are ω max = 0 . ω min = 0 . µ is increased further (solid red curve), the two cyclotronorbits become closer ( ω max = 4 . ω min = 3 . µ = 0. B. Impurity and Finite Temperature Effects
In the presence of impurities, the magnetic oscilla-tions are damped by a Dingle factor which dependson the scattering probability. This effect is includedby convolving M osc ( µ ) with P Γ ( µ − Γ) as is shown inSec. IV B. For gapped graphene, this has been workedout analytically , and can be extended to the doubly5 FIG. 11. (Color online) Magnetic oscillations as a functionof 1 /B . (a) dHvA effect for µ = 15 meV (above all thegaps). For finite ∆ z , two orbit areas exist, and a beating phe-nomenon emerges. (b) M osc ( B ) for ∆ z = 4 meV, and varying µ . For clarity, the curves have been offset from oscillatingaround zero. gapped case of silicene. Thus, in the presence of impuri-ties, and at finite temperature [see Eqn. (8.17) of Ref. which we repeat here for completeness],Ω ξσ osc ( µ, Γ , T ) = eBE hµ ∞ X k =1 πk ) cos πk ( µ − ∆ ξσ − Γ ) E ! × R D R T , (60)with the Dingle factor R D = exp (cid:18) − πk µ Γ E (cid:19) , (61)and temperature factor R T = kλ sinh( kλ ) , (62) where λ = 2 π T µE . (63)For both impurity, and temperature effects, the dampingof the oscillations is dependent on µ making the dHvAeffect difficult to observe for large values of chemicalpotential . VI. CONCLUSIONS
The application of a perpendicular electric field to alow-buckled honeycomb lattice creates an onsite potentialdifference (∆ z ) between the two sublattices. The result-ing electronic band structure consists of four valley-spinpolarized bands which result from a large spin-orbit in-teraction. An example material is silicene (a monolayerof silicon atoms). The Dirac fermions involved are mas-sive with the magnitude of the two resulting gaps modu-lated by ∆ z . By varying the electric field, a phase tran-sition from a topological insulator, through a valley-spinpolarized metal, to a trivial band insulator is induced.We have studied the magnetization of such bands in anattempt to find distinctive features associated with thethree different phases. Particular attention is given toidentifying the boundary between the TI, and BI.The derivative of the magnetization ( M ) with respectto the chemical potential ( µ ) is calculated, and found togive a series of steps as a function of µ . The onset of eachnew plateau is associated with the energies of the Lan-dau levels involved. Through direct calculation of theDC Hall conductivity ( σ H ), we have verified explicitlythat the Streda formula e ( ∂M/∂µ ) = σ H holds. In theabsence of Zeeman splitting, the Hall conductivity (inunits of e /h ) has filling factors ν = 0 , ± , ± , ± , ± , ... for both the TI, and BI; while, for the VSPM, ν = ± , ± , ± , ± , ... . Including a finite Zeeman interactionsplits the valley-spin degeneracy of the ν = ± , ± , ... fill-ing factors, and the resulting Hall conductivity can takeany integer multiple of e /h . For moderate values of Zee-man splitting, the four steps associated with each of the N = 1 , , , ... levels become spin-polarized. Therefore,as an example, while the ν = 2 Hall conductivity is spin-degenerate, the ν = 3 , ν = 5 ,
6, andthen regained for ν = 7 ,
8, etcetera. Near the chargeneutral point, σ H can also display spin, and valley po-larization even when the Zeeman splitting is neglected.We emphasize that the boundary between the TI, andBI supports a finite charge Hall conductivity providinga signature of the VSPM. This conductivity goes from − e /h to e /h at µ = 0 while it is zero for the other twophases. The finite charge Hall conductivity near µ = 0 ofthe VSPM is lost as the Zeeman interaction is increased.Steps analogous to those found in the derivative ofthe magnetization also exist in the integrated density of6states [ I ( ω max )] to energy ω max , as well as in the opti-cal spectral weight [ W (Ω c )] accumulated below a cutofffrequency Ω c . In the first instance, the energy at whichthe new steps onset is determined by the energies of thevarious Landau levels. In the optical case, it is the en-ergy of the allowed optical transitions which matters; thisinvolves the sum of two Landau level energies for the in-terband transitions, and the difference for the intrabandtransitions. In both cases, we present simple analytic al-gebraic expressions for the height of the various plateaus;these are not integral in fundamental constants but in-volve only the Landau level energies, and the band gaps.For the quasiparticle density of states in the TI, and BIregimes, I ( ω max ) is zero for ω max less then the two gaps.It is finite in the VSPM for all ω max = 0. This pro-vides a clear signature of the phase transition betweenthe TI, and BI phases. For W (Ω c ), conservation of spec-tral weight is noted for all µ . While the magnitude of thechemical potential strongly affects the height of the firstplateau, for Ω c much larger than the gaps, and chemicalpotential, all curves merge into the B = 0 background;this has a linear dependence on Ω c .The magnetic susceptibility ( χ ) as a function of the on-site potential difference is found to exhibit a singularityas ∆ z goes through the spin-orbit band gap ∆ so . Thisresults from the lowest band gap closing as the system en-ters the VSPM phase, and the well understood divergentgraphene result emerges. The singularity is not presentfor any other ∆ z , and can be removed by finite tem-perature or by impurity scattering (Γ). For parameterscharacteristic of silicene, we find that to observe a resid-ual signature of the divergence, it is necessary to haveΓ ≤ . z providingonly a kink or change in slope at ∆ z = ∆ so . This couldstill be employed to identify the boundary between thenontrivial, and trivial topological phase. In the presenceof doping away from charge neutrality, the finite chemi- cal potential provides a cutoff on the contribution to thesusceptibility that comes from the dispersion branch withthe minimum gap ∆ min . It is entirely eliminated for allvalues of ∆ z such that µ > (1 / | ∆ z − ∆ so | . This fact canbe used to obtain an estimate of the chemical potentialas well as the magnitude of the spin-orbit gap ∆ so .Finally, the de-Haas van-Alphen magnetic oscillationsare discussed. Depending on the location of the chemi-cal potential, there can be zero, one, or two fundamentalcyclotron frequencies; these occur when µ is below bothgaps, between the gaps, or above both gaps, respectively.For the VSPM, there is only one gap; thus, there willalways be at least one fundamental frequency. Whenthere are two fundamental cyclotron frequencies, a beat-ing pattern emerges in the oscillations. For large valuesof µ , where the Dirac fermion dispersion curves merge,the beating is lost. For no external potential, the systemis a topological insulator, and there is a single degeneratecyclotron frequency which can be split by the applicationof a small ∆ z .In summary, we have examined the magnetic responseof Dirac fermions in a low-buckled honeycomb lattice.We have found signatures which should allow for an ex-perimental verification of the two insulating regimes (TI,and BI) and, the VSPM through tuning by an externalelectric field. Note added.
During the preparation of thismanuscript, a preprint appeared which discusses theeffect of finite µ on the susceptibility of related 2D sys-tems. ACKNOWLEDGMENTS
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