Tunneling Conductance in a Two-dimensional Dirac Semimetal Protected by Non-symmorphic Symmetry
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Tunneling Conductance in a Two-dimensional Dirac Semimetal Protected byNon-symmorphic Symmetry
Tetsuro Habe
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan (Dated: March 12, 2018)We theoretically study a tunneling effect in a two-dimensional Dirac semimetal with two Diracpoints protected by non-symmorphic symmetries. The tunnel barrier can be arranged by a magneticexchange potential which opens a gap at the Dirac points which can be induced by a magneticproximity effect of a ferromagnetic insulator. We found that the tunnel decay length increases witha decrease in the strength of the spin-orbit coupling, and moreover the dependence is attributed tothe correlation of sublattice and spin degree of freedoms which lead to symmetry-protected Diracpoints. The tunnel probability is quite different in two Dirac points, and thus the tunnel effect canbe applied to the highly-selective valley filter.
PACS numbers: 73.22.-f
Dirac semimetal has the gapless energy spectrum ofelectrons with a point node at which the conduction andthe valence bands are touched and such a node, so-calledDirac point, emerges at a symmetrical point in the Bril-louin zone . The electric excitation energy has a lineardependence on the wave number from the Dirac pointat which the even number of electric states are degen-erated. One can control the low energy spectrum by asymmetry-breaking external field and change the trans-port phenomena.In recent years, several kinds of candidates and the-oretical predictions of the realistic Dirac semimetal inthree-dimensions have been proposed . The wealthof candidates in three-dimensions is associated withthe simple necessary condition for realization of theDirac semimetal which requires the band inversionand crystal symmetry C . In two-dimensions, on theother hand, there are few candidate compounds withDirac points. Graphene had been regarded as a two-dimensional(2D) Dirac semimetal and studied as a testground for researching the unique features of masslessDirac fermions . However, the linear dispersion ingraphene is an approximation even in a clean system be-cause the spin-orbit interaction opens an energy gap andmakes it to be a topological insulator .Recently, it was proposed that non-symmorphic sym-metries are necessary for Dirac points to be stable in a2D system , and these symmetries also play an impor-tant role in one- and three-dimensional semimetals .In practice, it is shown that the Dirac points in such a2D Dirac semimetal are preserved even in the presenceof the spin-orbit interaction . However, the spin-orbitinteraction in graphene is quite small ∼ K , and thus itseems that there is no difference in ordinary experimen-tal observations between graphene and the symmetry-protected 2D Dirac semiemtal. Thus it is important tosuggest the different physical property of the electrons inthe symmetry-protected Dirac semimetal from the nearlymassless excitation in graphene.In this paper, we discuss the tunneling electric trans-port in the symmetry-protected 2D Dirac semimetal with two Dirac points, i.e. the valley degree of freedom. Thetunnel barrier is fabricated by a magnetic exchange po-tential, e.g. which is induced by the magnetic proxim-ity effect in a junction with a ferromagnetic insulator.We show that the perpendicular component of the ex-change field opens a gap in the 2D Dirac semimetal sim-ilar to the surface states of three-dimensional topologicalinsulators or the sublattice-symmetry-breaking po-tential in graphene . However, the exchange potential,unlike that in the three-dimensional topological insula-tor, is not a simple mass term for massless Dirac fermionsin the 2D Dirac semimetal because of the spin-sublatticecorrelation associated with non-symmorphic symmetry.We find that the tunnel decay length, unlike ordinary 2DDirac fermions, becomes longer with a decrease in thespin-orbit coupling because of the sublattice-spin corre-lation in the symmetry-protected 2D Dirac semimetal,and propose that this tunneling system plays a role ofthe highly-selective valley filter.We consider a model Hamiltonian proposed by Youngand Kane for describing electric states in the symmetry-protected 2D Dirac semimetal with a square lattice in-cluding two atoms in a unit cell , and the simplest formis given by, H =2 tτ x cos k x k y t (cos k x + cos k y )+ t so τ z ( σ y sin k x − σ x k y ) , where τ and σ are Pauli matrices in the sublattice andspin spaces, respectively. The electric states have threeDirac points M = ( π, π ), X = ( π, X = (0 , π )in the first Brillouin zone. The energy of node at X and X is equal to each other but it is different fromthat at M because of the chiral symmetry breaking term t , and thus there is no dip of density of states in thismodel. However, the C screw symmetry breaking term,which is introduced to simulate iridium oxide superlatticeproposed by Chen and Kee , V = ∆ sin k x k y τ x , opens a gap only at the M point. The Hamiltonian H = H + V represents a rigorous 2D Dirac semimetal.The electric transport is associated with the electricstates around the Fermi energy, and thus we can discussthe transport phenomenon for the Fermi energy near theDirac node by using the effective Hamiltonian based on kp theory around the X j as H X ( p ) = − t so τ z ( σ y p x + σ x p y ) + τ x ( − tp x + ∆ p y ) H X ( p ) = t so τ z ( σ y p x + σ x p y ) + τ x ( − tp y + ∆ p x ) , with a relative wave vector p = ( p x , p y ) from the X j . Inwhat follows, we discuss the tunneling effect at only the X , however the result is applicable to the case at the X with t so → − t so , − t → ∆ , and ∆ → − t .First, we consider the effect of a magnetic exchange po-tential on the electric states in the symmetry-protected2D Dirac semimetal and we show that the potential en-ables us to open the energy gap at the Dirac node. Forinstance, such an exchange potential m µ σ µ can be in-duced by a magnetic proximity effect of the junction witha ferromagnetic insulator, and the coupling constants m µ can be controlled by changing the magnetization in theferromagnetic insulator. To analyze the effect of the ex-change potential, we rewrite the potential in the basis ofthe eigenvectors for H X , U † θ,φ H X U θ,φ = q t so p + ( − tp x + ∆ p y ) τ z σ z , where the unitary operator U θ,φ consists of two rotationmatrices R σ,θ in the spin subspace and R τ,φ in the sub-lattice subspace, U θ,φ = R σ,θ √ iτ x σ z ) R τ,φ , with p = ( p cos θ, p sin θ ), sin φ = ( − tp x + ∆ p y ) /ε , and R σ,θ = 1 √ σ z + σ y cos θ + σ x sin θ ) . In this basis, the spin operators are transformed into U † θ,φ σ z U θ,φ =( − σ x cos θ + σ y sin θ ) × ( τ z sin φ − ( τ x cos φ + τ y sin φ ) cos φ ) U † θ,φ σ x U θ,φ = σ z sin θ − cos θ ( σ y cos θ + σ x sin θ ) × ( τ z sin φ − ( τ x cos φ + τ y sin φ ) cos φ ) U † θ,φ σ y U θ,φ = σ z cos θ + sin θ ( σ y cos θ + σ x sin θ ) × ( τ z sin φ − ( τ x cos φ + τ y sin φ ) cos φ ) . One can find a particular angle θ for any exchange po-tential coupling to an in-plane spin to be the identity inthe sublattice space, and thus the in-plane component ofthe exchange field m µ preserves the gapless energy dis-persion where two Weyl nodes can be found on the line p with this angle θ . The potential coupling to the out-of-plane spin m z σ z , on the other hand, opens a gap in the energy spectrum because it provides non-zero com-ponent proportional to τ µ σ ν even with any angles θ and φ . We show the energy dispersion in the presence ofthe magnetic exchange potential coupling to out-of-planespin and in-plane spin in Fig.1. The effect of the in-planeexchange field is similar to the separation of the Diracnode into two Weyl nodes in the time-reversal breakingWeyl semimetal with a exchange potential . FIG. 1. The energy dispersion of the 2D Dirac semimetalaround X with a magnetic moment H m = m x σ x (a), m y σ y (b), and m z σ z (c). Next, we consider the tunnel junction arranged by theexchange potential mσ z in the symmetry-protected 2DDirac semimetal where the tunneling barrier can be fab-ricated by attaching a ferromagnetic insulator locally asshown in Fig.2. The junction system can be described by FIG. 2. The schematic picture of the tunneling junction ar-ranged by a ferromagnetic insulator. H = H ξ (0) θ ( − x ) + H ξ ( m ) θ ( x ) θ ( L − x )+ H ξ (0) θ ( x − L ) , (1)where the Hamiltonian can be represented in the basis ofthe eigenvectors for the glide mirror operator τ x σ z , H ξ ( m ) = ( ξm + tp x − ∆ p y ) s z − t so ( ξs y p x + s x p y ) , (2)with a Pauli matrix s and the eigenvalue of the glide mir-ror operator ξ = τ x σ z because the glide mirror symmetryis preserved even in the presence of mσ z . Here, the Paulimatrix s is the pseudo spin in the basis of eigenstate forthe glide mirror operator and it represents the staggeredalignment of spin at two sublattices in each spin axis.The eigenstate with the incident electron with the en-ergy ε for the Hamiltonian (1) can be written by a wavefunction consisting of three functions smoothly connectedat the boundaries of the three regions,Ψ = ψ + ξ,p + x e ip + x x + Rψ + ξ,p − x e ip − x x ( x < C ψ + ξ,q + x e iq + x x + C ψ + ξ,q − x e iq − x ( x − L ) (0 < x < L ) T ψ + ξ,p − x e ip + x ( x − L ) ( L < x ) , where ψ ± ξ,p x is the eigenfunction of Eq.(2). Here, T and R are the transmission and reflection coefficients. If the twoboundaries of the second region 0 < x < L are assumedto be parallel to each other, the wave number parallel tothe interface is preserved in the scattering, and the wavenumber perpendicular to the boundary is a function of m as p ± x = k ± ( m ) and q ± x = k ± (0) with k ± ( m ) = − t ( ξm − ∆ p y ) t + t so ± q ( t + t so )( ε − t so p y ) − t so ( ξm − ∆ p y ) t + t so , (3) for the eigenstate with the energy ε . The exchange po-tential provides a tunnel barrier to the electrons with theenergy | ε | < t so q p y + ( ξm − ∆ p y ) / ( t + t so ).In the tunnel junction, the analytic formulation ofthe transmission coefficient T ( p y ) can be obtained bysmoothly connecting the wave functions at the bound-aries x = 0 and x = L , and it can be represented by T ( p y ) − = − e − iq + x L (cid:18) − γ ( m, q − x , , p − x ) γ (0 , p + x , , p − x ) γ (0 , p + x , m, q + x ) γ ( m, q − x , m, q + x ) × (1 − e − i ( q − x − q + x ) L ) (cid:17) , (4)with γ ( m , k , m , k ) =( ε + m + tk − ∆ p y )( p y + iξk ) − ( ε + m + tk − ∆ p y )( p y + iξk ) , for each channel of p y . We show the tunneling probabil-ity as a function of the length of the barrier region L inFig.3. One can find that the mean value of the tunnel-ing probability | T X | shows the typical property of anordinary tunnel junction where the tunneling probabilitydecreases with an increase in L . FIG. 3. The tunneling probabilities at X as a function of L with ε F = 0 . t , ∆ = 0 . t , m = 0 . t , and L = t/ε F . However, the damping factor κ = Im[ q + x ] showsthe characteristic feature of the symmetry-protected 2DDirac semimetal unlike the ordinary 2D Dirac fermion.The damping factor can be written by a function of theratio r so of the spin-orbit coupling constant t so to thehopping matrix t , κ = s r so ( ξm/t − ∆ /tp y ) (1 + r so ) − ε /t − r so p y (1 + r so ) , and the tunnel decay length κ − drastically increaseswith a decrease in the spin-orbit coupling. This is be-cause the insulating gap induced by the magnetic ex-change potential reduces with a decrease in the spin-orbitinteraction. This dependence of the gap on the spin-orbitcoupling constant is quite different from the case of theordinary 2D Dirac fermion where the gap is fixed by themass term, i.e. the exchange potential, and does notchange with the strength of the spin-orbit coupling. Theextension of the tunnel decay length with the spin-orbitcoupling can be observed by controlling the strength t so which can be realized, e.g. by the effect of substrates.Finally, we discuss the difference between two Diracpoints X and X in the tunneling effect. We show theratio of tunneling probabilities at two Dirac points asa function of the relative strength of distortion of thelattice ∆ /t , which describes the non-equivalence in thetwo points, in Fig.4 and 5. When the distortion is small FIG. 4. The ratio of tunneling probabilities at X and X asa function of ∆ /t with ε F = 0 . t and t so = 0 . t . ∆ /t ≪
1, the tunneling at the X point are negligiblysmall compared with that at the X point. This is be-cause the sublattice degree of freedom works as a pseudospin and the reflection is suppressed at the X point bya mismatch of the pseudo spin between incident and re-flected waves. In Eq.(2), the direction of the pseudo spinis determined by two factors; the hopping matrix and thespin-orbit interaction. The hopping matrix between thesublattces with the hopping parameters t and ∆ cou-ples to s z in the subspace, and the spin-orbit interactionis represented by the in-plane component of the pseudospin s x and s y . Therefore, the pseudo spin is nearlyaligned in the z direction because the spin-orbit interac-tion is generally much smaller than the hopping matrix.With a small distortion ∆ /t ≪
1, the pseudo spin of the incident wave with 0 < p x and the reflected wave with p x < X . The incident andreflected waves at the X , on the other hand, have thenearly parallel pseudo spin because the sign of p y , whichis preserved in the tunneling process, is relevant to itsdirection. In practice, the tunneling probability is samein two valleys under the condition of ∆ /t = 1 where thecontribution of p x to the alignment of the pseudo spin isunchanged between the two points. FIG. 5. The ratio of tunneling probabilities at X and X asa function of ∆ /t with ε F = 0 . t and t so = 0 . t . The difference of tunneling probability in two valleyscan be enhanced with an increase in the exchange poten-tial m and the length of the insulating region as shownin Fig.4 and 5, respectively. The dominant valley for thetunneling can be selected by the direction of the tun-neling junction. The asymmetric tunneling effect in twovalleys produces the valley polarized current and givesa way to control the valley degree of freedom withoutvalley Hall effect.In conclusion, we studied the tunneling effect inthe non-symmorphic symmetry-protected 2D Diracsemimetal with a tunneling barrier arranged by the mag-netic exchange potential, and found that the tunnelingdecay length shows a quite different feature from ordi-nary 2D Diac fermions as a function of the strength ofthe spin-orbit interaction. The characteristic property isattributed to the C screw symmetry-breaking interac-tion which preserves the other non-symmorphic symme-try about a glide mirror operation and induces a rigorous2D Dirac semimetal. We also found that the tunnelingjunction works as a highly selective valley filter. S. Murakami, New. J. Phys. , 356 (2007). X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B , 205101 (2011). A. A. Burkov and L. Balents, Phys. Rev. Lett. , 127205 (2011). M. Neupane, S. Xu, R. Sankar, N. Alidoust, G. Bian,C. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin,A. Bansil, F. Chou, and M. Z. Hasan, Nature Communi-cations , 3786 (2014). S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy,B. Buechner, and R. J. Cava, arXiv: , 1309.7978 (unpub-lished). Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini,G. Xu, H. Weng, X. Dai, and Z. Fang,Phys. Rev. B , 195320 (2012). S. M. Young, S. Zaheer, J. C. Y. Teo,C. L. Kane, E. J. Mele, and A. M. Rappe,Phys. Rev. Lett. , 140405 (2012). S. Borisenko, Q. Gibson, D. Evtushinsky,V. Zabolotnyy, B. B¨uchner, and R. J. Cava,Phys. Rev. Lett. , 027603 (2014). Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,D. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai,Z. Hussain, and Y. L. Chen, Science , 864 (2014). A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109 (2009). C. L. Kane and E. J. Mele,Phys. Rev. Lett. , 226801 (2005). C. L. Kane and E. J. Mele,Phys. Rev. Lett. , 146802 (2005). S. M. Young and C. L. Kane,Phys. Rev. Lett. , 126803 (2015). Y. Zhao and A. P. Schnyder, arxiv , :1606.03698 (2016). L. M. Schoop, M. N. Ali, C. Straer, A. Topp,A. Varykhalov, D. Marchenko, V. Duppel,S. S. P. Parkin, B. V. Lotsch, and C. R. Ast,Nature Communications , 11696 (2016). J. Liu, D. Kriegner, L. Horak, D. Puggioni, C. Rayan Ser-rao, R. Chen, D. Yi, C. Frontera, V. Holy, A. Vish-wanath, J. M. Rondinelli, X. Marti, and R. Ramesh,Phys. Rev. B , 085118 (2016). T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. B , 121401(R) (2010). T. Habe and Y. Asano, Phys. Rev. B. , 195325 (2012). Y. Chen and H.-Y. Kee, Phys. Rev. B , 195145 (2014). T. Habe and Y. Asano, Phys. Rev. B89