Tunnelling theory of Weyl semimetals in proximity to a metallic band

We consider a minimal Hamiltonian which captures the Fermi arc feature. This can be achieved either by breaking $\mathcal{T}$ while preserving $\mathcal{I}$ or vice-versa. In order to work with smaller matrices, we choose the former. Explicitly, then, our Hamiltonian must satisfy $H\left(\mathbf{k}\right)=\sigma_{z}H\left(-\mathbf{k}\right)\sigma_{z}$ and $H\left(\mathbf{k}\right)\neq\sigma_{y}H^{*}\left(-\mathbf{k}\right)\sigma_{y}$. A simple tight-binding Hamiltonian which abides by these symmetries is ($\hbar=\text{lattice constant}=1$) [1]

Here, $\mathbf{c}_{\mathbf{k}}=\left(c_{\mathbf{k},\uparrow},c_{\mathbf{k},\downarrow}\right)^{\top}$ is an annihilation operator in momentum space, $t_{s}$ ($s=x,y,z$) is the strength of hopping in the $s$-direction and ${\sigma}$ are the Pauli spin matrices. We further set $t_{x}=t_{y}=t_{z}=t>0$ for simplicity. The Hamiltonian (2) admits the bulk energies

These vanish at $\mathbf{k}^{\pm}_{w}=\left(0,0,\pm\arccos{\gamma}\right)\equiv\left(0,0,\pm k_{w}\right)$ – the aforementioned Weyl nodes. We emphasize the importance of the $\cos{k_{x,y}}$ terms, without which there would be more than two nodes in the BZ for a given $\gamma$.

These gapless bulk momenta $\mathbf{k}^{\pm}_{w}$ suggest that $H_{w}$ exhibits different phases that depend solely on the arc length parameter $\gamma$. For $\gamma>1$, $m\left(\mathbf{k}\right)>0$ for all $\mathbf{k}$ and the system is trivially gapped. As $\gamma$ decreases to $1$, a pair of Weyl nodes appear at the origin and move outward along $k_{z}$ as $\gamma$ decreases further. This defines a gapless topological phase whereby a nonzero Berry flux flows within the momentum range $|k_{z}|<k_{w}$ from the node of negative chirality to the one of positive chirality. Consequently, the Chern number – defined for a fixed $k_{z}$ – is nonzero between the nodes, and zero beyond them. When $\gamma\leq-1$, the Weyl nodes reach the BZ boundaries and disappear, leaving the bulk dispersion with an inverted band gap. Between $-5<\gamma<-1$, the same process occurs for Weyl nodes with $(k_{x},k_{y})=(0,\pi)$, $(\pi,0)$, and $(\pi,\pi)$, until $\gamma<-5$ where the system is again gapped and trivial for all $\mathbf{k}$. In all numerical results that follow, we take $\gamma=0$ ($k_{w}=\pi/2$), well within the gapless topological regime and with a Fermi arc length $k_{\mathrm{arc}}=\pi$. The bare WSM’s surface spectrum, Fermi arc, and topological phases are shown in Fig. 1.

To draw out the tunnelling properties of the Weyl semimetal, we couple it to a simple parabolic band via non-magnetic surface tunnelling. The band’s Hamiltonian is spin-independent and reads

where $t_{m}$ is the hopping amplitude, $\mu$ the chemical potential and $\mathbf{d}_{\mathbf{k}}=\left(d_{\mathbf{k},\uparrow},d_{\mathbf{k},\downarrow}\right)^{\top}$ is an annihilation operator in momentum space. For brevity, we equivalently refer to this non-magnetic parabolic band as “metal”, though one may of course tune $\mu$ to achieve a semi-conductor or an insulator, as discussed in App. B, where we also consider two parabolic bands.

We now introduce a tunnelling Hamiltonian which couples the surface of the WSM to the surface of the metal. We proceed with open boundary conditions in the $y$-direction and keep well-defined momenta perpendicular to the surface, $\mathbf{k}_{\perp}=\left(k_{x},k_{z}\right)$. The WSM (metal) side runs from $y=-L_{y}+1$ to $0$ ($y=1$ to $L_{y}$), defining an interface between the WSM’s $y=0$ and metal’s $y=1$ sites. The Hamiltonian for the full (finite-sized) system is therefore

where

and $\mathcal{H}^{\mathrm{open}}$ is the partial-in-$y$ Fourier transform of $\mathcal{H}^{\mathrm{bulk}}$. The full form of Eq. (5) is shown in App. A. The surface tunnelling term is also non-magnetic and takes the form $\left(T\right)_{y,y^{\prime}}=\Delta\delta_{0,L_{y}-1}$, or

where, for simplicity, we have assumed that $\Delta$ is a real constant that modulates the tunnelling strength between interface sites $y=0$ and $y=1$. Physically, the tunnelling strength can be modified either by varying the metal bandwidth $t_{m}$ or changing the interface thickness, as suggested by Fig. 2. There are therefore two competing energy scales at the WSM’s interface: the interlayer hopping $t$ pulling the electron towards the bulk and the tunnelling strength $\Delta$ pulling the electron towards the metal.

Before moving on to the finite lattice simulations, we note that the metal dynamics can be exactly integrated out to make way for a modified WSM propagator [21, 22]. More precisely, the effective Green’s function becomes

where $\omega_{n}$ is the Matsubara frequency and $G_{w,m}=\left(i\omega_{n}-\mathcal{H}^{\mathrm{open}}_{w,m}\right)^{-1}$ are the bare Green’s functions. Substituting in Eqs. (4) and (7) and yields, after some algebra,

where $h_{m}=-2t_{m}\left(\cos{k_{x}}+\cos{k_{z}}\right)-\mu$. Thus, surface tunnelling simply shifts the same-site hopping of the last site (Fig. 2c).

As a first test of validity, we take the $\Delta=0$ case and recover the chiral state and Fermi arc of the finite lattice model. We do not impose any boundary conditions at the $y$-termination but instead just look for exponentially localized states which solve the bulk difference equations. In a lattice formalism, $\mathcal{H}^{\mathrm{open}}_{w}\bm{\varphi}_{w}=E\bm{\varphi}_{w}$ produces a set of coupled difference equations relating $\bm{\varphi}_{w}(y)$ to its nearest neighbours $\bm{\varphi}_{w}(y\pm 1)$ [25]:

Plugging in Eq. (11), we obtain the matrix equation

where $\sigma_{\pm}=\left(\sigma_{x}\pm i\sigma_{y}\right)/2$. Setting the determinant of Eq. (14) to zero yields the ratio of spins and energy, respectively:

Guided by the previous section we make the assumption $\phi^{\uparrow}_{w}/\phi^{\downarrow}_{w}=0$ (spin in the $-x$ direction) and find a solution with $\ell=t/g_{3}$ and $E=-g_{1}$. To satisfy the boundary condition at $-\infty$, we impose $\mathrm{Re}\left(\ell\right)>1$, or $g_{3}<t$. The familiar Fermi arc condition $\gamma<\cos{k_{z}}$ then follows naturally¹¹1For a bulk state of energy $E$, $\mathrm{Re}\left(\ell\right)>1$ implies $-E_{\mathrm{bulk}}<E<E_{\mathrm{bulk}}$ where $E_{\mathrm{bulk}}(\mathbf{k}_{\perp})=\left[g_{1}^{2}+(g_{3}-t)^{2}\right]^{\frac{1}{2}}$ is the bulk edge.. We have therefore recovered the aforementioned chiral state: a uni-directional interface state on the Fermi arc.

We now allow for tunnelling at the interface between the $y=0$ and $y=1$ sites. There are then four difference equations, one for each type of site: the Weyl bulk, Weyl interface, metal interface, and metal bulk. Substituting in the supposed forms of $\bm{\varphi}_{w,m}(y)$, the difference equations are, respectively:

Eq. (16c) which has no matrix structure and is hence the same for both components of the spinor requires the spinor direction to be the same on both sides of the interface. Moreover, it determines the magnitude ratio:

Together with Eq. (16b), a final relation is obtained:

Eq. (18) is similar in form and purpose to the effective surface Green’s function (8) except it is purely in spin space since the ansatze and vanishing boundary conditions took care of position dependencies. It can be interpreted as an eigenvalue problem for the matrix $g_{1}\sigma_{z}-g_{3}\sigma_{x}+\ell^{-1}\sigma_{+}$ whose eigenvalues are

With this, its energy bands are twofold and defined by the implicit equation

where $\eta=\pm$ is the band index and $\ell$ ($\ell_{m}$) is itself a function of energy through Eq. (16a) [Eq. (16d)]:

Here, $Q=\frac{g_{1}^{2}+g_{3}^{2}+t^{2}-E^{2}}{2g_{3}t}$ and $P=\frac{h_{m}-E}{2t_{m}}$. While it may seem at first glance that the energies are symmetric in $k_{x}$ due to the even parity of both $\ell$ and $\ell_{m}$ with respect to $k_{x}$, one must be careful in choosing the appropriate branch $\eta$, such that the state indeed decays away from the interface. In general, the branch may vary as a function of $\mathbf{k}_{\perp}$. The infinite lattice theory Eq. (20) is compared to the finite model in Fig. 3.

Another remarkable consequence of tunnelling is spin canting. At first glance, one should not expect that non-magnetic tunnelling to a non-magnetic metal should cause the polarized spins at the interface to cant. Indeed, this plain intuition is seemingly supported by Eq. (17) and agrees with the finite lattice model for $\Delta\lesssim t$. To dress a more complete picture, however, we must consider the ratio of spins of the interface state which, in light of Eq. (18), is

If $\Delta>0$ and $E<h_{m}+t_{m}\ell_{m}^{-1}$, we expect the interface spin to cant away from ${\phi^{\uparrow}_{w}}/{\phi^{\downarrow}_{w}}=0$ ($\sigma_{x}=-1$) towards ${\phi^{\uparrow}_{w}}/{\phi^{\downarrow}_{w}}=-1$ ($\sigma_{z}=+1$). Solving for ${\phi^{\uparrow}_{w}}/{\phi^{\downarrow}_{w}}$ together with Eq. (20) yields the spins of the chiral state as they vary with tunnelling, shown in Fig. 4. We therefore conclude that non-magnetic surface tunnelling to a non-magnetic metal can in fact induce a change in the spins of the WSM’s chiral states. It is also of note that without $H_{w}$’s $\cos{k_{y}}$ term, spin canting is absent and the interface state will remain in a $\sigma_{x}=-1$ eigenstate independent of tunnelling.

We first consider the $\Delta=0$ case, a semi-infinite WSM slab in the continuum limit. Keeping $\mathcal{O}\left(k_{y}^{2}\right)$ terms in the WSM Hamiltonian, letting $k_{y}=-i\partial_{y}$, and multiplying by $i\sigma_{y}$ throughout yields the differential equation ($t=1$):

where $h_{z}\equiv g_{3}-t$. To hone in on the objective interface states, we take the interface to be at $y=0$ and make the ansatz ${\bm{\psi}}\propto e^{\kappa y}\bm{\phi}$, where $\bm{\phi}$ is an unspecified spinor and $\mathrm{Re}\left(\kappa\right)>0$ such that $\bm{\psi}\to 0$ as $y\to\infty$. The differential equation (23) admits four solutions for $\kappa$, of which two have a putatively positive real part:

For a state of energy $E$, $\mathrm{Re}\left(\kappa\right)>0$ translates to $E^{2}<\sin^{2}{k_{x}}+h_{z}^{2}$, in agreement with the infinite lattice theory. Eq. (24) sheds light on the fact that for a given eigenvector with energy $E$ satisfying Eq. (23), there is a distinct eigenvector with equal and opposite energy $-E$ which is also a solution. Therefore, there are two $\kappa$ values per energy.

To determine which solution is correct, we impose the boundary condition $\bm{\psi}(0)=0$. In general, one has the superposition ${\bm{\psi}}\propto e^{\kappa_{+}y}\bm{\phi}_{\kappa_{+}}+\alpha e^{\kappa_{-}y}\bm{\phi}_{\kappa_{-}}$. Therefore, $\alpha=-1$ and $\bm{\phi}_{\kappa_{+}}=\bm{\phi}_{\kappa_{-}}$. Equating the ratio of spins, the latter condition can be surmised as

After some algebra, we recover the aforementioned chiral state of energy $E=-t\sin{k_{x}}$, leading to the decay parameters $\kappa_{\pm}=1\pm\sqrt{1+2h_{z}}$ and spin in the negative $x$-direction:

The condition of $\mathrm{Re}\left(\kappa\right)>0$ leads to $\gamma<\cos{k_{z}}$, which is the familiar arc condition. At the surface BZ origin $\mathbf{k}_{\perp,0}=\left(0,0\right)$, the chiral state’s decay length is on the order of a lattice length, pointing to a strongly localized state which may therefore well be described by a continuum interface theory. At the surface Weyl points $\mathbf{k}^{\pm}_{\perp,w}=(0,\pm k_{w})$, however, $\kappa_{-}=0$ and the chiral state’s decay length diverges, as expected from the absence of such surface states at the Weyl node.

In order to get a simple analytical result, we imagine coupling the WSM to a quantum dot of energy $M$. Here, we model the metal as a flat band since it is well above the WSM and only states with the same energy are relevant. The continuum Hamiltonian reads

Once again, we focus on solutions bound to the interface $\bm{\psi}_{w}\propto e^{\kappa y}\bm{\phi}_{w}$ ($\bm{\psi}_{m}\propto e^{-\kappa_{m}y}\bm{\phi}_{m}$), leading to four differential equations. The first two restrict the metal spins to be identical to the Weyl spins up to a scalar factor:

The remaining two equations reduce to a $2\times 2$ matrix equation expressed in the basis of Weyl spins $\bm{\phi}_{w}$:

which is the continuum form of Eq. (18). When $\Delta=0$ and $E\neq M$, it is not difficult to see that the bare WSM surface chiral state is recovered. For $\Delta>0$, the physics are identical to the $\Delta=0$ case with the substitution $E\to E-{\Delta^{2}}/(E-M)=E_{\Delta}$ ²²2In fact, the effective surface propagator Eq. (9) exactly reduces to $-\Delta^{2}/\left(E-M\right)$ when $t_{m}=0$ and $\mu=-M$.. For instance, the decay parameters are now

where we have re-inserted the energy scale $t$ and the lattice constant $a$.

The continuum interface theory therefore hints at a straightforward interpretation of the energy shift upon tunnelling. Indeed, seeing as the only effect of $\Delta$ was to shift the energies, the chiral band’s energy in the continuum theory is defined by $E_{\Delta}=-t\sin{k_{x}}$, or

In regimes where the decay lengths $\kappa^{-1}_{\pm}\sim a$, Eq. (31) is in agreement with finite lattice simulations, as shown in Fig. 5. As for the chiral state’s spin, it remains unchanged due to Eq. (25) still being satisfied and equal to $-1$ when $E\to E_{\Delta}=-t\sin{k_{x}}$³³3In the infinite lattice theory, the replacement $$E\to E_{\Delta}=-t\sin{k_{x}}=-g_{1}$$ in Eq.(22) also leads to a spin $\sigma_{x}=-1$ ($r_{\mathrm{interface}}=0$).. Therefore, the validity of Eq. (31) will depend wholly on whether or not the state is in a $\sigma_{x}=-1$ eigenstate, and any deviations in the bandstructure must reflect a changing spin in the lattice model. Since the spins do in fact cant for $\Delta\gtrsim t$, this is the root of the continuum theory’s inaccuracy in this regime.

Another aspect captured by the continuum theory is the localization of bulk states at the interface to produce the emergent interface state, a typical feature of systems with boundary topologies [25]. Simply put, the lowering of energy with tunnelling will give the bulk state’s decay parameter a positive real part.

We will now turn to the transport consequences of the previously described theory and numerics. We begin by analyzing the current along the interface, travelling in the $x$-direction. We fix $k_{z}$ and analyze transport in 2D, summing over all momenta at the end.

At $\Delta=0$, the conductance at the Weyl node should vanish due to the gap closure and subsequent absence of uni-dimensional current-carrying states. For $\Delta>0$, however, the presence of interface states near the Weyl node and the resulting spectral asymmetry in $k_{x}$ [Fig. 3 (row 3, col. b)] suggests a jump in group velocity $\partial_{k_{x}}E$ across the Weyl point, leading to a nonzero conductance.

We verify our reasoning numerically via the Landauer-Büttiker formalism, where conductance along the interface $\mathcal{G}_{\parallel}$ is defined as [26]

Here, $G^{R}$ is the usual retarded Green’s function

with the lead self-energy $\Sigma^{R}=\Sigma^{R}_{l}+\Sigma^{R}_{r}$ giving the quasiparticles a finite lifetime. The $\Gamma_{l}$ and $\Gamma_{r}$ operators describe the loss of electrons into the left and right leads, respectively:

In the simplest case, we place two leads, one on each of the $x$-boundaries, which span the entire sample in the $y$-direction. Since the leads are (the interface is) in the plane perpendicular to $x$ ($y$), our construction forces the sample to be open in both the $x$-direction and the $y$-direction while still remaining periodic in $z$. For any $k_{z}$, $\Sigma^{R}_{l}$ takes the form

where $\tau$ is the quasiparticle’s lifetime. For its part, $\Sigma^{R}_{r}$ admits a similar form with $\delta_{x0}$ replaced by $\delta_{x,L_{x}-1}$. The tunnelling matrix $T$, while unchanged in the $y$-direction, now adopts a new diagonal sub-component in the $x$-direction:

For $\Delta=0$ (Fig. 6, panels a and b, top row), the $e^{2}/h$ quantized conductance for $|k_{z}|<k_{w}$ can be understood in the context of the quantum anomalous Hall effect, treating each constant $k_{z}$ plane as a 2D quantum spin Hall insulator with one-dimensional edge states carrying $\mathcal{G}_{\parallel}=e^{2}/h$ [27].

At $k_{z}=0$ (Fig. 6a), the surface tunnelling localizes a bulk state to within the gap, allowing for both left- and right-moving carriers to produce a “bump” in the conductance. One can reason by examining the juxtaposed spectrum. Above and below the bump energies (denoted by pink and green lines), there is only one left-moving state, whereas within it there are two left- and one right-mover. Without scattering between left and right movers, these states should contribute $2e^{2}/h$ to the conductance in one direction and $e^{2}/h$ in the other direction. On the other hand, scattering may reduce the conductance since a left and right mover can hybridize. In our case, the scattering is provided by the leads and therefore the resulting conductance is between 1 and 2 quanta of conductance.

The effect of tunnelling is perhaps most pronounced at the Weyl node (Fig. 6b). As discussed, the bulk gap closes and the subsequent absence of interface states at $\Delta=0$ leads to zero conductance at zero energy. However, with tunnelling there are now two interface states in the spectrum: the left-moving chiral state and the right-moving emergent interface state. The former will terminate at an energy $E_{\mathrm{term}}$ (green line), the intersection of $E_{\mathrm{bulk}}=\left(t^{2}\sin^{2}{k_{x}}+h_{z}^{2}\right)^{\frac{1}{2}}$ with Eq. (20). Below $E_{\mathrm{term}}$, there are no uni-directional carriers and the conductance is unchanged. For $E_{\mathrm{term}}<E<0$, only the chiral state is present, and there is a conductance $e^{2}/h$. Note the deviation from $e^{2}/h$ due to the small amount of bulk states present near zero energy. Above this range, both the chiral and the emergent interface state are present and move in opposite directions – their sum is null (modulo scattering), and only bulk states contribute.

When experimentally measuring transport between leads, the measured quantity is a sum over all $k_{z}$ momenta. We therefore define the total conductance along the interface,

Summing over quantized conductance contributions on the $z$-projected Fermi arc $k_{\mathrm{arc}}^{z}$, Eq. (37)’s minimum is fixed (Fig. 7):

To probe this signature, we vary the arc length along the $k_{z}$-direction, as shown by Fig.7b. In the minimal model, this can be done by applying a strong Zeeman-like magnetic field $b_{z}\mathbf{e}_{z}$ coupling to spin degrees of freedom, bringing the arc length to $k_{\mathrm{arc}}^{z}\to 2\arccos{(\gamma+b_{z})}$ provided $b_{z}$ is small enough not to change the overall topological phase and that its orbital effects may be neglected.

We complete our study with a simple analytical model for electron tunnelling across the interface. We set out to derive an expression for the conductance across the interface $\mathcal{G}_{\perp,\sigma}=dI_{\sigma}/dV$ of a particle polarized with spin $\sigma$. A detailed derivation is included in App. D.

We begin by expressing the current $I_{\sigma}$ of a particle with spin $\sigma$ in terms of the retarded correlation function $U_{R}^{\sigma\sigma^{\prime}}$ [28, 29, 30]:

${U}^{\sigma\sigma^{\prime}}_{R}\left(-eV\right)$ is found by computing the Matsubara correlation function ${\mathcal{U}}^{\sigma\sigma^{\prime}}(i\omega_{n})$ and analytically continuing $i\omega_{n}\to-eV+i0^{+}$. At finite temperature $\beta^{-1}$, we have

where $\mathbf{k}$ ($\mathbf{q}$) is the momentum in the WSM (metal), $T_{\mathbf{k}\mathbf{q}}$ is the tunnelling matrix element, $\omega_{n}$ ($p$) is a bosonic (fermionic) Matsubara frequency, and $\bm{g}_{w}$ ($\bm{g}_{m}$) is the Matsubara Green’s function for the bare WSM (metal). Since states bound to the interface will not contribute to tunnelling across of it, we may consider only bulk states. The bulk Green’s functions $\bm{g}_{m,w}$ are therefore

with the WSM (metal) dispersion $\xi_{w}$ ($\xi_{m}$). Setting $|T_{\mathbf{k}\mathbf{q}}|^{2}=\Delta^{2}\delta(\mathbf{k}_{\perp}-\mathbf{q}_{\perp})$, we perform the Matsubara frequency summation $\sum_{ip}\left({ip-\xi}\right)^{-1}=\beta n_{F}\left(\xi\right)$ [29], where $n_{F}$ is the fermionic distribution, by splitting the denominator into partial fractions. Using $\mathrm{Im}\,(-eV+i0^{+}-\xi)^{-1}=-\delta(-eV-\xi)$, Eq. (39) becomes

where $\xi_{\pm}=\xi_{m}\pm\xi_{w}$ and

Note that we have chosen the quantization axis in the $x$-direction for simplicity. More generally, the second term in Eqs. (43) is an odd function of $k_{x}$, $k_{z}$, and $\xi_{w}$ and will vanish when integrated over, leaving the current spin-independent.

To proceed, we imagine placing the metal band’s Fermi level $\mu_{m}$ in the WSM’s upper band and largely above to parabolic band minimum. At low energies,

where $m=1/2t_{m}$ and $\tilde{\mu}=\mu+\mu_{m}+6t_{m}$ (the lattice constant is still $a=1$). The latter expression is found by expanding near $\xi_{m}$’s intercept with $\mu_{m}$ along $q_{y}$, the metal’s $y$-momentum. We further consider a small positive applied voltage such that particles tunnel from the upper WSM band to the metal. Thus, only the first term of Eq. (VI.2) contributes. Replacing the sums by integrals, changing variables from $k_{y}$ to $\xi_{w}$ and $q_{y}$ to $\xi_{m}$, and re-inserting $\hbar$, the current is now

Note that we have applied the low-temperature limit $n_{F}(\xi_{w})=-\theta(\xi_{w})$. The integral over $d^{2}\mathbf{k}_{\perp}$ can be done analytically, yielding the conductance across the interface:

For $eV\ll\sqrt{2mv^{2}\tilde{\mu}}\equiv\varepsilon$, the leading order term is quadratic in $V$:

Eq. (47) maintains that tunnelling measurements with featureless metals reveal the density of states at the tunnelling energy, since the three-dimensional WSM’s linear dispersion corresponds to a density of states proportional to $E^{2}$.