Evolution of Edge States and Critical Phenomena in the Rashba Superconductor with Magnetization
aa r X i v : . [ c ond - m a t . s up r- c on ] J a n Evolution of Edge states and Critical Phenomena in the Rashba Superconductor with Magnetization
Ai Yamakage, Yukio Tanaka, and Naoto Nagaosa
2, 3 Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan Cross Correlated Materials Research Group (CMRG), ASI, RIKEN, WAKO 351-0198, Japan (Dated: September 30, 2018)We study Andreev bound states (ABS) and resulting charge transport of Rashba superconductor (RSC) wheretwo-dimensional semiconductor (2DSM) heterostructures is sandwiched by spin-singlet s -wave superconductorand ferromagnet insulator. ABS becomes a chiral Majorana edge mode in topological phase (TP). We clarifythat two types of quantum criticality about the topological change of ABS near a quantum critical point (QCP),whether ABS exists at QCP or not. In the former type, ABS has a energy gap and does not cross at zero energyin non-topological phase (NTP). These complex properties can be detected by tunneling conductance betweennormal metal / RSC junctions. Topological quantum phenomena and relevant quantumcriticality have been an important concept in condensed matterphysics [1, 2]. Recently, stimulated by the issue of Majoranafermion in condensed matter physics [3–6], topological quan-tum behavior of superconductivity becomes a hot topic [7–12]. One of the most crucial point is the property of the non-trivial edge modes in topological phase where edge modes areprotected by the bulk energy gap.The edge state of superconductor has been known from thestudy of Andreev bound state (ABS) in unconventional su-perconductors [13–15]. In high T C cuprate, dispersionlesszero energy ABS ubiquitously appears [14, 15] due to the signchange of the pair potential on the Fermi surface. The zero en-ergy state manifests itself as a zero bias conductance peak intunneling spectroscopy [15, 16]. Subsequently, the presenceof ABS with linear dispersion has been clarified in chiral p -wave superconductor [17] realized in Sr RuO , where timereversal symmetry is broken [18]. On the other hand, in thepresence of spin-orbit(SO) coupling with time reversal sym-metry, it has been revealed that spin-singlet s -wave pairingand spin-triplet p -wave one can mix each other due to the bro-ken inversion symmetry [19–21]. ABS appears as a helicaledge mode appears for ∆ p > ∆ s where we denote s -waveand p -wave pair potentials as ∆ s and ∆ p , respectively, with ∆ s > and ∆ p > [21, 22].The critical behavior of ABS has been discussed in spin-triplet chiral p -wave pairing [3]. By changing the chemicalpotential µ of spin-triplet chiral p -wave superconductor frompositive to negative, ABS as a chiral Majorana mode disap-pears. The corresponding quantum critical point is µ = 0 .Although, such a quantum phase transition can be possible in ν = 5 / fractional quantum Hall system [3] and cold atom[23, 24], it is significantly difficult to obtain superconductingstate for negative µ in electronic superconductors.In all of above works, ABS is generated from unconven-tional pairing with non-zero angular momentum. On the otherhand, in the presence of strong SO coupling with broken timereversal symmetry, chiral Majorana modes can be generatedfrom spin-singlet s -wave pairing [25, 26]. Fu and Kane haverevealed the presence of chiral Majorana mode at the bound-ary between ferromagnet and superconductor generated on the surface of topological insulator (TI). After that manipulatingMajorana mode in TI [26] and in semiconductor hetero struc-tures based on conventional spin-singlet s -wave superconduc-tor have been proposed in several contexts [27–29]. Sau etal. has proposed a unique Rashba superconductor where two-dimensional electron gas (2DEG) is sandwiched by conven-tional spin-singlet s -wave superconductor and ferromagneticinsulator [28]. These systems are really promising for futureapplication of quantum qubit since host superconductor is ro-bust against impurity scattering.Although there have been several theoretical studies aboutthe present RSC [30–32], the feature of the Andreev boundstate (ABS) and its relevance to the topological quantumphase transition has not been revealed at all. It is known thatABS emerges as a chiral Majorana edge mode in TP, however,the evolution of ABS in the non-topological phase (NTP) andits connection to quantum phase transition have not been clar-ified yet. To reveal these problems is indispensable to under-stand the tunneling spectroscopy of normal metal /RSC junc-tion system and future applications of quantum device.In this Letter, we study energy dispersions of ABS in RSCcomposed of 2DEG sandwiched by spin-singlet s -wave su-perconductor and ferromagnetic insulator. It is clarified thatthere are two types of quantum criticality for ABS, i.e. , quan-tum phase transition with or without ABS corresponding totype I and type II, respectively. In type I, ABS can existeven at critical point where bulk energy gap closes and inthe NTP. Nonzero ABS generated in the NTP does not crossat zero energy. These features are completely different fromthose in type II where edge states become absent both at thecritical point and in the NTP. The conventional criticality ofspinless spin-triplet chiral p -wave superconductor belongs totype II [3, 24]. The conductance between normal metal / RSCjunction shows wide variety of line shapes reflecting on thesenovel quantum criticalities. We also show the drastic jump ofthe conductance at critical point.A Hamiltonian of Rashba superconductor with magnetiza-tion is given by the following form [27–29] : H ( k ) = H ( k ) + H R ( k ) + H Z + H S , (1)where kinetic energy H , Rashba spin–orbit interaction E k k − µ c + µ c (a) mλ < | V z | − µ c + µ c (b) mλ > | V z | FIG. 1. (color online) Energy spectra of the normal ( ∆ = 0 ) states.(a)Zeeman (Rashba spin–orbit) interaction is dominant with mλ < | V z | . (b)Rashba spin–orbit interaction is dominant with mλ > | V z | .The critical value of chemical potential for the transition betweentopological and non–topological superconductors is given by ± µ c = ±√ V z − ∆ . (see discussion below eq. (6)) (RSOI) H R , Zeeman interaction H Z by exchange field fromFM insulator, and spin-singlet s –wave pair potential H S in-duced by proximity effect are H ( k ) = ξ k s τ z , H R ( k ) = λ ( s x τ k y − s y τ z k x ) , H Z = V z s z τ z , H S = − ∆ s y τ y , where s and τ are Pauli matrices, s and τ are × unit matrices, de-scribing electron spin and particle–hole degrees of freedom,respectively. We take the explicit form of kinetic energy as ξ k = k / m − µ with µ being chemical potential, for simplic-ity. The exchange energy in a 2DEG can be tuned by changingthe material of ferromagnetic insulator, or tuning the barrierthickness between the ferromagnetic insulator and the 2DEG.In the normal states ( ∆ = 0 ), there are two types of the energybands as shown in Fig. 1. For Zeeman interaction dominantcase with mλ < | V z | , there are two parabolic dispersions(Fig. 1(a)). On the other hand, for RSOI dominant case with mλ > | V z | , the shape of the energy band is wine–bottle like(Fig. 1(b)). As we shall see later, the difference between thesetwo types of energy bands in normal state becomes important.The eigenvalues of the Hamiltonian for the infinite sys-tem are given by E a ( k x , k y ) = √ η k + ζ k , E b ( k x , k y ) = −√ η k + ζ k , E c ( k x , k y ) = √ η k − ζ k , and E d ( k x , k y ) = −√ η k − ζ k with η k = ξ k + λ k + V z + ∆ ,ζ k = 2 q ( λ k + V z ) ξ k + V z ∆ , (2)where k is defined by k = q k x + k y with real k x and k y forthe plane wave. The corresponding eigenvectors u α ( k x , k y ) with α = a, b, c , and d are also obtained analytically.Let us now consider a semi–infinite RSC in x > with flatsurface at x = 0 . The wave function in the present system isgiven by ψ k y ,E ( x >
0) = X i =1 t i u i ( q i , k y ) e iq i x e ik y y . (3)When q i is a real number, the corresponding wave function ex-presses propagating wave, i.e. , scattering state. On the otherhand, when q i is a complex number, it describes an evanescentwave. Energy E and y –component of momentum k y are good FM insulator2DEG + Rashbas-wavesuperconductor
Metal xyz
FIG. 2. (color online) Normal metal (N) / Rashba superconductor(RSC) junction. Andreev bound state as edge state can exist denotedby the (red) arrow. quantum numbers. To obtain q i , we solve k for fixed E = E a ( k x , k y ) and E = E c ( k x , k y ) for E > [ E = E b ( k x , k y ) and E = E d ( k x , k y ) for E < ]. q i is given by q i = k x by postulating the constraints ∂E α ( q i , k y ) /∂q i > for scat-tering state, and Im q i > for evanescent state. Note herethat, in general, k and q i become complex numbers which canbe obtained by analytical continuation. The coefficient t i isdetermined by the confinement condition as ψ k y ,E (0) = 0 .Tunneling conductance of normal metal (N) / RSC junctionas shown in Fig. 2 is calculated based on the standard way[15, 33]. Suppose that the normal metal has no spin–orbitinteraction, i.e. , the Hamiltonian reads H M ( k ) = ( k / m − µ M ) s τ z , where µ M = µ − ǫ with ǫ being the energy ofbottom of the energy band, which is negative, and the interfacepotential is given by H I = Hs τ δ ( x ) . The wave function inN is given by ψ k y ,E,s ( x < " χ s e e ik ex x + X s ′ τ ′ r ss ′ τ ′ χ s ′ τ ′ e − iτ ′ k τ ′ x x e ik y y , (4)where the first term denotes an incident electron with spin s ,and χ sτ is the eigenvector of spin s for electron ( τ = +1 )or hole ( τ = − ), and k e x = q m ( µ M + E ) − k y and k h x = q m ( µ M − E ) − k y are momenta of reflected elec-tron and hole, respectively. On the other hand, the wave func-tion in RSC ( x > ) obeys the same form as in eq. (3).The boundary condition at the interface located on x = 0 is given by the following two expressions [34]. ψ ( −
0) = ψ (+0) , v (+0) ψ (+0) − v ( − ψ ( −
0) = − i Hτ z ψ (0) , wherevelocity in x –direction is v ( x ) = ∂H/∂k x | k x →− i∂ x . Solvingthe above equations, we obtain reflection (transmission) coef-ficient r ( t ). Charge conductance G normalized by its value G N in the normal state ( ∆ = 0 ) with V z = 0 , µ/mλ = 4 , µ M /mλ = 2 × , and Z = mH /µ M = 10 , whichcorresponds to the case of Figs. 3(h) and 3(k) with ∆ = 0 , at E / ∆ κ Topological Phase κ (b) κ (c) Critical Point E / ∆ κ (g) κ (h) Non–Topological Phase κ (i) G / G N eV/ ∆(j) eV/ ∆(k) Non–Topological Phase eV/ ∆ G / G N eV/ ∆(d) Topological Phase eV/ ∆(e) eV/ ∆(f)
Critical Point -1-0.500.51 -4 -2 0 2(a) -4 -2 0 2 -4 -2 0 2 4-1-0.500.51 -4 -2 0 2 -4 -2 0 2 -4 -2 0 2 400.511.52 -1 -0.5 0 0.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 1(l)00.511.52 -1 -0.5 0 0.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 1
FIG. 3. (color online) Energy spectra and tunneling conductances asa function of bias voltage ( eV / ∆ ) of the Rashba superconductor. Thehorizontal axis denotes the normalized momentum κ = k y / √ m ∆ .Zeeman interaction and Rashba spin–orbit interaction are fixed as V z / ∆ = 2 , mλ / ∆ = 0 . . The chemical potential is set as fol-lows. (a),(d): µ/ ∆ = 0 , (b),(e): µ/ ∆ = 1 . , (c),(f): µ/ ∆ = √ ,(g),(j): µ/ ∆ = 1 . , (h),(k): µ/ ∆ = 2 , (i),(l): µ/ ∆ = 2 . . zero bias voltage ( eV = 0 ) is given by G/G N = X s Z k F − k F dk y T s ( k y , E ) (cid:30) X s Z k F − k F dk y T s ( k y , , (5)with T s ( k y , E ) = 2 − P s ′ τ ′ τ ′ | r ss ′ τ ′ | and µ M = 2 mk F .Hereafter, the parameters are fixed as Z = 10 , µ M / ∆ = 10 ,and all the conductances G are normalized by the same valueof G N .We discuss the energy spectra and the tunneling conduc-tances, focusing on the difference of the criticality betweentwo RSCs with different chemical potential with µ > (Fig.3) and µ < (Fig. 4) for | V z | > mλ .In TP (Fig. 3(a) and Fig. 4(a)), ABS appears as a chiralMajorana edge mode, where | V z | > p µ + ∆ is satisfied. E / ∆ κ Topological Phase κ Critical Point (b) κ (c) Non–Topological Phase G / G N eV/ ∆(d) Topological Phase eV/ ∆(e)
Critical Point eV/ ∆ Non–Topological Phase -1-0.500.51 -4 -2 0 2(a) -4 -2 0 2 -4 -2 0 2 400.10.20.3 -1 -0.5 0 0.5 -1 -0.5 0 0.5 -1 -0.5 0 0.5 1(f)
FIG. 4. (color online) Energy spectra (upper) and tunneling conduc-tances (lower) of the Rashba superconductor for negative chemicalpotentials. (a),(d): µ/ ∆ = − , (b),(e): µ/ ∆ = −√ , (c),(f): µ/ ∆ = − . The other parameters are the same as in Fig. 3 Due to the presence of this mode, the corresponding tunnelingconductance has a zero bias peak as shown in Fig. 3(d) andFig. 4(d). For µ > , near the QCP [Fig. 3(b)], althoughABS appears as a chiral Majorana mode, the corresponding G has a zero bias dip as shown in Fig. 3(e) due to the presenceof a parabolic dispersion of bulk energy spectra near k y = 0 .At QCP (Fig. 3(c)), it is noted that ABS remains althoughthe bulk energy gap closes at k y = 0 . This feature is quitedifferent from µ < , where ABS is absent at QCP (Fig. 4.(b)). The resulting G has a V –shaped zero energy dip bothfor two cases shown in Figs. 3(f) and 4(e). For µ > , ABSstill remains even in the NTP as shown in Figs. 3(g), 3(h), and3(i). ABS has an energy gap and is absent around k y = 0 . Thetunneling conductance shows a gap structure around eV = 0 [Fig. 3(j)]. With the increase of µ , i.e. , away from QCP, theadditional non-zero ABS around k y = 0 [Fig. 3(h) and 3(i)]with the almost flat dispersion are generated. As a result, G has two peaks at the corresponding voltages inside the bulkenergy gap (Fig. 3(k) and 3(l)). On the other hand, for µ < ,ABS is absent in NTP as shown in Fig. 4(c). The resulting G is almost zero inside the bulk energy gap (Fig. 4(f)). Based onthese results, we can classify two types of criticality whetheredge state exists at QCP or not. We denote former type as typeI and the latter one as type II in the following.We have also studied for | V z | ≤ mλ . The energy spectra atQCP with positive µ (Fig. 5(a)) and negative µ (Fig. 5(b)) areshown. In this case, irrespective of the value of µ , ABS existsat QCP. Therefore, the resulting criticality is always type I.Type I and II transitions can be distinguished experimen-tally by the line shape of G . In type I transition, line shapeof G becomes almost symmetric with respect to eV = 0 asshown in Figs. 3(f), 5(c), and 5(d) as compared to that in typeII as shown in Fig. 4(f). Furthermore, G at type I transitiontakes one order of magnitude larger value than that at type II,due to contribution from the edge states.It is noted that the small value of Z does not qualitatively E / ∆ κ κ G / G N eV/ ∆(c) eV/ ∆(d)-2-1012 -6 -4 -2 0 2 4(a) -6 -4 -2 0 2 4 6(b)012345 -2 -1 0 1 -2 -1 0 1 2 FIG. 5. (color online) Energy spectra and tunneling conductancesof the Rashba superconductor for mλ > | V z | at quantum criticalpoint. (a),(c): µ/ ∆ = √ . , (b),(d): µ/ ∆ = −√ . . The otherparameters are taken as follows. mλ / ∆ = 5 , V z / ∆ = 1 . . change the results of the paper In the low transparency limit,the contribution from edge states becomes dominant for theconductance G , then the resulting line shape of G becomesinsensitive to the parameters of the normal metal, i.e., Z , µ M , and m . In the present case, the transmission probabil-ity in the normal state ( ∆ = 0 ) becomes sufficiently smallwith G N /G ∼ , where G denotes the maximum valueof G N , even for Z = 0 since the magnitudes of Fermi mo-menta in left normal metal ( x < ) and right RSC ( x > ) aremuch different with µ M /µ > .Here, we mention the criticality of ABS in spinless chiral p -wave superconductor. Hamiltonian of spinless chiral p –wavesuperconductor is given by H p ( k ) = (cid:18) k / m − µ ∆ p k − ∆ p k + − k / m + µ (cid:19) . (6)It is known that QCP is located at µ = 0 . ABS appears as achiral Majorana mode in TP ( µ > ) while it is absent in NTP µ < , respectively [3]. ABS disappears at QCP. In the lightof our classification, quantum criticality of spineless chiral p -wave superconductor belongs to the type II.To understand the difference of two types of criticality, wefocus on the energy dispersions in the normal state shown inFig. 1. Here we introduce the critical value of transition be-tween TP and NTP ± µ c = ± p V z − ∆ . The ABS is gener-ated from − k F to + k F , where the magnitude of k F is almostthe same with that of the large Fermi surface. First, we focuson the case with mλ < | V z | . The type I quantum phase tran-sition occurs at µ = µ c , shown in Fig. 3. In this case, thelarge Fermi surface survives as shown in Fig. 1(a). On theother hand, as shown in Fig. 4, type II quantum phase tran-sition occurs at µ = − µ c . In contrast to the type I, the largeFermi surface vanishes in the NTP as shown in Fig. 1(a). For mλ > | V z | , the quantum criticality always belongs to type I.Actually, as shown in Fig. 1(b), the large Fermi surface sur-vives both at µ = µ c and µ = − µ c . For type I, the number ofFermi surfaces is 2 in NTP and 1 in TP. On the other hand, for µ / ∆-3-2-10123 V z / ∆ G / G N TPTPNTP
FIG. 6. (color online) Conductance as a function of chemical poten-tial µ/ ∆ and Zeeman interaction V z / ∆ . Rashba spin–orbit interac-tion is taken as mλ / ∆ = 0 . . The transition of type I (II) occursat positive (negative) µ . The solid (broken) line indicates the criticalline of type I (II) transition. type II, the number of Fermi surface is 0 in NTP and 1 in TP.Above rich behavior of quantum criticality in RSC originatesfrom the simultaneous existence of the Rashba spin-orbit cou-pling and the Zeeman interaction.Finally, we show the zero–bias tunneling conductance ofRSC as a function of µ and V z in Fig. 6. The quantum phasetransition from NTP to TP occurs with tuning the parameter V z or µ . In accordance with this transition, the conductanceincreases by about three orders of magnitude, due to the con-tribution from zero energy ABS at k y = 0 .In this letter, we have calculated the energy spectrum andthe tunneling conductance of RSC and clarified its quantumcriticality. Quantum phase transition between topological andnon–topological superconductors has two types of criticalitywhether ABS survives or not at QCP. It is remarkable thatABS can remain at QCP in RSC distinctly from spinless chi-ral p –wave superconductor which is a prototype of topologicalsuperconductor. This stems from the structures of Fermi sur-faces which are spin–split by Rashba spin–orbit interaction inthe normal state. This results can provide a new perspective ofquantum criticality for topological superconductors. We haveconsidered only the spin-singlet s -wave superconductor. It isinteresting to study in the case of unconventional supercon-ductor where much richer quantum criticality can be expected[35–37].This work is supported by Grant-in-Aid for Scientific Re-search (Grants No. 17071007, No. 17071005, No. 19048008,No. 19048015, No. 22103005, No. 22340096, and No.21244053) from the Ministry of Education, Culture, Sports,Science and Technology of Japan, Strategic International Co-operative Program (Joint Research Type) from Japan Scienceand Technology Agency, and Funding Program for World-Leading Innovative RD on Science and Technology (FIRSTProgram). [1] X. G. Wen: Quantum Field Theory of Many-Body Systems (Ox-ford University Press, Oxford, 2004).[2] G. R. Volovik:
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