Transport across junctions of a Weyl and a multi-Weyl semimetal
We study transport across junctions of a Weyl and a multi-Weyl semimetal separated by a region of thickness which has a barrier potential . We show that in the thin barrier limit ( and with kept finite, where is velocity of low-energy electrons and is Planck’s constant), the tunneling conductance across such a junction becomes independent of . We demonstrate that such a barrier independence is a consequence of the change in the topological winding number of the Weyl nodes across the junction and point out that it has no analogue in tunneling conductance of either junctions of two-dimensional topological materials such as graphene or topological insulators or those made out of Weyl or multi-Weyl semimetals with same topological winding numbers. We study this phenomenon both for normal-barrier-normal (NBN) and normal-barrier-superconducting (NBS) junctions involving Weyl and multi-Weyl semimetals and discuss experiments which can test our theory.
A Weyl semimetal (WSM) hosts a three-dimensional (3D) gapless topological state whose wavefunction carries a non-zero topological winding number arising out of singularity in space weylrev (); ashvin1 (); ybk1 (); exp1 (). These singularities occur at Weyl points where the conduction and the valence bands touch. The low-energy effective Hamiltonian of these WSMs around these Weyl points is given by , where is the wave vector, is the velocity of the electrons near the Weyl point which depends on material parameters, and denotes Pauli matrices. These Weyl nodes occur in pairs and are protected due to either time-reversal or inversion symmetry breaking weylrev (). Such isotropic Weyl nodes are characterized by a topological winding number which takes values depending on the chirality of the electrons around the node. The electron around such nodes display spin momentum locking; this property along with the linear dispersion and a non-zero topological winding number distinguishes WSMs from ordinary metals. This distinction is manifested in several unconventional features associated with transport, magneto-transport and edge physics of these materials weylrev (); transport1 (); transport2 (); transport3 (); edge1 ().
More recently, materials with Weyl points having anisotropic dispersion in two transverse direction (chosen to be and in this work) has been discovered msm1 (). Such materials are termed as multi-Weyl semimetals (MSMs) since their anisotropic dispersion occurs due to merger of two or more Weyl nodes with same chirality. Such a merger is found to be topologically protected by point group symmetries (such as and rotational symmetries) msm2 (). The low energy dispersion of the electrons in MSMs remain linear in the symmetry direction (chosen to be in this work) but vanishes as (where ) with in the transverse directions: [msm1, ; msm2, ; msm3, ]. The winding number of these anisotropic Weyl points is given by and most MSMs discovered so far has or . The presence of a winding number different from unity modifies the helicity properties and the density of states of the electrons in these materials msmheli (). The signature of being different from unity also shows up in several optical and transport quantities such as longitudinal optical conductivity, anomalous Hall conductivity, collective modes, and magnetoresistance msmoptical (); msmcollective (); msmmagneto ().
It is well known that transport measurement across junctions of topological materials provides access to their topological properties and unravels several unconventional features that have no analog in standard metals geim1 (); beenakker1 (); ks1 (); ks2 (); beenakker2 (); weyl1 (). In 2D topological materials such as graphene, the tunneling conductance across graphene normal metal-barrier-normal metal (NBN) junctions, display oscillatory behavior and a transmission resonance as a function of the barrier potential geim1 (). Similar behavior is also seen in subgap tunneling conductance of graphene NBS junctions, where superconductivity is induced in graphene via a proximate -wave superconductor ks1 (). Such an oscillatory behavior and the transmission resonance phenomenon turns out to be a signature of the Dirac quasiparticles in graphene; they do not occur in standard metals whose quasiparticles obey Schrodinger equation. Similar behavior is also seen for quasiparticles on the surface of a topological insulator ks2 (). More recently tunneling conductance across NBN and NBS junctions of WSMs have also been studied weyl1 (); weyl2 (); weyl3 (); weyl4 (). In particular, it was found that the NBS junctions of Weyl semimetals may host a universal zero-bias conductance value of ; moreover, the subgap tunneling conductance displays oscillatory behavior as a function of the barrier strength which is expected in standard Dirac materials weyl3 ().
In this work, we study the tunneling conductance across NBN and NBS junctions between either a WSM () and a MSM () or two MSMs with separated by a barrier of width and a potential . Such junctions differ from their previously studied WSM counterparts in the sense that the topological winding number of the system changes across these junctions. The main results obtained from our study are as follows. First, we show that the tunneling conductance of these junctions becomes independent of the barrier potential in the thin barrier limit where and with being held fixed. We note that this behavior is in contrast to that found in junctions of both ordinary Schrodinger metals (where is a monotonically decaying function of ) and Dirac or WSM materials (where oscillates with ). We demonstrate that this independence is a consequence of difference of winding numbers between the WSM and MSM (or two MSMs) on two sides of the junction. Second, we find that the subgap tunneling conductance of the NBS junction depends crucially on the topological winding numbers. It vanishes if superconductivity is induced on the MSM with higher topological winding number; in contrast, it is finite when superconductivity is induced on the WSM or MSM with lower topological winding number. Third, we analyze the fate of the tunneling conductance for NBN junctions away from the thin barrier limit. We find that they display weak oscillatory dependence on the barrier potential for finite barrier thickness ; the amplitude (period) of these oscillations decreases (increases) with for any finite . For large , becomes independent of leading to the thin barrier result. Finally, we discuss experiments which can test our theory.
The plan of the rest of the paper is as follows. In Sec. II, we analyze the transport in NBN junctions between a WSM and a MSM or two MSMs with different winding numbers. This is followed by a similar analysis for NBS junctions in Sec. III. Finally, we discuss our main results, point out relevant experiments which may test our theory, and conclude in Sec. IV.
Ii NBN junctions
In this section we shall derive the conductance of a NBN junction between a WSM and a MSM or two MSMs with different winding numbers. The geometry of the setup is sketched in Fig. 1. The Hamiltonian of the system is given by
where is the Heaviside step function. The Hamiltonians and are given by
where and are the topological winding numbers in regions I and II as shown in Fig. 1, , and is the energy scale in which all energies are measured. In the rest of this work, we shall take this energy scale to be upper cutoff up to which the low-energy continuum Hamiltonians (Eq. 2) hold. Here and are the Fermi velocities for electrons in region I and III, is the momentum scale chosen to make all momenta dimensionless, and and are material specific constants whose precise numerical value is not going to alter our main results. We shall further choose a common chemical potential across the junction. In what follows, we shall apply a voltage across the junction and compute as a function of .
To compute , we first consider the electron wavefunction in region I. A straightforward calculation shows that the wavefunction for right(R) and left(L) moving electrons in region I in the presence of an applied voltage is given by weyl3 ()
where , , and we have measured all energies (wavevectors) in units of ().
The wavefunction in region I can be written in terms of and as
where is the amplitude of reflection from the barrier. We note here that leading to ; thus the azimuthal angle dependence of the wavefunction in region I can be interpreted as a spin rotation by an angle of about the axis.
In region II, the electrons see an additional applied potential . The right and the left moving electron wavefunction in this regime can be written as
where and . We note that when . Thus the wavefunction in region II can be written as
where and denotes amplitudes of right and left moving electrons in region II. We note that and are related by
In region III, the right moving electrons have a wavefunction given by
where , , and is the measure of the Fermi velocity mismatch across the junction. We note that and , for any given voltage , are related by
The wavefunction in region III is thus given by
where is the transmission amplitude across the junction. We note that .
To obtain the reflection and transmission amplitude across the barrier, we match the wavefunctions at and , where constitutes the width of the barrier in units of . This requires and and leads to
Solving for from these equations one obtains where
The expression of the transmission and hence the conductance can be obtained using Eq. 12 as and
Here denote the total number of transverse modes around a Weyl node up to the cutoff for which the continuum Weyl model used here holds and is the largest momentum channel participating in current transport across the junction. Note that is determined by the condition that both and must be real for a particular momentum channel to conduct.
Next, we note that in contrast to junctions between WSMs with or two similar MSMs with , , and hence possess non-trivial dependence for . To understand this phenomenon better, we now move to the thin barrier limit. In this limit, it is easy to see that , and . The boundary conditions can then be written as
We note that this implies that the dimensionless barrier potential induces a rotation by in spin space about the axis. For , this leads to oscillatory dependence of the conductance on . In contrast, for , since , the rotation induced by the barrier can be offset by changing , where . Thus the junction conductance, which involves a sum over all azimuthal angles, is expected to become barrier independent in the thin barrier limit.
To verify this expectation, we first substitute , and in Eq. 12 and obtain, after a few lines of algebra,
From Eq. 15, we find that in the presence of a change in winding number across region I and III (), appears as a phase shift to the azimuthal angle . Since , , and are independent of , the integration over in Eq. 13 is straightforward and yields
Thus becomes independent of in the thin barrier limit according to our earlier expectation. This independence is a direct consequence of dependence of which happens for . Thus such a barrier independence of requires a change in the topological winding number across the junction; consequently, this effect would not show up in junctions between WSMs or MSMs with . We would like to point out that this phenomenon can only occur in where there are more than one transverse directions; thus it does not have an analogue in 2D topological materials.
Next, we provide numerical support to our finding. To this end, we first obtain by numerically integrating over and . For all numerical plots we shall choose ; we have checked that the numerical values of these quantities do not alter qualitative nature of the results presented. The corresponding results are shown in Fig. 2 and 3. In Fig. 2, we show the variation of as a function of the applied voltage in the thin barrier limit for with and and for two representative values of . We have checked that the behavior of is identical for and qualitatively similar for for same . The different behavior of as a function of for and can be understood as follows. We note from Eq. 15 and 16 that for small , (Eq. 9). Consequently one may approximate
The integral over can then be analytically performed and leads to , where is a constant. Thus is a parabolic (linear) function of the applied voltage for and . An exactly similar behavior emerges when since is symmetric under the interchange of and . Note that for finite and , will always vary linearly with .
For both and , from Fig. 2, we find that is independent of . This independence can be more directly seen from Fig. 3(a). We also note that such a barrier independence is absent if ; this is easily seen from Fig. 3(b), where oscillates with for a junction between two WSMs () or MSMs (). We note that these numerical results confirm our earlier analytical expectation that the independence of is a consequence of the change in topological winding number across the junction.
Next, we investigate the fate of as a function of away from the thin barrier limit for several representative values of . To this end, we numerically compute from Eq. 12 and use Eq. 13 to obtain . Fig. 4 shows a plot of as a function of for several representative values of . We note that has small oscillatory dependence on ; the amplitude of these oscillations decay as is increased and becomes independent of for large . This is consistent with our earlier results in the thin barrier limit.
Iii NBS junctions
In this section, we study the transport through a NBS junction between a WSM and a MSM or two MSMs with different topological winding numbers. Throughout this section we shall work in the regime where the chemical potentials on the normal and superconducting regions ( and ) are large compared to the applied voltages but are small compared to so that one can use linearized model Hamiltonians to analyze the junction transport. The schematic representation of such a junction is given by Fig. 5. The MSM (or WSM) in region III has a proximate wave superconductor. In this section, we shall consider, following Ref. weyl3, , the case where the induced superconductivity is wave and the Cooper pairing connects two isotropic or anisotropic Weyl nodes with same chirality. The basis of this choice is the observation made in Ref. weyl3, that the inter-orbital superconducting pairing between nodes of opposite chirality is suppressed at low energy. We note that necessitates the presence of a at least four Weyl or multi-Weyl nodes in region III. With this model of induced superconductivity the Hamiltonian in region III is given by a matrix
where for denote Pauli matrices in particle-hole space, is the chemical potential, and , given by Eq. 2, may represent a WSM or a MSM depending on the value of . Here we shall choose the phase of the superconducting condensate to be zero without any loss of generality. The basic excitations of are Bogoliubov quasiparticles and quasiholes. The wavefunction of such right-moving quasiparticles, which would be necessary for our computation, are given by
where . In Eq. 19, correspond to electron-[hole-]like quasiparticles and are given by (for )
where we have scaled all energy scales by . We note that the wavefunctions of the quasiparticles and quasiholes retain the property .
The computation of tunneling conductance for such a junction follows the standard BTK formalism btk1 () applied to topological materials beenakker1 (); ks1 (); weyl1 (). To this end, we consider a right moving electron in region I approaching the barrier. Upon reflection (Andreev) from the barrier, a left moving electron (hole) propagates to the left. The wavefunctions of these electron and holes are given by
where , , and we shall choose for the all numerical evaluations. In region I, the wavefunction can then be written as
where and denotes amplitude or ordinary and Andreev reflections respectively. We note that so that for . Moreover, we find that ; thus for both electrons and holes, one can interpret dependence of the wavefunctions as a rotation in spin space about the axis.
In region II, the wavefunctions of right/left moving electrons and holes are given by Eq. LABEL:ehwav1 with , , , and . The wavefunctions of left and right-moving electrons and holes are therefore given by
where , and . We note that is related to by the relation . The wavefunction in region II is thus given by
In region III, the wavefunctions constitutes a superposition of electron-like and hole-like quasiparticles are given by
where are wavefunctions of electron- and hole-like quasiparticles given by Eq. 19. We note that one can express and in terms of and as
Also, we find that .
To compute and , we need to match the boundary conditions on the wavefunctions at and (Fig.5): and . The boundary condition at leads to
while that at yields