Magnetic breakdown spectrum of a
Kramers-Weyl semimetal
Thesis
submitted in partial fulfillment of the requirements for the degree of
Master of Science in
Physics
Author : Alvaro Don´ıs Vela´
Student ID : s2284839
Supervisor : Prof. Dr. C. W. J. Beenakker
2ndcorrector : Dr. V. Cheianov
Magnetic breakdown spectrum of a
Kramers-Weyl semimetal
´
Alvaro Don´ıs Vela
Instituut-Lorentz, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
July 8, 2020
Abstract
We calculate the Landau levels of a Kramers-Weyl semimetal thin slab in a perpendicular magnetic field B. The coupling of Fermi arcs on opposite surfaces broadens the Landau levels with a band width that oscillates periodically in 1/B.
We interpret the spectrum in terms of a one-dimensional superlattice induced by magnetic breakdown at Weyl points. The band width oscillations may be observed
as 1/B-periodic magnetoconductance oscillations, at much weaker fields than the oscillations due to Landau level quantization. No such spectrum appears in a
generic Weyl semimetal, the Kramers degeneracy at time-reversally invariant momenta is essential.
Contents
1 Introduction 1
1.1 Preface 1
1.2 Weyl semimetal 3
1.3 Fermi arc surface states 6
1.4 Kramers-Weyl semimetal 7
1.5 Magnetic breakdown 10
2 Dispersive Landau levels in a Kramers-Weyl semimetal with
slab geometry 13
2.1 Introduction 13
2.2 Boundary condition for Kramers-Weyl fermions 14
2.3 Fermi surface of Kramers-Weyl fermions in a slab 16
2.3.1 Dispersion relation 16
2.3.2 Fermi surface topology 18
2.4 Resonant tunneling between open and closed orbits in a magnetic
field 19
2.5 Dispersive Landau bands 20
2.6 Magnetoconductance oscillations 22
2.7.1 Hamiltonian 24
2.7.2 Folded Brillouin zone 24
2.8 Tight-binding model results 26
2.9 Conclusion 29
A Suplementary material 31
A.1 Coupling of time-reversally invariant momenta by the boundary 31 A.2 Criterion for the appearance of surface Fermi arcs 33 A.3 Calculation of the dispersive Landau bands due to the coupling
of open and closed orbits 34
Chapter
1
Introduction
1.1
Preface
The goal of condensed matter physics is to describe systems of electrons at low energies. The simplest case we can think of is the one of independent elec-trons in vacuum, which is described by the Schr¨odinger Hamiltonian H = 2mp2
e.
However, the key element that makes condensed matter problems interesting is the presence of a medium in which electrons exist. A crystalline structure restricts the symmetries present in vacuum bringing about all kinds of interst-ing phenomena. Startinterst-ing from a system with certain dimensionality and a set of symmetries we can write down a low energy effective Hamiltonian. For its complete description we would also have to take interactions into account. In the general case, studying systems of interacting electrons is a difficult task, and it is not always directly possible. Fortunately, topological classification al-lows us to obtain a lot of information about a system even with a non-interacting electron description. The idea is that if we manage to find a feature of a system that is linked to its topology, no interaction or perturbation that keeps it in the same topological class can remove that feature. In this way, we can use a non-interactive Hamiltonian and draw conclusions about the whole topological class to which it belongs.
Indeed, symmetry and topology are enough to explain a plethora of fascinating phenomena. Remarkably, this approach allows to identify a kind of systems in which electrons appear to behave like massless fermions. The topic of study of this thesis is a specific type of this kind of systems characterised by chiral and time reversal symmetries, the so called Kramers-Weyl semimetals – a subcate-gory of Weyl semimetals, which in some sense can be thought of as a 3D version of Dirac semimetals (like graphene). Chiral crystals are those in which there is a well defined handedness due to the absence of any roto-inversion symmetries [1]. In the presence of a strong enough spin-orbit coupling, these symmetries give
Figure 1.1: Ab initio calculation of the surface spectral weight on the (001) surface for CoSi. The predicted bulk chiral fermions project onto Γ and M, connected by a pair of topological Fermi arcs extending diagonally across the surface BZ. The colour bar limits correspond to high (H) and low (L) electron density. Source: [4]
rise to measurable massless low energy excitations at the time reversal invariant momenta that we call Kramers-Weyl fermions. One of the observable features of Kramers-Weyl semimetals is the presence of a certain type of surface states arising from the topology of the system called surface Fermi arcs [4] (see Fig. 1.1). Fermi arcs do not necessarily appear always in these systems, but when they do, they occupy a much larger part of the Brillouin zone than in ordinary Weyl semimetals.
The main result of this thesis is the prediction that the conductance of a Kramers-Weyl semimetal with a slab geometry in the presence of a perpen-dicular magnetic field oscillates periodically in 1/B under certain conditions. Here, it is important to point out that in our system there are actually two mechanisms that produce such oscillations. One of them is due to the Landau levels crossing the Fermi energy periodically in 1/B. This produces the well-known Shubnikov-de Haas (SdH) oscillations that appear in any system with closed electron orbits. However, here we report on a different mechanism based on the fact that our Landau levels are dispersive with a band width that os-cillates in 1/B. This effect relies on the presence of Fermi arc states spanning across the whole Brillouin zone characteristic of Kramers-Weyl semimetals. The period of these oscillations is much larger than the one due to SdH effect and can be observed at higher temperatures and lower field strenghts.
This thesis is structured in two chapters, in the first one, we go through the concepts and methods upon which this research stands: Weyl and Kramers-Weyl semimetals, Fermi arcs and magnetic breakdown. In the second chapter
1.2 Weyl semimetal 3
we give a detailed explanation on how the magnetoconductance oscillations appear in a KWS with slab geometry, as well as a comparison with a numerical implementation for a tight-binding realisation of such system as appears in [32]. At the end, an appendix is included with some proofs and calculations needed for the discussion.
1.2
Weyl semimetal
Weyl fermions take their name from the German physicist Hemann Weyl, who first derived a solution for the Dirac equation for massless fermions. In such case, the 4-component Dirac equation can be separated into two independent 2-component spinor equations by making a wise choice of the γ matrices (Weyl basis of γ matrices). Such equations are
i~∂tψ = ±cp · σψ (1.1)
The Hamiltonian H = cp · σ is easily diagonalizable in a basis of plane waves and its dispersion relation is E = ±cp, the same as that of photons. We see that it is linear with slope c and does not present a mass gap. This means that fundamental Weyl fermions would be ultrarelativistic particles, behaving in some sense like photons. For some time, it was believed that neutrinos would satisfy this equation until they were observed to be massive.
Iterestingly, in 2011 it was shown that the Weyl equation equation could ef-fectively describe electrons in certain solids with restricted symmetries [33]. Namely, time reversal or inversion symmetry must be broken. Indeed, this was later confirmed experimentally [34] and since then the interesting topological properties of Weyl semimetals have drawn a lot of attention.
Let us use an example to illustrate the properties of Weyl semimetals. In crys-tals, translational symmetry ensures that crystal momentum is a good quantum number and imposes periodicity of the dispersion relation in the Brillouin zone. With this, the following 2-band Hamiltonian [35] as a function of crystal mo-mentum k can give rise to Weyl fermions as we shall now see.
H(k) = −[tx(cos kx−cos k0)+m(2−cos ky−cos kz)]σx+tysin kyσy+tzsin kzσz.
(1.2) If we diagonalise it, we find that the two bands only touch at two points in the reciprocal space k = ±k0x (Fig. 1.2). They are called Weyl points and aroundˆ
them the dispersion relation is linear. If the valence band is completely filled, the density of states goes to zero at the Fermi energy (EF = 0 in this case) so
it is indeed a semimetal.
Expanding H around the Weyl points we obtain these expressions in terms of q±= (kx∓ k0, ky, kz) for each cone respectively.
H±(q±) = ± vxqxσx+ vyqyσy+ vzqzσz
-π -π 2 π 2 π kx Energy
Figure 1.2: Band structure of the hamiltonian H of equation 1.2 for several values of ky , with tx = ty = tz = m, kz = 0 and k0 = π/3. The thick lines correspond to
ky= 0. Notice that H is symmetric under the exchange ky↔ kz.
with vx = txsin k0, vy = ty and vz = tz. This can be seen as an anisotropic
version of the Weyl Hamiltonian of equation 1.1 (vx, vy, vz → c). Therefore,
the low energy excitations in this system behave like Weyl fermions of chirality κ±= ±sign(vxvyvz).
Probably the most interesting aspect of Weyl semimetals is that the Weyl nodes are topologically protected. This means that a small perturbation or modifica-tion of the parameters of the Hamiltonian cannot open a gap as long as the crystal symmetry remains present. This makes a big difference with respect to graphene, whose low energy Hamiltonian is a 2-dimensional version of the one in equation 1.3 but in which an arbitrarily small perturbation in the direction af the third Pauli matrix could open a gap.
Let us show this by finding a topological index associated to a Weyl point [35]. Consider the low energy Hamiltonian of a Weyl point H(k) = vk · σ. Notice that the following discussion also works for anisotropic systems in which case we just have to make the change of variables ki→ (v/|vi|)ki. Since we have k as
a parameter of the Hamiltonian explicitly, we can define the Berry connection and Berry curvature
A(k) = −iψ†(k)∇kψ(k)
B(k) = ∇k× A,
(1.4) being ψ(k) the eigenstate of H(k) corresponding to the lower energy band. There is a gauge freedom in the choice of ψ(k) since eiφ(k)ψ(k) provides an
equivalent description of the state. It is easy to check that whereas the Berry connection changes as A → A − ∇kφ under gauge transformations, the Berry
curvature is gauge invariant. This fact already indicates that it is convenient to see the Berry curvature as a kind of magnetic field defined in the Brillouin zone derived from the Berry connection.
1.2 Weyl semimetal 5
the flux of the Berry curvature through a surface enclosing the Weyl point, which equals 2πκ, being κ the chirality of the Weyl cone. Following the magnetic field interpretation of the Berry curvature, we can therefore see the Weyl points as magnetic monopoles of charge κ in the Brillouin zone.
In order to see this, we can calculate the low energy band obtained from the isotropic Weyl hamiltonian
ψ(k) = sinθ2 − cosθ 2e iϕ (1.5) where the (k, θ, ϕ) describe k in spherical coordinates. Notice that ψ has a singularity at the Weyl point k = 0 so we cannot argue that ∇k· B = ∇k·
(∇k× A) = 0 to state that the flux of B through a surface enclosing the Weyl
point vanishes. Instead, we can calculate explicitly A = −cos 2(θ/2) k sin θ ϕˆ B = 1 2k2kˆ (1.6)
and integrate the flux of B through a sphere, which easily yields 2π. It is trivial to check that this procedure yields −2π for a cone with negative chirality. We have then proved that a Weyl cone carries a a chiral charge that can only take discrete values. -π -π 2 0 π 2 π -π -π 2 0 π 2 π kx ky
Figure 1.3: Berry curvature B obtained from the Hamiltonian 1.2. The left panel shows a projection of a cut in the kz = 0 plane, and the right one is a 3-dimensional
representation focusing around the Weyl points.
There are two important conclusions that we can draw from this fact. The first one comes from the fact that the reciprocal space is periodic, thus the flux of B accross the Brillouin zone edges must vanish and therefore any system must have an even number of Weyl cones such that the total chiral charge is zero. The second conclusion is that a small perturbation cannot make the Berry flux through a sphere around the Weyl point switch to zero in a discrete step, so there must always be a Weyl node inside it in which the gap closes, in other words, the gap closing is topologically protected. The only effect of a perturbation is
to move the node around the Brillouin zone and the only way to open a gap is to tune the perturbation so that Weyl nodes with opposite chiral charge are brought together and annihilate each other.
1.3
Fermi arc surface states
A second aspect resulting from the topological nature of Weyl semimetals is the presence of surface states. The existence of these states is subject to transla-tional symmetry and is justified by mapping the problem to separate 2D prob-lems for different cuts in the reciprocal space each of them representing a 2D insulator.
In the theory of topological insulators, edge and surface states arise from the fact that the bulk Hamiltonian and vacuum belong to different topological classes. Since we cannot transition between two distinct topological phases continuously, there must be a gap closing at the interface, this is, a state on the surface or edge of the system. Here we must point out that the topological index classifying 2-dimensional insulators is the Chern number, defined as the Berry flux through the 2D Brillouin zone torus
Ch = 1 2π
Z
2dBZ
B(k)d2k (1.7)
where now, since we have a 2D system, the Berry curvature is just
B = ∂kxAky− ∂kyAkx. (1.8)
Now consider a Weyl semimetal with two nodes separated along the kx axis
at k = ±k0x and suppose we cut it in the xy-plane as in [35].ˆ Since the
system is still translationally invariant in the x (and y) direction, we know that kx is a good quantum number and we can consider eigenstates of the form
ψ(x, y, z) = eikxxf
kx(y, z) and solve the following problems independently:
Hkxfkx(y, z) = εkxfkx(y, z) (1.9)
For a given kx 6= k0 cut in the Brillouin zone, the bulk Weyl Hamiltonian is
gapped, so equation 1.9 represents a 2-dimensional insulator with an edge along the y axis. We can calculate the Chern number corresponding to this insulator with formula 1.7. The bulk Hamiltonian of the 2D topological insulator is given by Hkx(ky, kz) = HW eyl(kx, ky, kz), so B appearing in 1.7 coincides with the
Berry curvature of the Weyl Hamiltonian evaluated at the corresponding kx
plane.
Looking at Fig.1.4.a it is possible to understand that if two different kx cuts
have a Weyl node in between, then their corresponding Chern numbers must differ. This means that at least one of them must represent a 2D topological insulator which will have edge states. This edge states will exist for all the kx
values between a pair of Weyl nodes, so they will form a so called Fermi arc (Figs.1.4.b and 1.5.b).
1.4 Kramers-Weyl semimetal 7
If the system is cut along two parallel planes forming a slab, the reasoning is analogous and there will be surface states on both surfaces, as shown in Fig.1.5 for the associated 2D system.
Figure 1.4: a) Two different kxcuts in the Brillouin zone with different Chern
num-bers. b) Surface states and Fermi arc in a Weyl semimetal with a single surface. Source:[35]
Figure 1.5: Left: Band structure of a 2D topological insulator with two edges. Right: Fermi surface showing Fermi arcs (bright yellow lines) connecting Weyl points at the zone boundaries in CoSi. Source: [2]
1.4
Kramers-Weyl semimetal
In order for a system to host Weyl fermions, at least one of time reversal or some spatial roto-inversion symmetry must be broken. In Kramers-Weyl semimetals time reversal symmetry is kept. They are 3D systems, but we can illustrate some aspects of them by discussing how these symmetries affect the band structure of a Hamiltonian in 1D. The symmetries of a crystalline system are described by the group of spatial symmetries present in it. In 1D, there are only two such
possible symmetries: the identity and inversion symmetry. Therefore, there are only two possible groups, the trivial one and the group with both symmetries. Consider a one-dimensional spin-1/2 system. Inversion and time reversal trans-form the Hamiltonian respectively as
P : k → −k, σi→ σi
T : k → −k, σi→ −σi
(1.10)
Following this, it is possible to derive some conditions on the dispersions of systems with these symmetries:
Inversion → Eσ(k) = Eσ(−k)
Time reversal → Eσ(k) = E−σ(−k)
Both → Eσ(k) = E−σ(k)
(1.11)
We see that inversion symmetry imposes that both bands have even parity. For time reversal, it enforces that both bands are each other’s symmetric with re-spect to k = 0, which makes them cross at that point, so it imposes a degeneracy. If both symmetries are present the bands are degenerate everywhere.
From the conditions in expression 1.10 we can also deduce the lowest order terms in the expansion of the Hamiltonian around k = 0. If both symmetries are present, the first allowed term is k2σ
0 (appart from a constant offset). If
we allow for time reversal to be broken, we can include a lower order term proportional to σi. On the other hand, if time reversal symmetry remains but
inversion is broken, we can include a spin-orbit coupling term proportional to kσi. The effect on the band structure of these terms is shown in Fig.1.6. We
see how breaking only inversion symmetry keeps the point at k = 0 degenerate and the dispersion around it is linear in a similar way as in the Weyl cones.
k E a) k E b) k E c)
Figure 1.6: Dispersions around k = 0 for Hamiltonians with different symmetries. a): H = k2σ
0, keeps inversion and time reversal symmetries; b): H = (1/10)σz+ k2σ0,
keeps inversion and breaks time reversal; c): H = (1/2)kσz+ k2σ0, breaks inversion
and keeps time reversal.
There is a fundamental reason why the bands are degenerate at the time reversal invariant momentum (TRIM) k = 0. Kramers theroem states that in any time-reversal symmetric system with half integer total spin, for every state there exists at least another one with the same energy. These states are each other’s time reversed and since a TRIM is invariant under time reversal, the bands must cross by pairs at every TRIM.
1.4 Kramers-Weyl semimetal 9
In Fig.1.7 we show the band structure of a tight-binding system associated with the Hamiltonian represented in Fig.1.6.c for lattice constant a = 1 where t parametrizes the strength of the spin-orbit coupling
H(k) = t
2sin kσz+ 1
2(1 − cos k)σ0. (1.12) We can see how the bands do not only cross at k = 0, but also at the other TRIM at k = π. Notice how the spin-orbit coupling must be strong enough for the cones to be perceptible.
-π πk E a) -π πk E b) -π πk E c)
Figure 1.7: Band structure of the tight-binding Hamiltonian of expression 1.12 for different spin-orbit coupling strengths. a) t = 0, b) t = 1/20, c) t = 1/2.
If we want to extrapolate this to the case of 3D systems we have to include a lot more possible symmetries. There are 230 symmetry groups in 3D, so finding which of them represent systems hosting Weyl fermions at the time-reversal invariant momenta is not that simple. In [1], the authors report their finding that in the 65 groups without symmetries that switch chirality there are Weyl points at the TRIMs and they also find representative chiral materials in 33 of them.
The group SG16 is given as an example of chiral group. The tight-binding Hamiltonian obtained by including all symmetry-allowed nerest-neighbor hop-pings in this group is
H(k) = X
i=x,y,z
t1i cos ki+ tsisin kiσi, (1.13)
where tsi parametrizes the spin-orbit coupling and t1i the s-orbital-like hopping. The band structure of this hamiltonian is represented in Fig.1.8 in absence (tsi = 0) and presence (tsi 6= 0) of spin orbit coupling. Without SOC, both spin bands are completely degenerate and when SOC is switched on, all of the 8 TRIMs in the Brillouin zone become Weyl points of alternating chiralities protected by Kramers degeneracy.
Several things make Kramers-Weyl semimetals an especially interesting case of Weyl semimetals. First of all, there are plenty of materials with chiral symmetry in which KW fermions are relevant for the low energy physics [1]. Also, no perturbation preserving time reversal symmetry can open a gap, since Kramers degeneracy protects the band crossings at the TRIMs. If the perturbation breaks TR symmetry, it still needs to be strong enough to bring together pairs of nodes that are as far from each other as possible in the Brillouin zone. For the same reason, if there are Fermi arcs, they will be large, spanning along all the Brillouin
Figure 1.8: a) Crystalline structure of right and left-handed YSb2, it is an example
of chiral crystal with well-defined handedness. b) Weyl nodes at the TRIMs corre-sponding to the Hamiltonian of expression 1.13 colored according to their chirality. The actual chirality of all the nodes can switch depending on the signs of ts
i, but they
will always alternate. c) and d) Band structure along a certain line in the BZ without and with spin-orbit coupling respectively. Source: [1]
zone. This last feature is essential to obtain the open orbits that allow for the dispersive Landau bands reported in this thesis.
1.5
Magnetic breakdown
In the previous sections, we have presented the system in which we predict magnetoconductance oscillations, but it remains to explain the key concept to understand how they arise. In several occasions the behavior of electrons can be accurately described by semiclassical equations in which the expectation values of certain observables satisfy equations resembling classical ones.
An example of this is the case of a metal in a moderate constant magnetic field. An electron wave packet under these circumstances satisfies
˙k = e ~ ˙r × B ˙r = 1 ~∇kε(k) (1.14)
The first expression is analogous to the formula of the Lorentz force of a clas-sical electron in a magnetic field, and the second one just gives the velocity of
1.5 Magnetic breakdown 11
an electron wave packet. Notice that this expression is given in terms of the dispersion ε(k), so it is valid independently of the specific band structure of the system. In our case we will be dealing with a two-dimensional system in a perpendicular magnetic field. Let us take that simplification already since the discussion becomes more clear but it is conceptually the same as in the 3D case. Assuming B = Bˆz is perpendicular to the system, we can deduce from 1.14 that k will evolve in the reciprocal space in a direction that is always orthogonal to the gradient of the dispersion, this is, it will move along equienergy contours, which will be given by the Fermi surface at the corresponding energy. Also, it is immediate that ˙k ⊥ ˙r and ˙r = (~/eB) ˙k. This means that the trajectory of the electron in real space will have the same shape as in the reciprocal space but rotated by π/2 and rescaled by a factor l2
m= ~/eB.
Figure 1.9: Quasiclassical orbits of an electron in reciprocal and real spaces. The orbit in reciprocal space coincides with the Fermi surface at the corresponding energy.
In a simple case the trajectory C may be a closed orbit like in Fig.1.9, which gives rise to Landau levels quantized by the semiclassical rule
I C e ~ Adr = 2π(n − ν) → BS(ε) ≡ Φ(ε) =h e(n − ν) (1.15) which imposes that the gauge phase accumulated along the orbit is a multiple of 2π, being ν a constant offset that cannot be accounted for with semiclassi-cally. This is equivalent to the quantization of the magnetic flux through the surface defined for the orbit at that energy S(ε). For a 2DEG of non-relativistic electrons, this rule is actually exact in the quantum limit.
However, we must keep in mind that this semiclassical approach assumes adia-batic evolution. If we interpret the system as a Hamiltonian dependent on time through the parameter k(t), the assumption that the electron will remain in the band representing the original orbit is only true for sufficiently slow evolution. The limit for what is meant by sufficiently slow is set by the energy separation of bands according to the adiabatic theorem.
Figure 1.10: Trajectories in the reciprocal space corresponding to bands 1 and 2. The velocities of the corresponding real orbits are also depicted. The magnetic breakdown point between quasiclassical regions (- and +) allows for the electron to tunnel between trajectores. Source: [26]
Therefore, if two bands become too close for some k, the adiabatic assumption will fail and the electron may tunnel between different orbits. This phenomenon is called magnetic breakdown. Fortunately, the Landau-Zener formula gives us the probability that a transition between the two bands happens when they become close to each other [26].
TM B= e−Bc/B (1.16)
where Bc depends on the specific dispersion relation but is proportional to the
square of the energy difference between bands. In this way, it is possible to accout for the possibility that the electron does not stay in a single orbit. It is illustrative to mention already the type of orbits that we encounter in our system. As is shown in Fig.2.5 and explained in the next chapter, the electrons in our Kramers-Weyl slab will be tunneling between the orbits corresponding to the Fermi arcs and the lowest energy bulk band around the Weyl points. The fact that the Fermi orbits span along the whole Brillouin zone allows for open trajectories for the electron. As is explained in more detail in the appendix, it is possible to use the scattering formalism to understand how magnetic breakdown between open and close orbits results in dispersive Landau bands.
Chapter
2
Dispersive Landau levels in a
Kramers-Weyl semimetal with
slab geometry
2.1
Introduction
In the previous chapter, we explained that Kramers-Weyl fermions are mass-less low-energy excitations that may appear in the Brillouin zone near time-reversally invariant momenta (TRIM). Their gapless nature is protected by Kramers degeneracy, which enforces a band crossing at the TRIM. Crystals that support Kramers-Weyl fermions have strong spin-orbit coupling and be-long to one of the chiral point groups, without reflection or mirror symmetry, to allow for a linear rather than quadratic band splitting away from the TRIM. The materials are called topological chiral crystals or Kramers-Weyl semimetals — to be distinguished from generic Weyl semimetals where Kramers degeneracy plays no role. Several candidates were predicted theoretically [1, 2] and some have been realized in the laboratory [3–7].
These recent developments have motivated the search for observables that would distinguish Kramers-Weyl fermions from generic Weyl fermions [8–10]. Here we report on the fundamentally different Landau level spectrum when the semimetal is confined to a thin slab in a perpendicular magnetic field.
Generically, Landau levels are dispersionless: The energy does not depend on the momentum in the plane perpendicular to the magnetic field B. In contrast, we have found that the Landau levels of a Kramers-Weyl semimetal are broadened into a Landau band. The band width oscillates periodically in 1/B, producing an oscillatory contribution to the magnetoconductance.
Figure 2.1: Electron orbits in a thin slab geometry perpendicular to a magnetic field (along the x-axis), for a generic Weyl semimetal [16, 17] (at the left) and for a Kramers-Weyl semimetal (at the right). In each case we show separately a front view (in the x–y plane, to show how the orbits switch between top and bottom surfaces of the slab) and a top view (in the y–z plane, to indicate the magnetic flux enclosed by the orbits). The Kramers-Weyl semimetal combines open orbits (red arrows) with closed orbits enclosing either a large flux Φ or a small flux δΦ. Open and closed orbits are coupled by a periodic chain of magnetic breakdown events, spaced by l2m/a0 (with a0
the lattice constant and lm=
p
~/eB the magnetic length). The open orbits broaden the Landau levels into a band, the band width varies from minimal to maximal when δΦ is incremented by h/e. Because δΦ ∝ Bl4
m∝ 1/B, the band width oscillations are
periodic in 1/B.
The phenomenology is similar to that encountered in a semiconductor 2D elec-tron gas in a superlattice potential [11–15]. In that system the dispersion is due to the drift velocity of cyclotron orbits in perpendicular electric and magnetic fields. Here the surface Fermi arcs provide for open orbits, connected to closed orbits by magnetic breakdown at Weyl points (see Fig. 2.1).
No open orbits appear in a generic Weyl semimetal [16, 17], because the Weyl points are closely separated inside the first Brillouin zone, so the Fermi arcs are short and do not cross the Brillouin zone boundaries (a prerequisite for open orbits). The Landau band dispersion therefore directly ties into a defining property [1] of a Kramers-Weyl semimetal: surface Fermi arcs that span the entire Brillouin zone because they connect TRIM at zone boundaries.
In the next two sections 2.2 and 2.3 we first compute the spectrum of a Kramers-Weyl semimetal slab in zero magnetic field, to obtain the equi-energy contours that govern the orbits when we apply a perpendicular field. The resonant tun-neling between open and closed orbits via magnetic breakdown is studied in Sec. 2.4. With these preparations we are ready to calculate the dispersive Lan-dau bands and the magnetoconductance oscillations in Secs. 2.5 and 2.6. The analytical calculations are then compared with the numerical solution of a tight-binding model in Secs. 2.7 and 2.8. We conclude in Sec. 2.9.
2.2
Boundary condition for Kramers-Weyl fermions
The first step in our analysis is to characterize the surface Fermi arcs in a Kramers-Weyl semimetal, which requires a determination of the boundary con-dition for Kramers-Weyl fermions. This is more strongly constrained by
time-2.2 Boundary condition for Kramers-Weyl fermions 15
reversal symmetry than the familiar boundary condition on the Dirac equation [18]. In that case the confinement by a Dirac mass Vµ= µ( ˆnk· σ) generates a
boundary condition
Ψ = ( ˆn⊥× ˆnk) · σΨ. (2.1)
The unit vectors ˆnk and ˆn⊥ are parallel and perpendicular to the boundary,
respectively.
Although σ 7→ −σ upon time reversal, the Dirac mass may still preserve time-reversal symmetry if the Weyl fermions are not at a time-time-reversally invariant momentum (TRIM). For example, in graphene a Dirac mass +µ at the K-point in the Brillouin zone and a Dirac mass −µ at the K0-point preserves time-reversal symmetry. In contrast, for Kramers-Weyl fermions at a TRIM the Vµ term in the Hamiltonian is incompatible with time-reversal symmetry. To
preserve time-reversal symmetry the boundary condition must couple two Weyl cones, it cannot be of the single-cone form (2.1).
In App. A.1 we demonstrate that, indeed, pairs of Weyl cones at a TRIM are coupled at the boundary of a Kramers-Weyl semimetal. Relying on that result, we derive in this section the time-reversal invariant boundary condition for Kramers-Weyl fermions.
We consider a Kramers-Weyl semimetal in a slab geometry, confined to the y–z plane by boundaries at x = 0 and x = W . In a minimal description we account for the coupling of two Weyl cones at the boundary. To first order in momentum k, measured from a Weyl point, the Hamiltonian of the uncoupled Weyl cones is H±(k) = H0(k) + ε 0 0 ±H0(k) − ε , H0(k) =Pα=x,y,zvαkασα. (2.2)
The ± sign indicates whether the two Weyl cones have the same chirality (+) or the opposite chirality (−). The two Weyl points need not be at the same energy, we allow for an offset ε. We also allow for anisotropy in the velocity components vα.
The σα’s are Pauli matrices acting on the spin degree of freedom. We will also
use τα Pauli matrices that act on the Weyl cone index, with σ0 and τ0 the
corresponding 2 × 2 unit matrix. We can then write
H+= H0τ0+ ετz, H− = H0τz+ ετz. (2.3)
The current operator in the x-direction is j+= vxσxτ0for H+and j−= vxσxτz
for H−. The time-reversal operation T does not couple Weyl cones at a TRIM,
it only inverts the spin and momentum:
T H±(k)T−1= σyH±∗(−k)σy= H±(k). (2.4)
An energy-independent boundary condition on the wave function Ψ has the general form [18]
in terms of a Hermitian and unitary matrix M±. The matrix M±anticommutes
with the current operator j± perpendicular to the surface, to ensure current
conservation. Time-reversal symmetry further requires that
σyM±∗σy = M±. (2.6)
These restrictions reduce M± to the single-parameter form
M+(φ) = τyσycos φ + τyσzsin φ,
M−(φ) = τxσ0cos φ + τyσxsin φ.
(2.7)
The angle φ has a simple physical interpretation in the case H+, M+ case of
two coupled Weyl cones of the same chirality: It determines the direction of propagation of the helical surface states (the Fermi arcs). We will take φ = 0 at x = 0 and φ = π at x = W . This produces a surface state that is an eigenstate of τyσy with eigenvalue +1 on one surface and eigenvalue −1 on the opposite
surface, so a circulating surface state in the ±y-direction. (Alternatively, if we would take φ = ±π/2 the state would circulate in the ±z-direction.)
Notice that these are helical rather than chiral surface states: The eigenstates Ψ of τyσy with eigenvalue +1 contain both right-movers (σyΨ = +Ψ) and
left-movers (σyΨ = −Ψ). This is the key distinction with surface states in
a magnetic Weyl semimetal, which circulate unidirectionally around the slab [19–22].
In the case H−, M− that the coupled Weyl cones have the opposite chirality
there are no helical surface states and the physical interpretation of the angle φ in Eq. (2.7) is less obvious. Since our interest here is in the Fermi arcs, we will not consider that case further in what follows.
2.3
Fermi surface of Kramers-Weyl fermions in
a slab
2.3.1
Dispersion relation
We calculate the energy spectrum of H+ with boundary condition M+ from
Eq. (2.7) along the lines of Ref. [23]. Integration in the x-direction of the wave equation H+Ψ = EΨ with kx = −i~∂/∂x relates the wave amplitudes at the
top and bottom surface via Ψ(W ) = eiΞΨ(0), with
Ξ = W ~vx
σx(E − vykyσy− vzkzσz− ετz). (2.8)
As discussed in Sec. 2.2 we impose the boundary condition Ψ = M+(0)Ψ on the
x = 0 surface and Ψ = M+(π)Ψ on the x = W surface.
The round-trip evolution
2.3 Fermi surface of Kramers-Weyl fermions in a slab 17
Figure 2.2: Dispersion relation E(ky, kz) as a function of kzfor fixed ky= 1/W (left
panel) and as a function of kyfor fixed kz = 1/W (right panel), calculated from Eq.
(2.11) for vx= vy= vz ≡ vFand ε = ~vF/W . The surface states are indicated in red.
The avoided crossings at kz= 0 become real crossings for ε = 0.
then gives the determinantal equation
Det 1 + τyσye−iΞτyσyeiΞ = 0, (2.10)
which evaluates to [E2− ε2+ (v zkz)2− (vyky)2] sin w−sin w+ q−q+ = 1 + cos w−cos w+, (2.11)
with the definitions
q±2 = (E ± ε)2− (vyky)2− (vzkz)2, w±=
W ~vx
q±. (2.12)
In the zero-offset limit ε = 0 Eq. (2.11) reduces to the more compact expression vzkz q tan W q ~vx 2 = 1, q2= E2− (vyky)2− (vzkz)2, (2.13)
which is a squared Weiss equation [23, 24].
The dispersion relation E(ky, kz) which follows from Eq. (2.11) is plotted in Fig.
2.2. The surface states (indicated in red) are nearly flat as function of kz, so
they propagate mainly in the ±y direction. In the limit ε → 0 the bands cross at kz= 0, this crossing is removed by the energy offset.
Figure 2.3: Solid curves: equi-energy contours E(ky, kz) = EF for ε = 0 at three
values of W (in units of ~vF/EF with EF > 0): W = π/2 = Wc (red curve in left
panel), W = 1.4 < Wc (blue curve in left panel), and W = 1.8 > Wc (blue curve in
right panel). The calculations are based on Eq. (2.11) with vx = vy= vz ≡ vF. The
red dashed curve in the right panel shows the effect of a nonzero ε = 0.1 EF: The
intersecting contours break up into two open and one closed contour, separated at kz = 0 by a gap δky. The dotted arrows, perpendicular to the equi-energy contours,
point into the direction of motion in real space. The assignment of the bands to the upper and lower surface is in accord with the time-reversal symmetry requirement that a band stays on the same surface when (ky, kz) 7→ −(ky, kz).
2.3.2
Fermi surface topology
The equi-energy contours E(ky, kz) = EF are plotted in Fig. 2.3 for several
values of W . The topology of the Fermi surface changes at a critical width Wc= π 2 ~vx EF + O(ε). (2.14)
At W = Wc the surface bands from upper and lower surface touch at the Weyl
point ky = kz = 0, and for larger widths the upper and lower surface bands
decouple from a bulk band, in the interior of the slab.
For ε = 0 the surface and bulk bands intersect at kz= 0 when W > Wc. The
gap δky which opens up for nonzero ε is
δky=
4 πvy
|ε| + O(ε2), W > Wc. (2.15)
For later use we also record the area S0enclosed by the bulk band,
S0= 43π
√
2(W/Wc− 1)3/2k2F+ O(W/Wc− 1)2+ O(ε), (2.16)
where we have defined the 2D Fermi wave vector of the Weyl fermions via EF= ~kF
√
2.4 Resonant tunneling between open and closed orbits in a magnetic field 19
Figure 2.4: Electron orbits in a magnetic field perpendicular to the slab, following from the Fermi surface in Fig. 2.3 (W > Wc, ε > 0). The tunneling events
(mag-netic breakdown) between open and closed orbits are indicated. These happen with probability TMB given by Eq. (2.18). Backscattering of the open orbit via the closed
orbit happens with probability R given by Eq. (2.19). The area Sreal∝ 1/B2 of the
closed orbit in real space determines the 1/B periodicity of the magnetoconductance oscillations via the resonance condition BSreal= nh/e.
2.4
Resonant tunneling between open and closed
orbits in a magnetic field
Upon application of a magnetic field B in the x-direction, perpendicular to the slab, the Lorentz force causes a wave packet to drift along an equi-energy contour. Because ˙k = e ˙r × B the orbit in real space is obtained from the orbit in momentum space by rotation over π/2 and rescaling by a factor ~/eB = l2m
(magnetic length squared).
Inspection of Fig. 2.3 shows that for W > Wc closed orbits in the interior of
the slab coexist with open orbits on the surface. The open and closed orbits are coupled via tunneling through a momentum gap δky (magnetic breakdown
[25, 26]), with tunnel probability TMB = 1 − RMB given by the Landau-Zener
formula
TMB= exp(−Bc/B), Bc' (~/e)δk2y' (~ε/evF)2. (2.18)
In the expression for the breakdown field Bca numerical prefactor of order unity
is omitted [26, 27].
The real-space orbits are illustrated in Fig. 2.4: An electron in a Fermi arc on the top surface switches to the bottom surface when the Fermi arc terminates at a Weyl point [16]. The direction of propagation (helicity) of the surface electron may change as a consequence of the magnetic breakdown, which couples a right-moving electron on the top surface to a left-right-moving electron on the bottom surface. This backscattering process occurs with reflection probability
R = TMB 1 − RMBeiφ 2 = T 2 MB T2 MB+ 4RMBsin2(φ/2) . (2.19)
The phase shift φ accumulated in one round trip along the closed orbit is de-termined by the enclosed area S0 in momentum space,
Figure 2.5: Equi-energy contours in the ky–kz plane, showing open orbits coupled
to closed orbits via magnetic breakdown (red dotted lines). The closed contours en-circle Weyl points at K = (0, 0) and K0 = (0, π/a0) — periodically translated by
the reciprocal lattice vector G = (0, 2π/a0). Arrows indicate the spectral flow in a
perpendicular magnetic field. The large area SΣ (yellow) determines the spacing of
the Landau bands, while the small area S0 and the magnetic breakdown probabilities
TMB, TMB0 determine the band width.
with ν ∈ [0, 1) a magnetic-field independent offset.
Resonant tunneling through the closed orbit, resulting in R = 1, happens when φ is an integer multiple of 2π. We thus see that the resonances are periodic in 1/B, with period ∆(1/B) = 2πe ~S0 ≈ e h(W/Wc− 1) −3/2k−2 F . (2.21)
(We have substituted the small-ε expression (2.16) for S0.)
The Shubnikov-de Haas (SdH) oscillations due to Landau level quantisation are also periodic in 1/B. Their period is determined by the area SΣ≈ 2πkF/a0 in
Fig. 2.5, hence ∆(1/B)SdH= 2πe ~SΣ ≈ ea0 ~kF . (2.22)
Comparison with Eq. (2.21) shows that the period of the SdH oscillations is smaller than that of the magnetic breakdown oscillations by a factor kFa0(W/Wc−
1)3/2, which is typically 1.
2.5
Dispersive Landau bands
Let us now discuss how magnetic breakdown converts the flat dispersionless Landau levels into dispersive bands. The mechanism crucially relies on the fact that the surface Fermi arcs in a Kramers-Weyl semimetal connect Weyl points at time-reversally invariant momenta. Consider two TRIM K and K0 in the (ky, kz) plane of the surface Brillouin zone. We choose K = (0, 0) at the
zone center and K0 = (0, π/a0) at the zone boundary, with G = (0, 2π/a0) a
2.5 Dispersive Landau bands 21
Figure 2.6: Equi-energy contours in the ky–kzplane for surface Fermi arcs coupled by
magnetic breakdown (left panel, schematic) and for the bulk cyclotron orbit of a Weyl fermion (right panel). The quantization condition for the enclosed area is indicated, to explain why the Landau level spacing is ∝ B for the Fermi arcs, while it is ∝√B for the cyclotron orbit.
In the periodic zone scheme, the Weyl points can be repeated along the kz-axis
with period 2π/a0, to form an infinite one-dimensional chain (see Fig. 2.5). The
perpendicular magnetic field B induces a flow along this chain in momentum space, which in real space is oriented along the y-axis with period
L = (2π/a0)l2m= 2πvy/ωc, ωc= eBvya0/~. (2.23)
In the weak-field regime lm a0 the period L of the magnetic-field induced
superlattice is large compared to the period a0 of the atomic lattice. We seek
the band structure of the superlattice.
We distinguish the Weyl points at K and K0by their different magnetic break-down probabilities, denoted respectively by TMB= 1−RMBand TMB0 = 1−RMB0 .
We focus on the case that TMBand TMB0 are close to unity and the areas S0and
S00 of the closed orbits are the same — this is the small-ε regime in Eqs. (2.16) and (2.18). (The more general case is treated in App. A.3.)
The phase shift ψ accumulated upon propagation from one Weyl point to the next is gauge dependent, we choose the Landau gauge A = (0, −Bz, 0). For simplicity we ignore the curvature of the open orbits, approximating them by straight contours along the line ky= E/~vy. The phase shift is then given by
ψ = E ~vy π a0 l2m= πE ~ωc , (2.24)
the same for each segment of an open orbit connecting two Weyl points. The quantization condition for a Landau level at energy En is 2ψ + φ = 2πn,
n = 1, 2, . . ., which amounts to the quantization in units of h/e of the magnetic flux through the real-space area SΣl4m. Since SΣ S0the Landau level spacing
is governed by the energy dependence of ψ,
En+1− En≈ π(dψ/dE)−1= ~ωc. (2.25)
The Landau level spacing increases ∝ B and not ∝ √B, as one might have expected for massless electrons. The origin of the difference is explained in Fig. 2.6.
Figure 2.7: Dispersion relation of the slab in a perpendicular magnetic field B, calculated from Eqs. (A.13) and (A.14) (for W = 20 a0, TMB = 0.85, TMB0 = 0.95,
S0=S00, ν = 0). In the left panel B is chosen such that the phase φ accumulated by
a closed orbit at E = 0.08 ~vF/a0 equals 11π, in the right panel φ = 10π. When φ
is an integer multiple of 2π the magnetic breakdown is resonant, all orbits are closed and the Landau bands are dispersionless. When φ is a half-integer multiple of 2π the magnetic breakdown is suppressed and the Landau bands acquire a dispersion from the open orbits.
The Landau levels are flat when TMB = TMB0 = 1, so that there are no open
orbits. The open orbits introduce a dispersion along ky, see Fig. 2.7. Full
expressions are given in App. A.3. For RMB, R0MB 1 and S0 = S00 we have
the dispersion E(ky) = (n − ν)~ωc± (~ωc/π) sin(φ/2) × RMB+ R0MB+ 2 q RMBR0MBcos kyL 1/2 , (2.26)
where the phase φ is to be evaluated at E = (n − ν)~ωc.
Each Landau level is split into two subbands having the same band width |E(0) − E(π/L)| = 2(~ωc/π)| sin(φ/2)| min( √ RMB, √ R0 MB). (2.27)
The band width oscillates periodically in 1/B with period (2.21).
2.6
Magnetoconductance oscillations
The dispersive Landau bands leave observable signatures in electrical conduc-tion, in the form of magnetoconductance oscillations due to the resonant cou-pling of closed and open orbits. These have been previously studied when the
2.7 Tight-binding model on a cubic lattice 23
open orbits are caused by an electrostatic superlattice [11–15]. We apply that theory to our setting.
From the dispersion relation (2.26) we calculate the square of the group velocity V = ∂E/~∂ky, averaged over the Landau band,
hV2i = L 2π Z 2π/L 0 dE(ky) ~dky 2 dky = 2vy2sin2(φ/2) min(RMB, R0MB). (2.28)
For weak impurity scattering, scattering rate 1/τimp ωc, the effective diffusion
coefficient [15],
Deff = τimphV2i, (2.29)
and the 2D density of states N2D= (π~vya0)−1 of the Landau band, determine
the oscillatory contribution δσyy to the longitudinal conductivity via the Drude
formula for a 2D electron gas, δσyy = e2N2DDeff =4e 2 h vyτimp a0 sin2(φ/2) min(RMB, R0MB). (2.30)
The magnetoconductance oscillations due to magnetic breakdown (MB) coexist with the Shubnikov-de Haas (SdH) oscillations due to Landau level quantization. Both are periodic in 1/B, but with very different period, see Eqs. (2.21) and (2.22).
The difference in period causes a different temperature dependence of the mag-netoconductance oscillations. A conductance measurement at temperature T corresponds to an energy average over a range ∆E ≈ 4kBT (being the
full-width-at-half-maximum of the derivative of the Fermi-Dirac distribution). The oscillations become unobservable when the energy average changes the area S0
or SΣ by more than π/l2m. This results in different characteristic energy or
temperature scales, ∆ESdH= π l2 m ∂SΣ ∂E −1 '1 2~ωc, (2.31a) ∆EMB= π l2 m ∂S0 ∂E −1 ' 1 4 √ 2(W/Wc− 1)−1/2 ~ω c kFa0 . (2.31b)
(In the second equation we took W/Wc& 1.) For kFa0 1 and W/Wc close to
unity we may have ∆ESdH ∆EMB, so there is an intermediate temperature
regime ∆ESdH. 4kBT . ∆EMB where the Shubnikov-de Haas oscillations are
suppressed while the magnetic breakdown oscillations remain.
2.7
Tight-binding model on a cubic lattice
We have tested the analytical calculations from the previous sections numeri-cally, on a tight-binding model of a Kramers-Weyl semimetal [1]. In this section
we describe the model, results are presented in the next section.
2.7.1
Hamiltonian
We take a simple cubic lattice (lattice constant a, one atom per unit cell), when the nearest-neighbor hopping terms are the same in each direction α ∈ {x, y, z}. There are two terms to consider, a spin-independent term ∝ t0 that is even in
momentum and a spin-orbit coupling term ∝ t1σα that is odd in momentum,
H = t0 X α cos(kαa) + t1 X α σαsin(kαa) − t0. (2.32)
The offset is arbitrarily fixed at −t0.
There are 8 Weyl points (momenta k in the Brillouin zone of a linear dispersion), located at kx, ky, kz ∈ {0, π} modulo 2π. The Weyl points at (kx, ky, kz) =
(0, 0, 0), (π, π, 0), (π, 0, π), (0, π, π) have positive chirality and those at (π, π, π), (π, 0, 0),(0, π, 0),(0, 0, π) have negative chirality [1].
The geometry is a slab, with a normal ˆn in the x–z plane at an angle φ with the x-axis (so the normal is rotated by φ around the y-axis). The boundaries of the slab are constructed by removing all sites at x < 0 and x > W . In the rotated basis aligned with the normal to the slab one has
k0 x k0z = cos φ sin φ − sin φ cos φ kx kz , ky0 = ky. (2.33)
We will work in this rotated basis and for ease of notation omit the prime, writing kx or k⊥ for the momentum component perpendicular to the slab and
(ky, kz) = kk for the parallel momenta.
2.7.2
Folded Brillouin zone
The termination of the lattice in the slab geometry breaks the translation invari-ance in the perpendicular x-direction as well as in the z-direction parallel to the surface. If the rotation angle φ ∈ (0, π/2] is chosen such that tan φ = M/N is a rational number (M and N being coprime integers), the translational invariance in the z-direction is restored with a larger lattice constant a0 = a√N2+ M2,
see Fig. 2.8. There are then N2+ M2atoms in a unit cell.
In reciprocal space the enlarged unit cell folds the Brillouin zone. Relative to the original Brillouin zone the folded Brillouin zone is rotated by an angle φ around the y-axis and scaled by a factor (N2+ M2)−1/2 in the x and z-directions, see
Fig. 2.9. The reciprocal lattice vectors in the rotated basis are
ex= (2π/a0)ˆx, ey = (2π/a)ˆy, ez= (2π/a0)ˆz. (2.34)
2.7 Tight-binding model on a cubic lattice 25
Figure 2.8: Slice at y = 0 through the cubic lattice, rotated around the y-axis by an angle φ = arctan(M/N ) with M = 1, N = 2. The enlarged unit cell (red square), parallel to a lattice termination at x = 0 and x = W , has volume a0× a0 × a = (N2+ M2)a3.
Figure 2.9: Slice at ky= 0 through the Brillouin zone of the rotated cubic lattice, for
rotation angles φ = arctan(M/N ) with M = 1, N = 0, 1, 2, 3. Weyl points of opposite chirality are marked by a green or red dot. The panel for N = 3 shows how translation by reciprocal lattice vectors (blue arrows) folds two Weyl points onto each other.
coordinates π
a(cos φ + sin φ, cos φ − sin φ, 0) = π
a0(N + M, N − M, 0)
in the rotated lattice. Upon translation over a reciprocal lattice vector this is folded onto the center of the Brillouin zone (the Γ point) when N + M is an even integer, while it remains at a corner for N + M odd. The midpoints of a zone boundary, the X and Z points, are folded similarly, as summarized by
M 7→ Γ, Γ 7→ Γ, X 7→ M, Z 7→ M, for N + M even, M 7→ M, Γ 7→ Γ, X 7→ X, Z 7→ Z, for N + M odd.
Since the Weyl points at Γ and M have the same chirality, for N + M even we are in the situation that the surface of the slab couples Weyl points of the same chirality — which is required for surface Fermi arcs to appear (see Sec. 2.2). For N + M odd, in contrast, the Weyl points at the Γ and X points of opposite chirality are coupled by the surface, since these line up along the k⊥axis. Then
surface Fermi arcs will not appear. In App. A.2 we present a general analysis, for arbitrary Bravais lattices, that determines which lattice terminations support Fermi arcs and which do not.
2.8
Tight-binding model results
We present results for M = N = 1, corresponding to a φ = π/4 rotation of the lattice around the y-axis. The folded and rotated Brillouin zone has a pair of Weyl points of + chirality at K = (0, 0, 0) and a second pair of − chirality at K0= (π/a0, 0, π/a0) in the rotated coordinates (see Fig. 2.9, second panel, with a0= a√2). There is a second pair translated by ky = π/a.
Each Weyl point supports a pair of Weyl cones of the same chirality, folded onto each other in the first Brillouin zone. The Weyl cones at K have energy offset ε = |2t0|, while those at K0 have ε0 = 0. We may adjust the offset by adding
a rotational symmetry breaking term δH = δt0cos kza to the tight-binding
Hamiltonian (2.32). This changes the offsets into
ε = |2t0+ δt0|, ε0= |δt0|. (2.35)
In Fig.2.10 we show how the Fermi arcs appear in the dispersion relation con-necting the Weyl cones at kz = 0 and kz= π/a0. This figure extends the local
description near a Weyl cone from Fig. 2.2 to the entire Brillouin zone. The corresponding equi-energy contours are presented in Fig. 2.11. Increasing the spin-independent hopping term t0 introduces more bands, but the qualitative
picture near the center of the Brillouin zone remains the same as in Fig. 2.3 for W > Wc.
Figure 2.10: Dispersion relations of a slab (thickness W = 10√2 a in the x-direction, infinitely extended in the y–z plane) in zero magnetic field. The plots are calculated from the tight-binding model of Sec. 2.8 (with t0= 0.04 t1, δt0= −0.02 t1,
correspond-ing to ε = 0.06 t1, ε0 = 0.02 t1). The left and right panels show the dispersion as a
function of kz and ky, respectively. The curves are colored according to the electron
density on the surfaces: red for the bottom surface, blue for the top surface, with bulk states appearing black.
2.8 Tight-binding model results 27
Figure 2.11: Panels a (full Brillouin zone) and b (zoom-in near ky= 0) show
equi-energy contours at E = 0.167 t1 (when W ≈ 1.5 Wc), for the same system as in Fig.
2.10. In panels c and d the spin-independent hopping term t0 is increased by a factor
5 (at the same δt0= −0.02 t1).
The effect on the dispersion of a magnetic field B, perpendicular to the slab, is shown in Fig. 2.12 (see also App. A.4). The field was incorporated in the tight-binding model via the Peierls substitution in the gauge A = (0, −Bz, 0), with coordinate z restricted to |z| < L/2. Translational invariance in the y-direction is maintained, so we have a one-dimensional dispersion E(ky). The boundaries
of the system at z = ±L/2 introduce edge modes, which are visible in panel a as linearly dispersing modes near ky = ±12L/l2m (modulo π/a). Panels b,c,d
focus on the region near ky= 0, where these edge effects can be neglected. The
effect on the dispersion of a variation in ε and ε0 is qualitatively similar to that
obtained from the analytical solution of the continuum model, compare the four panels of Fig. 2.12 with the corresponding panels in Fig. A.2.
The width δE of the dispersive Landau bands (from maximum to minimum energy) is plotted as a function of 1/B in Fig. 2.13 and the periodicity ∆(1/B) is compared with the predicted Eq. (2.21) in Fig. 2.14. To remove the rapid Shubnikov-De Haas (SdH) oscillations we averaged over an energy interval ∆E around EF. This corresponds to a thermal average at effective temperature
Teff = ∆E/4kB. From Eq. (2.31), with kFa ≈ 0.2, W/Wc ≈ 1.5, we estimate
that the characteristic energy scale at which the oscillations average out is five times smaller for the SdH oscillations than for the oscillations due to magnetic
breakdown, consistent with what we see in the numerics.
Figure 2.12: Dispersion relation of a strip (cross-section W × L with W = 10a0and L = 30a0) in a perpendicular magnetic field B = 0.00707 (h/ea2) (magnetic length lm = 4.74 a). The four panels correspond to t0/t1, δt0/t1 equal to 0, 0 (panel a),
0.04, −0.02 (panel b), 0.04, −0.04 (panel c), 0.16, −0.16 (panel d). The surface Fermi arcs near ky= 0 form closed orbits in panel a, producing flat Landau levels, while in
panel d they form open orbits with the same linear dispersion as in zero field. Panels b,c show an intermediate regime where magnetic breakdown between closed and open orbits produces Landau bands with an oscillatory dispersion.
0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 Bc/B 0.000 0.001 0.002 0.003 0.004 0.005 La nd au le ve l b an d w idt h / t1 t0=0.04t1 t0=0.2t1 t0=0.4t1
Figure 2.13: Band width of the Landau levels versus inverse of magnetic field for W = 10a0, L = 500a0, δt0 = −0.02 t1 and three different values of t0. The band
widths are averaged over an energy window ∆E = 0.004 t1 around the Fermi energy
EF = 0.167 t1. The rapid Shubnikov-de Haas oscillations are averaged out, only the
2.9 Conclusion 29
Figure 2.14: Periodicity in 1/B of the Landau band width oscillations as a function of the Fermi energy, for W = 10a0, L = 500a0, t0 = 0.04 t1, and δt0 = −0.02 t1. The
filled data points are obtained numerically from the Landau band spectrum, similarly to the data shown for one particular EF in Fig. 2.13. The open circles are calculated
from the area S0 of the closed orbit in momentum space (as indicated in Fig. 2.11b),
using the formula ∆(1/B) = 2πe/~S0.
2.9
Conclusion
In conclusion, we have shown that Kramers-Weyl fermions (massless fermions near time-reversally invariant momenta) confined to a thin slab have a funda-mentally different Landau level spectrum than generic massless electrons: The Landau levels are not flat but broadened with a band width that oscillates pe-riodically in 1/B. The origin of the dispersion is magnetic breakdown at Weyl points, which couples open orbits from surface Fermi arcs to closed orbits in the interior of the slab.
The band width oscillations are observable as a slow modulation of the conduc-tance with magnetic field, on which the rapid Shubnikov-de Haas oscillations are superimposed. The periodicities are widely separated because the quantized areas in the Brillouin zone are very different (compare the areas S0 and SΣ in
Fig. 2.5). This is a robust feature of the band structure of a Kramers-Weyl semimetal, as illustrated in the model calculation of Fig. 2.11. Since generic Weyl fermions have only the Shubnikov-de Haas oscillations, the observation of two distinct periodicities in the magnetoconductance would provide for a unique signature of Kramers-Weyl fermions.
The dispersive Landau band is interpreted as the band structure of a one-dimensional superlattice of magnetic breakdown centra, separated in real space by a distance L = (eBa0/h)−1 — which in weak fields is much larger than the
atomic lattice constant a0. Such a magnetic breakdown lattice has been
stud-ied in the past for massive electrons [26], the Kramers-Weyl semimetals would provide an opportunity to investigate their properties for massless electrons.
Acknowledgements
This work has been carried out at the Theoretical Nanophysics group in the Lorentz Institute. I would like to express my gratitude to Micha l, Jakub, Carlo and especially Gal for their help and guidance in acquiring the knowledge nec-essary to accomplish the project and introducing me to the research world. The tight-binding model calculations were performed using the Kwant code [29].
Appendix
A
Suplementary material
A.1
Coupling of time-reversally invariant
mo-menta by the boundary
The derivation of the boundary condition for Kramers-Weyl fermions in Sec. 2.2 relies on pairwise coupling of Weyl cones at a TRIM by the boundary. Let us demonstrate that this is indeed what happens.
Consider a 3D Bravais lattice and its Brillouin zone. A time-reversally-invariant momentum (TRIM) is by definition a momentum K such that K = −K + G with G a reciprocal lattice vector, or equivalently, K = 12G. Now consider the restriction of the lattice to x > 0, by removing all lattice points at x < 0. Assume that the restricted lattice is still periodic in the y–z plane, with an enlarged unit cell. Fig. 2.8 shows an example for a cubic lattice.
The enlarged unit cell will correspond to a reduced Brillouin zone, with a new set of reciprocal lattice vectors ˜G. The original set K1, K2, K3, . . . of TRIM
is folded onto a new set ˜K1, ˜K2, ˜K3, . . . in the reduced Brillouin zone. The
folding may introduce degeneracies, such that two different K’s are folded onto the same ˜K. The statement to prove is this:
Each TRIM ˜K in the folded Brillouin zone is either degenerate (because two K’s were folded onto the same ˜K), or there is a second TRIM ˜K0 along the kx-axis.
Fig. 2.9 illustrates that this statement is true for the cubic lattice. We wish to prove that it holds for any Bravais lattice.
into ˜a1, ˜a2, ˜a3. The two sets are related by integer coefficients nij, ˜ ai= 3 X j=1 nijaj, nij ∈ Z. (A.1)
The corresponding primitive vectors b, ˜b in reciprocal space satisfy
bi· aj = 2πδij, ˜bi· ˜aj = 2πδij. (A.2)
Any momentum k can thus be expanded as k = 1 2π 3 X i=1 (˜ai· k)˜bi= 1 2π 3 X i,j=1 nij(aj· k)˜bi. (A.3)
A TRIM Kαin the first Brillouin zone of the original lattice is given by
Kα=12 3
X
i=1
mα,ibi, mα,i∈ {0, 1}. (A.4)
The index α labels each TRIM, identified by the 8 distinct triples (mα,1, mα,2, mα,3) ∈
Z2⊗ Z2⊗ Z2. Subsitution into the expansion (A.3) gives
Kα= 12 3 X l=1 mα,l 1 2π 3 X i,j=1 nij(aj· bl)˜bi = 1 2 3 X i,j=1 mα,jnij˜bi. (A.5) mα,1mα,2mα,3 ni1ni2ni3 (mod 2) 000 001 010 011 100 101 110 111 000 0 0 0 0 0 0 0 0 001 0 12 0 12 0 12 0 12 010 0 0 12 12 0 0 12 12 011 0 12 12 0 0 12 12 0 100 0 0 0 0 12 12 12 12 101 0 12 0 12 12 0 12 0 110 0 0 12 12 12 12 0 0 111 0 12 12 0 12 0 0 12
Table A.1: Values of να,i calculated from Eq. (A.6), for each triple ni1ni2ni3 and
each triple mα,1mα,2mα,3 (both ∈ Z2 ⊗ Z2 ⊗ Z2). If we select any two rows and
intersect with any column to obtain an ordered pair of values ν, ν0, we can then find a second column with the same ν, ν0 at the intersection.
We now fold Kα7→ ˜Kαinto the first Brillouin zone of the ˜b reciprocal vectors,
˜ Kα= 3 X i=1 να,i˜bi, να,i∈ [0, 1), να,i=12 3 X j=1 mα,jnij (mod 1). (A.6)
A.2 Criterion for the appearance of surface Fermi arcs 33
In Table A.1 we list for each TRIM and each choice of (ni1, ni2, ni3) ∈ Z2⊗ Z2⊗
Z2the corresponding value of να,i∈ {0,12}.
We fix the y and z-components of ˜Kαby specifying να,2 and να,3 ∈ {0,12} and
ask how many choices of α remain, so how many values of α satisfy the two equations να,2=12 3 X i=1 n2imα,i(mod 1), να,3=12 3 X i=1 n3imα,i(mod 1). (A.7)
Inspection of Table A.1 shows that the number of solutions is even. More specifically, there are
8 solutions if n21, n22, n23 and n31, n32, n33both equal 000 mod 2;
4 solutions if only one of n21, n22, n23 and n31, n32, n33 equals 000 mod 2;
4 solutions if n21, n22, n23and n31, n32, n33are identical and different from
000 mod 2;
2 solutions otherwise.
The multiple solutions correspond to pairs Kα and Kβ that are either folded
onto the same ˜Kα= ˜Kβ (if det n = 0 mod 2), or onto ˜Kα and ˜Kβ that differ
only in the x-component (if det n = 1 mod 2). These are the TRIM that are coupled by the boundary normal to the x-axis.
A.2
Criterion for the appearance of surface Fermi
arcs
When the boundary couples only Weyl cones of the same chirality, these persist and give rise to surface Fermi arcs. If, however, opposite chiralities are coupled, then the boundary gaps out the Weyl cones and no Fermi arcs appear. Which of these two possibilities is realized can be determined by using that the parity of mα1+ mα2+ mα3determines the chirality of the Weyl cone at Kα.
Table A.2 identifies for each choice of n21, n22, n23 and n31, n32, n33 how many
pairs of Weyl cones of opposite chirality are folded onto the same point of the surface Brillouin zone. We conclude that surface Fermi arcs appear if either
n2i+ n3i= 1 mod 2 for each i, or
n21, n22, n23= 111 mod 2, or
n31n32n33 (mod 2) n21n22n23 (mod 2) 000 001 010 011 100 101 110 111 000 4 2 2 2 2 2 2 0 001 2 2 1 1 1 1 0 0 010 2 1 2 1 1 0 1 0 011 2 1 1 2 0 1 1 0 100 2 1 1 0 2 1 1 0 101 2 1 0 1 1 2 1 0 110 2 0 1 1 1 1 2 0 111 0 0 0 0 0 0 0 0
Table A.2: Number of pairs of opposite-chirality Weyl cones that are coupled by a surface termination characterized by the integers n2i, n3i, i ∈ {1, 2, 3}. When this
number equals 0 the surface couples only Weyl cones of the same chirality and surface Fermi arcs will appear. If the number is different from zero the surface does not support Fermi arcs.
A.3
Calculation of the dispersive Landau bands
due to the coupling of open and closed
or-bits
To calculate the effect of the coupling of open and closed orbits on the Landau levels we apply the scattering theory of Refs. [15, 26, 28] to the equi-energy contours shown in Fig. A.1. We distinguish the two Weyl points at kz= 0 and
kz = π/a0 by their different magnetic breakdown probability, denoted
respec-tively by TMB= 1 − RMB and TMB0 = 1 − R0MB. The areas of the closed orbits
may also differ, we denote these by S0 and S00 and the corresponding phase
shifts by φ = S0l2m+ 2πν and φ0= S00lm2 + 2πν.
The coupling of the closed and open orbits at these two Weyl points is described
Figure A.1: Equi-energy contours in the ky–kz plane. The labeled wave amplitudes
A.3 Calculation of the dispersive Landau bands due to the coupling of open and
closed orbits 35
Figure A.2: Dispersion relation of the slab in a perpendicular magnetic field, calcu-lated from Eqs. (A.13) and (A.14) for W = 1.8 a0, S0=S00, ν = 1/2, B = 0.1 ~/ea20.
The four panels correspond to different choices of the magnetic breakdown probabili-ties TMB and TMB0 at the two Weyl points. At the two extremes of strong and weak
magnetic breakdown we see dispersionless Landau levels (left-most panel) and linearly dispersing surface modes (right-most panel).
by a pair of scattering matrices, given by b− L b+R =r t t r ·b + L b−R , r = TMBe iφ/2 1 − RMBeiφ , (A.8a) t = −√RMB+ TMB √ RMBeiφ 1 − RMBeiφ , (A.8b)
for the Weyl point at kz= 0, and similarly for the other Weyl point at kz = π/a0
(with TMB 7→ TMB0 , φ 7→ φ0). The coefficients can be rearranged in an
energy-dependent transfer matrix, b+ R b−R = T (E)b + L b−L , T =t − r 2/t r/t −r/t 1/t , (A.9)
and similarly for T0 (with t 7→ t0, r 7→ r0). The transfer matrices are energy dependent via the energy dependence of S0 and hence of φ.
We ignore the curvature of the open orbits, approximating them by straight contours along the line ky= E/~vy. The phase shift accumulated upon
propa-gation from one Weyl point to the next, in the Landau gauge A = (0, −Bz, 0), is then given by ψ = E ~vy π a0 l2m= πE ~ωc , ωc = eBvya0/~. (A.10)
The full transfer matrix over the first Brillouin zone takes the form c+ R c−R = Ttotal(E) a+ R a−R , (A.11) Ttotal= t0− r02/t0 r0/t0 −r0/t0 1/t0 eiψ 0 0 e−iψ t − r2/t r/t −r/t 1/t eiψ 0 0 e−iψ , (A.12) tr Ttotal= eiφ− R MB eiφ0− R0 MB + 1 − e iφR MB 1 − eiφ0R0 MB − 2TMBTMB0 e 1 2i(φ+φ 0)+2iψ e2iψ eiφ− 1 eiφ0 − 1pRMBR0MB . (A.13)
Because det Ttotal= 1, the eigenvalues of Ttotalcome in inverse pairs λ, 1/λ. The
transfer matrix translates the wave function over a period L in real space, so we require that λ = eiqLfor some real wave number q, hence λ+1/λ = eiqL+e−iqL,
or equivalently [28]
tr Ttotal(E) = 2 cos qL. (A.14)
(In the main text we denote q by ky, here we choose a different symbol as a
reminder that q is a conserved quantity, while the zero-field wave vector is not.) A numerical solution of Eq. (A.14) is shown in Figs. 2.7 and A.2.
For TMB and TMB0 close to unity an analytical solution En(q) for the dispersive
Landau bands can be obtained. We substitute ψ = π(n − ν) − (φ + φ0)/4 + πδE/~ωc into Eq. (A.13) and expand to second order in δE and to first order
in RMB, R0MB. Then we equate to 2 cos qL to arrive at
En±(q) = (n − ν)~ωc± δE(q), (A.15a) (πδE/~ωc)2= ρ + ρ0+ 2 p ρρ0cos qL, (A.15b) ρ = RMBsin2(φ/2), ρ0= R0MBsin 2(φ0/2), (A.15c)
where φ and φ0 are evaluated at E = (n − ν)~ωc. Corrections are of second
order in RMBand R0MBand we have assumed that the areas S0, S00 of the closed
orbit are small compared to kF/a0 — so that variations of φ and φ0 over the
Landau band can be neglected relative to the band spacing ~ωc.
A.4
Landau levels from surface Fermi arcs
As explained in Fig. 2.6, the spacing of Landau levels formed out of surface Fermi arcs varies ∝ B — in contrast to the √B dependence for unconfined massless electrons. In the tight-binding model of Sec. 2.8 we can test this by setting ε = ε0 = 0, so that there are only closed orbits and the Landau levels are dispersionless. The expected quantization is
En= (n − ν)~ωc, ωc= eBvFa0/~, n = 0, 1, 2, . . . (A.16)
with vFthe velocity in the surface Fermi arc, connecting Weyl points spaced by
π/a0. As shown in Fig. A.3, this agrees nicely with the numerics.
In an unconfined 2D electron gas, the offset ν equals 1/2 or 0 for massive or massless electrons, respectively. For the surface Fermi arcs we observe that ν depends on the parity of the number of unit cells between top and bottom surface: ν = 0 if W/a0 is odd, while ν = 1/2 if W/a0 is even. This parity effect suggests that the coupling of Fermi arc states on opposite surfaces, needed to close the orbit in Fig. 2.1, introduces a phase shift that depends on the parity of W/a0. We are not aware of such a phase shift for generic Weyl semimetals [16, 17, 30, 31], it seems to be a characteristic feature of Kramers-Weyl fermions that deserves further study.
