# Bloch-Lorentz magnetoresistance oscillations in delafossites

Kostas Vilkelis<EMAIL_ADDRESS>Qutech, Delft University of
Technology, Delft 2600 GA, The Netherlands  Kavli Institute of Nanoscience,
Delft University of Technology, Delft 2600 GA, The Netherlands  Lin Wang
Kavli Institute of Nanoscience, Delft University of Technology, Delft 2600 GA,
The Netherlands  Anton Akhmerov  Kavli Institute of Nanoscience, Delft
University of Technology, Delft 2600 GA, The Netherlands

(September 3, 2024)

###### Abstract

Recent measurements of the out-of-plane magnetoresistance of delafossites
(PdCoO2 and PtCoO2) observed oscillations which closely resemble the Aharanov-
Bohm effect. We develop a semiclassical theory of these oscillations and show
that they are a consequence of the quasi-2D dispersion of delafossites. We
observe that the Lorentz force created by an in-plane magnetic field makes the
out-of-plane motion of electrons oscillatory, similarly to Bloch oscillations.
Analysis of the visibility of these Bloch-Lorentz oscillations reveals the
mean-free path to be $l\approx$4.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ in
comparison to the literature in-plane mean-free path of
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The mean-free path is reduced as a
consequence of the out-of-plane relaxation and sample wall scattering. Our
theory offers a way to design an experimental geometry that is better suited
for probing the phenomenon and to investigate the out-of-plane dynamics of
ballistic quasi-two-dimensional materials.

##  I Introduction

####  I.0.0.1 Delafossites are anomalously clean layered materials that are
the subject of ongoing studies.

Known since the discovery of mineral CuFeO2 by Friedel in 1873, delafossites
are materials with the general formula ABO2 [1, 2]. Delafossites are naturally
occurring layered structures of alternating conductive A layer and insulating
BO2 layer with the overall $R\bar{3}m$ space group [3]. These materials are
considered to be 2D owing to their weak interlayer coupling which results in a
near cylindrical Fermi surface [4, 5]. Of particular interest are PdCoO2 and
PtCoO2 which were first synthesized and characterized at room temperature in
1971 by Shannon _et al._ [2, 3, 6]. Even though nearly 50 years have passed
since then, the area of research is still very active due to the delafossite’s
impressive electronic transport properties [7]. At room temperature, it was
shown that the conductivity of PdCoO2 is
$2.6\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$,
very close to that of elemental Copper [8]. Part of the reason for such large
conductivity is the high Fermi velocity
$7.5\text{\times}{10}^{5}\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ [8].
Another reason is their exceptional mean-free path at $4\text{\,}\mathrm{K}$
which exceed $20\text{\,}\mathrm{\SIUnitSymbolMicro m}$ [8]. Such value of
mean-free path is accredited mostly to anomalously clean nature of
delafossites and orbital-momentum locking [9, 10]. Overall, all of these
properties of delafossites make them a good platform to study mesoscopic
ballistic transport [11].

####  I.0.0.2 Recent experiment observed magnetoconductance oscillations in
the out-of-plane transport similar to the AB effect.

Figure 1: PdCoO2 magnetoresistance experimental set up (inset) and results
(solid lines) obtained by Putzke et al. [12]. The semiclassical fit (dashed
line) was obtained using the mean free path of
$l(0)\approx$4.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$

Recent experiments looked into the out-of-plane transport of PdCoO2 [12], the
setup and magnetoconductance results shown by Fig. 1. Magnetoconductance was
measured with the magnetic field applied in the plane of the delafossite
layers and the current passing out-of-plane. Surprisingly, the results showed
oscillations with a magnetic field whose nature is not clear. At first sight,
the oscillations follow similar behaviour to the Aharanov-Bohm effect. The
frequency is controlled by the flux quanta going through the loop defined
between the adjacent conducting atomic layers and the width of the sample.
Despite the apparent similarity with the Aharanov-Bohm, it is not able to
explain these observations. Firstly, it would require hopping between layers
only at the edges of the sample. Furthermore, these loops would need to be
somehow decoupled. Lastly, the temperature dependence shows these oscillations
persist up until $60\text{\,}\mathrm{K}$, which would imply an impressive
resilience of the coherence to temperature. A better explanation is required.
A calculation of the oscillations by Putzke _et al._ relies on a tight-binding
Hofstadter model with Kubo formula. The approach reproduces the results,
however, due to its numerical nature, offers limited insight into the nature
of the phenomenon [12].

####  I.0.0.3 Magnetoconductance oscillations originate as a consequence of
the shape of the semiclassical trajectories as opposed to an interference
pattern.

In the following paper, we use the semiclassical theory to explain the
magnetoconductance results. We explain the oscillations as a consequence of
the shape of the semiclassical trajectories rather than an interference
pattern. Therefore, the quasi-2D energy dispersion of delafossites is the main
ingredient needed in reproducing the oscillations observed in experiments.

####  I.0.0.4 By analysing the visibility of the oscillations we predict that
the out-of-plane and sample wall scattering contribute to the mean-free path
length.

Measurements on delafossites’ mean free path suggest a fully ballistic model
[8]. However, we find that such a model describes oscillations with full
visibility, in contradiction to the results by Putzke _et al._ [12]. We
propose out-of-plane momentum relaxation as a possible mechanism to resolve
this contradiction. The scattering at the sample walls also contributes to the
mean-free path. However, the model does not take that explicitly into account.

####  I.0.0.5 The semiclassical theory suggests an experimental sample design
better suited for probing the phenomenon in quasi-2D materials.

We find that boundaries alter the electronic trajectories and therefore
complicate the analysis. We predict that by increasing sample’s length along
the direction of the magnetic field, the results of the theory converge with
experimental findings.

##  II Ballistic in-plane model in the weak out-of-plane coupling limit

####  II.0.0.1 Because the system is metallic and has a large density of
states, we work with the semiclassical regime.

Delafossites have large carrier density and are highly conductive [13, 14].
Therefore it is a metallic system in which interference effects should not
play a deciding role.

####  II.0.0.2 We assume quasi-2D dispersion and ballistic inplane transport
to model the trajectories.

Delafossites’ conduction band is well approximated by the energy dispersion

$\varepsilon(\mathbf{k})=\varepsilon_{\parallel}(\mathbf{k_{\parallel}})-t_{z}\cos(k_{z}c^{\prime}),$
(1)

where $\varepsilon_{\parallel}(\mathbf{k_{\parallel}})$ is hexagonal in-plane
dispersion [8, 14, 15], $c^{\prime}$ is the interlayer distance and $t_{z}$ is
the interlayer hopping. The interlayer dispersion is weak [8, 4] i.e. $t_{z}$
is much smaller than the in-plane bandwidth. This motivates a perturbative
approach in terms of $t_{z}$ used throughout the paper.

####  II.0.0.3 We model the transport using a linearized Boltzmann equation.

We compute electron density $f(\bm{r},\bm{k},t)d^{3}\mathbf{k}$ at position
$\mathbf{r}$, time $t$ and momentum $\mathbf{k}$ by using the Boltzmann
equation. We separate electron density into the equilibrium part (Fermi-Dirac
distribution) $f^{0}$ and non-equilibrium part $\delta f$. Further
simplification is achieved by setting

$\delta f(\bm{r},\bm{k})=-g(\bm{r},\bm{k})\frac{\partial
f^{0}}{\partial\varepsilon},$ (2)

where at zero temperature $\frac{\partial f^{0}}{\partial\varepsilon}$ becomes
a Dirac delta function centered around Fermi energy. The resulting steady
state linearised Boltzmann equation reads [16]

$\bm{v}\cdot\bm{\nabla}_{\bm{\kern-1.39998ptr}}g-\frac{e}{\hbar}\left(\bm{v}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{\kern-1.39998ptk}}g+ev_{z}\mathcal{E}_{z}=\mathcal{L}g,$
(3)

where $\bm{v}$ is the velocity, $e$ is the elementary charge, $\hbar$ is the
reduced Planck constant, $\bm{B}$ is the magnetic field, $\mathcal{E}_{z}$ is
the electric field along the out-of-plane direction and $\mathcal{L}g$ is the
linearised collision integral. The boundary conditions at the boundaries $x=0$
($\xi=1$) and $x=W$ ($\xi=-1$) are

$|v_{x}(\bm{k_{\parallel}})|g(\bm{r_{B}},\bm{k})=\kern-15.00002pt\int\limits_{v_{x}(\bm{k^{\prime}_{\parallel}})\xi<0}\kern-15.00002ptK\left(\bm{k^{\prime}},\bm{k}\right)\\\
\times\left|v_{x}(\bm{k^{\prime}_{\parallel}})\right|g(\bm{r_{B}},\bm{k^{\prime}})d^{3}k^{\prime},$
(4)

where $\bm{r_{B}}$ is the location of the boundary and $K$ is the boundary
scattering kernel. Because Eqs (3) and (4) are translationally invariant along
$y$ and $z$, the solution $g(x,\bm{k})$ is invariant as well.

####  II.0.0.4 We linearise the Boltzmann equation with respect to $t_{z}$.

We expand $g$ in Eq. (3) as a series to first order in $t_{z}$:

$g(\bm{r},\bm{k})\approx g_{0}(\bm{r},\bm{k})+g_{1}(\bm{r},\bm{k}),$ (5)

where $g_{0}$ does not depend on $t_{z}$ and $g_{1}\propto t_{z}$. With the
magnetic field inside the $yz$ plane $\bm{B}=(0,B_{y},B_{z})$, the zeroth
order equation is

$v_{x}\frac{\partial g_{0}}{\partial
x}-\frac{e}{\hbar}(\bm{v_{\parallel}}\times\bm{B})\cdot\bm{\nabla}_{\bm{\kern-1.39998ptk}}g_{0}=\mathcal{L}g_{0},$
(6)

where $\bm{v_{\parallel}}$ is in-plane velocity. Equation (6) describes an
electron in a magnetic field with no external forces capable of generating a
steady non-equilibrium distribution. Furthermore, the scattering term ensures
that the steady state solution is $g_{0}=0$. Therefore, to first order in
$t_{z}$ linearised Boltzmann equation is

$v_{x}\frac{\partial g}{\partial
x}-\frac{e}{\hbar}\left(\bm{v_{\parallel}}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{\kern-1.39998ptk}}g+ev_{z}\mathcal{E}_{z}=\mathcal{L}g.$
(7)

Since $g\propto t_{z}$, it is sufficient to approximate $\mathcal{L}$ to
zeroth order in $t_{z}$.

####  II.0.0.5 No excess of in-plane electrons leads to simplified diffusive
boundary conditions.

Integrating Eq. (7) over $k_{z}$ within the 1st Brillouin zone, we obtain an
equation identical to Eq. (6), but with $g_{0}$ replaced by
$g_{\parallel}(\bm{r},\bm{k_{\parallel}})\equiv\int_{\textrm{BZ}}g(\bm{r},\bm{k})dk_{z}^{\prime}$.
Therefore, the in-plane current of electrons vanishes in the steady state:

$\int\limits_{BZ}g(\bm{r},\bm{k})dk_{z}=0.$ (8)

Because the samples in the work by Putzke _et al._ [12] are formed using
focused ion beam milling and thus have amorphous edges, and because the weak
out-of plane dispersion, we focus on the boundary conditions that randomize
the out-of-plane momentum $k_{z}$. In this limit the boundary scattering
kernel $K$ does not depend on $k_{z}$, and substitution of Eq. (8) in Eq. (4)
yields the following:

$\begin{split}&g(x_{B},y,z,\bm{k})=0,\\\
&x_{B}=\left\\{\begin{array}[]{ll}0&k_{x}>0\\\
W&k_{x}<0.\end{array}\right.\end{split}$ (9)

While this approximation simplifies the simulations, and is likely realistic,
we note that the appearance of oscillations in this limit is not sensitive to
the boundary conditions.

####  II.0.0.6 We compute $\sigma_{zz}$ to the lowest nonvanishing order in
$t_{z}$.

To analyse experimental observations, we compute the current along $z$

$I_{zz}=e\int\limits_{0}^{W}dx\int\limits_{0}^{L}dy\iiint\limits_{\textrm{BZ}}f(\bm{r},\bm{k})v_{z}(\bm{k})d{\bm{k}}.$
(10)

We express the out-of-plane conductivity
$\sigma_{zz}=I_{zz}/(\mathcal{E}_{z}WL)$ at zero temperature by substituting
Eq. (2) into Eq. (10):

$\sigma_{zz}=\frac{e}{W\mathcal{E}_{z}}\int\limits_{0}^{W}dx\iiint\limits_{\textrm{BZ}}\delta(\varepsilon-\varepsilon_{F})g(x,\bm{k})v_{z}(\bm{k})d{\bm{k}},$
(11)

where $\varepsilon_{F}$ is the Fermi energy. We compute the lowest
nonvanishing contribution in $t_{z}$ to conductivity. Since $g_{0}=0$, we
approximate the energy
$\varepsilon(k_{x},k_{y},k_{z})\approx\varepsilon_{\parallel}$ only to zeroth
order in $t_{z}$. We switch to cylindrical coordinates in k-space
$(x,k,\theta,k_{z})$ where $k$ is the in-plane wavevector length and $\theta$
is the azimuth. The conductivity to the lowest order in $t_{z}$ is

$\sigma_{zz}=\frac{e}{W\mathcal{E}_{z}\hbar}\int\limits_{0}^{2\pi}\\!\int\limits_{0}^{W}\\!\int\limits_{-\frac{\pi}{c^{\prime}}}^{\frac{\pi}{c^{\prime}}}\\!\frac{k_{F}(\theta)}{v_{R}(\theta)}g(x,k_{F}(\theta),\theta,k_{z})\\\
\times v_{z}(k_{z})d\theta dxd{k_{z}},$ (12)

where

$v_{R}(\theta)=\frac{1}{\hbar}\frac{\partial\varepsilon_{\parallel}}{\partial
k},$ (13)

and $k_{F}(\theta)$ is the Fermi wavevector
$\varepsilon(k_{F}(\theta),\theta)=\varepsilon_{F}$.

##  III Results

###  III.1 In-plane magnetic field

####  III.1.0.1 In-plane magnetic field reduces the degrees of freedom of the
Boltzmann equation.

Figure 2:  (A) Trajectories with oscillations commensurate to the sample width
due to an in-plane magnetic field. Different curves indicate the different
initial phase of the trajectory. (B) Same as in (A), but the in-plane magnetic
field is chosen to give incommensurate oscillations. (C) Same as in (A), but
with scattering present. (D) Trajectories due to an out-of-plane magnetic
field. Blue lines are boundary-to-boundary trajectories whereas the red lines
are edge-localized trajectories. Only the boundary-to-boundary trajectories
produce current oscillations due to a net $k_{z}$ drift throughout the
trajectory.

Because the mean free path of delafossites is larger than the sample size
[12], to illustrate the origin of the oscillations we first neglect scattering
$\mathcal{L}g=0$. With in-plane magnetic field $\mathbf{B}=(0,B_{y},0)$, the
Boltzmann Eq. (7) reduces to

$v_{x}\frac{\partial g(x,v_{x},k_{z})}{\partial
x}-v_{x}\frac{eB_{y}}{\hbar}\frac{\partial g(x,v_{x},k_{z})}{\partial
k_{z}}+e\mathcal{E}_{z}v_{z}=0,$ (14)

where $g(x,v_{x},k_{z})$ only depends on $k_{x}$ and $k_{y}$ through
$v_{x}(k_{x},k_{y})$ as seen in Eq. (14).

####  III.1.0.2 The conductance profile of each of the trajectories is
oscillatory.

Solution to Eq. (14) fulfilling the boundary conditions of Eq. (9) is

$g\left(x,v_{x},k_{z}\right)=\frac{t_{z}\mathcal{E}_{z}}{B_{y}v_{x}}\biggl{[}\cos(k_{z}c^{\prime})-\\\
\cos\left(k_{z}c^{\prime}+\frac{\omega B_{y}}{W}x_{B}\right)\biggr{]},$ (15)

with:

$\omega=\frac{e}{\hbar}c^{\prime}W,\quad
x_{B}=\left\\{\begin{array}[]{ll}x&\textrm{for }v_{x}>0\\\ x-W&\textrm{for
}v_{x}<0.\end{array}\right.$ (16)

We substitute Eq. (15) into Eq. (12), and obtain the conductivity along $z$

$\sigma_{zz}=\frac{e\pi t_{z}^{2}}{\omega\hbar
B_{y}^{2}}\left(1-\cos\left(\omega
B_{y}\right)\right)\int\limits_{0}^{2\pi}\frac{k_{F}(\theta)}{v_{x}(\theta)v_{R}(\theta)}d_{\theta}.$
(17)

####  III.1.0.3 Since all of the trajectories are straight in-plane and share
the same conductance profile, we predict oscillations of correct periodicity,
but with full visibility.

When the magnetic field is of the form $\mathbf{B}=(0,B_{y},B_{z})$, the
$k_{z}$ dependence on x is

$k_{z}(x)=k_{z0}+\frac{e}{\hbar}B_{y}x.$ (18)

This results in the electron trajectory

$z(x)=\frac{t_{z}}{\hbar
v_{x}}\left[\cos{\left(k_{z0}+\frac{e}{\hbar}B_{y}x\right)}-\cos{\left(k_{z0}\right)}\right].$
(19)

All trajectories have a similar oscillatory vertical displacement as a
function of $x$, shown in Fig. 2(A,B). _The universal trajectory shape is a
result of the $k_{z}$ advancing over the complete out-of-plane Brillouin zone,
similar to Bloch oscillations [17], however because of the magnetic origin of
the momentum drift, $k_{z}$ has a universal dependence on $x$ regardless of
the in-plane trajectory._ When the oscillation period is commensurate with the
sample width, all trajectories have a zero net vertical displacement over the
time of flight, and therefore carry no current as shown in Fig. 2(A). At the
same time, the vertical displacement of different trajectories—and therefore
the current—is maximal when a half-integer number of oscillation periods fits
into the sample width as shown in Fig. 2(B). Because the contribution of every
trajectory to the conductance has the same magnetic field dependence, as seen
in Eq. (17), this minimal model yields an oscillatory conductance with a
correct frequency, but full visibility of the oscillations in contrast to the
experimental data.

####  III.1.0.4 The only mechanism in this geometry that reduces visibility is
the out-of-plane momentum scattering.

In the regime of linearised Boltzmann equations with translational invariance
along $y$ and $z$, the only possible generalisation is addition of scattering
which randomizes the vertical displacement and suppresses the visibility of
the oscillations due to the trajectory shape as shown in Fig. 2 (C). However,
in-plane scattering is negligible due to the well-known ballistic transport in
delafossites [8]. Therefore, the scattering must be out-of-plane. We assume a
simple scattering model

$W_{\mathbf{k},\mathbf{k^{\prime}}}=\alpha(\theta)\delta(\theta-\theta^{\prime})\delta(\varepsilon(\mathbf{k})-\varepsilon(\mathbf{k^{\prime}})).$
(20)

We use Eq. (8) and $\varepsilon(k,\theta)\approx\varepsilon_{\parallel}$ to
compute the collision contribution to the Boltzmann equation

$\mathcal{L}g=-\frac{\alpha(\theta)k}{\hbar
v_{R}(\theta)}g(k,\theta,k_{z})\approx-\frac{g(k,\theta,k_{z})}{\tau},$ (21)

where for simplicity, in the last equality we reduced Eq. (21) to a relaxation
time approximation form with relaxation time $\tau$.

####  III.1.0.5 Accounting for out-of-plane scattering, the predicted
conductivity explains the oscillations qualitatively.

We now substitute Eq. (21) into the right hand side of Eq. (14) and obtain
linearised Boltzmann equation with relaxation whose solution is

$g\left(x,\theta,k_{z}\right)=\frac{\mathcal{E}_{z}\tau
ec^{\prime}t_{z}}{\hbar\left(B_{y}^{2}\phi^{2}+1\right)}\biggl{(}B_{y}\phi\cos\left(k_{z}c^{\prime}\right)+\sin\left(k_{z}c^{\prime}\right)\\\
-\exp\left(-\frac{x_{B}}{l}\right)\biggl{[}B_{y}\phi\cos\biggl{(}k_{z}c^{\prime}+\frac{\omega
B_{y}}{W}x_{B}\biggr{)}+\\\ \sin\left(k_{z}c^{\prime}+\frac{\omega
B_{y}}{W}x_{\theta}\right)\biggr{]}\biggr{)},$ (22)

with:

$l(\theta)=\tau
v_{x}(\theta),\quad\phi(\theta)=\frac{e}{\hbar}c^{\prime}l(\theta).$ (23)

We substitute Eq. (22) into Eq. (12), and obtain conductivity per unit azimuth

$\sigma_{zz}(B_{y})=\frac{\tau
e^{2}t_{z}^{2}c^{\prime}\pi}{\hbar^{2}}\int\limits_{0}^{2\pi}\frac{k_{F}(\theta)}{v_{R}(\theta)\left(B_{y}^{2}\phi^{2}+1\right)}\\\
\times\biggl{(}1+\frac{r}{\left(B_{y}^{2}\phi^{2}+1\right)}\biggl{[}B_{y}^{2}\phi^{2}-1-\exp{\left(-\frac{1}{r(\theta)}\right)}\\\
\times\biggl{(}(B_{y}^{2}\phi^{2}-1)\cos(\omega B_{y})+2B_{y}\phi\sin(\omega
B_{y})\biggr{)}\biggr{]}\biggr{)}d\theta,$ (24)

with

$r=l(\theta)/W.$ (25)

We recover a simple Drude model $B^{2}$ resistivity scaling [18] in Eq. (24)
by removing the boundaries, $W\to+\infty$, which removes the second term in
Eq. (24). We conclude that the addition of boundaries alters the overall
conductance profile. Numerical integration of Eq. (24) with
$l(0)\approx$4.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ gives the result
shown in Fig. 1. We share the numerical code and the resulting data in Ref.
[19]. The theory shows qualitative agreement with experimental results for low
magnetic fields but starts to break down for large magnetic fields. A likely
origin of the disagreement is the use of translational invariance along $y$.
The sample walls along the $y$ direction act as an additional source of
scattering besides the out-of-plane relaxation. We predict that repeating the
experiment with a longer sample length, $L$, would recover our theoretical
predictions. In addition, the larger aspect ratio of the sample would allow to
better probe the out-of-plane scattering mechanism. Another likely source of
disagreement is the choice of a simplistic scattering model.

###  III.2 Out-of-plane magnetic field

####  III.2.0.1 With the out-of-plane magnetic field, the trajectories follow
hexagonal orbits.

Figure 3: Numerical results from the semiclassical theory (dashed lines) with
$l(0)=$4.4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ compared to the
experimental (solid lines) magnetoresistance results by Putzke et al. [12] for
magnetic field tilted out-of-plane by $5^{\circ}$ steps. Black lines indicate
the critical field when the cyclotron orbit fits inside the sample. The
critical field value is determined by the shorter side of the sample. In the
experiment, this is the lenght of the sample $L$, whereas in the
semiclasssical fit it is the width of the sample $W$

.

To solve the Boltzmann equation in presence of out-of-plane magnetic field we
use that scattering Eq. (21) does not mix non-equilibrium electron densities
along different ballistic trajectories are independent. We therefore compute
$g$ along each ballistic electron trajectory separately. The in-plane
projection of each trajectory is a rotated and rescaled hexagonal Fermi
surface, while the out of plane motion follows the same oscillatory pattern,
with the resulting motion shown in Fig. 2 (D). The trajectories follow the
semiclassical equations of motion:

$\begin{split}&\hbar\frac{d\bm{r}(t)}{dt}=\bm{\nabla_{k}}\varepsilon(\bm{k}),\\\
&\hbar\frac{d\bm{k}(t)}{dt}=-e\bm{v}(t)\times\bm{B}.\end{split}$ (26)

We limit our consideration to the case when the cyclotron orbits are larger
than the sample width, and therefore each trajectory starts and ends at one of
the sample boundaries. We choose the starting point of each trajectory as
$t=0$, so that its initial conditions are $\bm{r_{0}}=(x_{0},y_{0},z_{0})$ and
$\bm{k_{0}}=(k_{0}\cos\theta_{0},k_{0}\sin\theta_{0},k_{z,0})$. Here $x_{0}=0$
and $-\pi/2<\theta_{0}<\pi/2$ at the left boundary, while $x_{0}=W$ and
$\pi/2>\theta_{0}>3\pi/2$ at the right boundary.

####  III.2.0.2 We write the Boltzmann equation along each trajectory.

Using Eq. (26), we obtain the evolution of $g$ along a single trajectory

$\frac{\partial g(t,{\bm{r_{0}},\bm{k_{0}}})}{\partial
t}=\bm{v}\cdot\bm{\nabla}_{\bm{\kern-1.39998ptr}}g-\frac{e}{\hbar}\left(\bm{v}\times\bm{B}\right)\cdot\bm{\nabla}_{\bm{\kern-1.39998ptk}}g.$
(27)

Substituting Eq. (27) and Eq. (21) into Eq. (7), we obtain the Boltzmann
equation along a single trajectory

$\frac{\partial g(t,{\bm{r_{0}},\bm{k_{0}})}}{\partial
t}+e\mathcal{E}_{z}v_{z}(k_{z}(t))=-\frac{g(t,\bm{r_{0}},\bm{k_{0}})}{\tau},$
(28)

with solution

$g(t,\bm{r_{0}},\bm{k_{0}})=e\mathcal{E}_{z}\int_{0}^{t}v_{z}\left(k_{z}(t^{\prime})\right)\\\
\times\exp{\left(-\frac{(t-t^{\prime})}{\tau}\right)}dt^{\prime}.$ (29)

By changing the variables in Eq. (12) to the trajectory coordinate system
$(t,\theta_{0},k_{z0})$, $\sigma_{zz}$ has the form

$\sigma_{zz}=\frac{e}{W\mathcal{E}_{z}\hbar}\int\limits_{0}^{2\pi}\int\limits_{0}^{t_{B}(\theta_{0})}\frac{k_{F}(\theta(\theta_{0},t))}{v_{R}(\theta(\theta_{0},t))}J(t^{\prime},\theta_{0})\\\
\times\int\limits_{-\pi/c}^{\pi/c}g(t^{\prime},\theta_{0},k_{z0})v_{z}(t^{\prime},\theta_{0},k_{z0})d\theta_{0}dt^{\prime}d{k_{z0}},$
(30)

with the Jacobian determinant $J$ is

$J(t^{\prime},\theta_{0})=\left(\frac{\partial\theta}{\partial
t^{\prime}}\bigg{\rvert}_{t^{\prime}=0}\right)^{-1}\frac{\partial\theta}{\partial
t^{\prime}}v_{x}(0,\theta_{0}).$ (31)

In Eq. (30), $t_{B}(\theta)$ is the time that the trajectory hits a boundary.
Because the integral over $\theta_{0}$ goes from $-\pi/2$ to $3\pi/2$, this
expression includes the contributions of all trajectories.

####  III.2.0.3 Only the boundary-to-boundary trajectories contribute to the
oscillations.

There are two types of trajectories along which Eq. (30) is integrated:
boundary-to-boundary and edge-localized trajectories as shown in Fig. 2(D).
These two trajectories have different contributions to the conductance
profile. Conductance oscillations are controlled only by the boundary-to-
boundary trajectory because it results in net $k_{z}$ drift according to Eq.
(18). On the other hand, the edge-localized trajectory does not affect the
conductance oscillations as a result of the absence of $k_{z}$ drift. Once the
cyclotron orbits become smaller than $W$, which happens at

$B_{z}>\left(\frac{2\hbar k_{F}}{eW}\right),$ (32)

with $k_{F}$ the Fermi wavevector, the boundary-to-boundary trajectories
disappear, and so does the oscillatory contribution to conductance.

####  III.2.0.4 Numerical results show qualitative agreement with experiment.

We perform numerical integration of Eq.(30), with the result shown in Fig. 3.
The model qualitatively agrees with the experimental data at the small tilt
angles from the $xy$-plane. However, the disagreement increases with $B_{z}$,
likely due to our calcuation assuming translational invariance along $y$
direction. This is likely a crude approximation because the sample length $L$
is shorter than $W$ in the experiment. The extension of the theory to a
realistic sample geometry is straightforward—especially since one may still
compute $g$ for every trajectory independely—but it strongly increases the
computational costs, and therefore we consider it unjustified for the purposes
of our study.

##  IV Summary

####  IV.0.0.1 Our findings are useful for probing ballistic quasi-2D
materials.

In summary, we demonstrated that the observed magnetoresistance of delafossite
materials is explained by the Bloch-like oscillations of the out-of-plane
electron trajectories. These Bloch-Lorentz oscillations arise from the
quasi-2D dispersion of these materials combined with the nearly ballistic
motion of the electrons. Our identification enables a way of probing the out-
of-plane momentum relaxation in quasi-2D materials as well as out-of-plane
dispersion relation.

## Acknowledgements

We would like to thank Andy Stern, Veronika Sunko, and Maja Bachmann for the
helpful discussions in the early stages of the project. Also, we are grateful
to Philip J.W. Moll and Carsten Putzke for sharing with us their experimental
findings and subsequent consultations. This work was supported by the NWO VIDI
Grant (016.Vidi.189.180), an ERC Starting Grant 638760 and European Union’s
Horizon 2020 research and innovation programme FE-TOpen Grant No. 828948
(AndQC).

## Author contributions statement

A.A. formulated the project idea. K.V. developed the theory, carried out the
numerical simulations, and analyzed the data with input from the other
authors. The manuscript was written jointly by K.V. and A.A. with input from
L.W.

## References

  * Friedel [1873] C. Friedel, Sciences Academy 77, 211 (1873).
  * Shannon  _et al._ [1971a] R. D. Shannon, D. B. Rogers, and C. T. Prewitt, Inorganic Chemistry 10, 713 (1971a).
  * Shannon  _et al._ [1971b] R. D. Shannon, C. T. Prewitt, and D. B. Rogers, Inorganic Chemistry 10, 719 (1971b).
  * Ong  _et al._ [2010] K. P. Ong, J. Zhang, J. S. Tse, and P. Wu, Phys. Rev. B 81, 115120 (2010).
  * Noh  _et al._ [2009] H.-J. Noh, J. Jeong, J. Jeong, E.-J. Cho, S. B. Kim, K. Kim, B. I. Min, and H.-D. Kim, Phys. Rev. Lett. 102, 256404 (2009).
  * Shannon  _et al._ [1971c] R. D. Shannon, D. B. Rogers, C. T. Prewitt, and J. L. Gillson, Inorganic Chemistry 10, 723 (1971c).
  * Mackenzie [2017] A. P. Mackenzie, Reports on Progress in Physics 80, 032501 (2017).
  * Hicks  _et al._ [2012] C. W. Hicks, A. S. Gibbs, A. P. Mackenzie, H. Takatsu, Y. Maeno, and E. A. Yelland, Physical Review Letters 109, 10.1103/physrevlett.109.116401 (2012).
  * Sunko  _et al._ [2020] V. Sunko, P. McGuinness, C. Chang, E. Zhakina, S. Khim, C. Dreyer, M. Konczykowski, H. Borrmann, P. Moll, M. König, and et al., Physical Review X 10, 10.1103/physrevx.10.021018 (2020).
  * Usui  _et al._ [2019] H. Usui, M. Ochi, S. Kitamura, T. Oka, D. Ogura, H. Rosner, M. W. Haverkort, V. Sunko, P. D. C. King, A. P. Mackenzie, and K. Kuroki, Phys. Rev. Materials 3, 045002 (2019).
  * Bachmann  _et al._ [2019] M. D. Bachmann, A. L. Sharpe, A. W. Barnard, C. Putzke, M. König, S. Khim, D. Goldhaber-Gordon, A. P. Mackenzie, and P. J. W. Moll, Nature Communications 10, 10.1038/s41467-019-13020-9 (2019).
  * Putzke  _et al._ [2019] C. Putzke, M. D. Bachmann, P. McGuinness, E. Zhakina, V. Sunko, M. Konczykowski, T. Oka, R. Moessner, A. Stern, M. König, S. Khim, A. P. Mackenzie, and P. J. W. Moll, $h/e$ oscillations in interlayer transport of delafossites (2019), arXiv:1902.07331 [cond-mat.mtrl-sci] .
  * Takatsu  _et al._ [2010] H. Takatsu, S. Yonezawa, S. Fujimoto, and Y. Maeno, Physical Review Letters 105, 10.1103/physrevlett.105.137201 (2010).
  * Takatsu  _et al._ [2013] H. Takatsu, J. J. Ishikawa, S. Yonezawa, H. Yoshino, T. Shishidou, T. Oguchi, K. Murata, and Y. Maeno, Phys. Rev. Lett. 111, 056601 (2013).
  * Prentice and Coldea [2016] J. C. A. Prentice and A. I. Coldea, Phys. Rev. B 93, 245105 (2016).
  * Ziman [1972] J. M. Ziman,  _Principles of the Theory of Solids_ (Cambridge University Press, 1972) p. 425.
  * Bloch [1929] F. Bloch, Zeitschrift für physik 52, 555 (1929).
  * Zhang  _et al._ [2019] S. Zhang, Q. Wu, Y. Liu, and O. V. Yazyev, Phys. Rev. B 99, 035142 (2019).
  * Vilkelis and Akhmerov [2020] K. Vilkelis and A. Akhmerov, Bloch-Lorentz magnetoresistance oscillations in delafossites (2020), This work was supported by the NWO VIDI Grant (016.Vidi.189.180), an ERC Starting Grant 638760 and European Union’s Horizon 2020 research and innovation programme FE-TOpen Grant No. 828948 (AndQC).