Scalable Sondheimer oscillations driven by commensurability between two quantizations

Abstract

The electrical conductivity of metallic crystals exhibits size effects when the electron mean free path exceeds the sample thickness. One such phenomenon, known as Sondheimer oscillations, was discovered decades ago. These oscillations, periodic in magnetic field, have been hitherto treated with no reference to Landau quantization. Here, we present a study of longitudinal and transverse conductivity in cadmium single crystals with thicknesses ranging from 12.6 to 475 μm, and demonstrate that the amplitude of the first ten oscillations is determined by the quantum of conductance and a length scale that depends on the sample thickness, the magnetic length and the Fermi surface geometry. We argue that this scaling is unexpected in semiclassical scenarios and it arises from the degeneracy of the momentum derivative of the cross-sectional area A along the orientation of the magnetic field $\frac{\partial A}{\partial {k}_{z}}$ in cadmium, which couples Landau quantization to the discretization of k_z imposed by the finite sample thickness. We show that the oscillating component of the conductivity is uniquely governed by fundamental constants and the ratio of two degeneracies, which acts as an inverted filling factor. Our conjecture is supported by the absence of such scaling in thin copper crystals.

Introduction

Confinement of electrons to small spatial dimensions is known to affect electrical transport properties^1,2,3,4. Sondheimer discovered one such size effect⁵, which occurs when the sample thickness becomes shorter than the bulk mean free path. These oscillations of conductivity, sometimes dubbed magneto-morphic, are periodic in the magnetic field. Available explanations of Sondheimer oscillations (SO) do not invoke Landau quantization, in contrast to Shubnikov-de Haas oscillations. The latter are periodic in the inverse of the magnetic field⁶ and are known to arise from commensurability in momentum space between the Landau tubes and the extremal areas of the Fermi surface. The prevailing understanding of Sondheimer oscillations invokes commensurability between the orbits of the charged carriers in real space and the sample thickness⁷. The phenomenon is classified as a semiclassical size effect, where neither the discrete energy levels introduced by the magnetic field nor those resulting from spatial confinement play a role.

Consider a magnetic field perpendicular to an electric current generating an electric field (Fig. 1a). Owing to their Fermi velocity, mobile electrons experience a Lorentz force, giving rise to a helical trajectory (Fig. 1b) whose axis is along the magnetic field. With increasing magnetic field, the radius of each turn of the helix decreases. Concomitantly, the distance between successive turns (i.e. the helix pitch, p) is reduced. At particular values of the magnetic field occurring with a fixed period, the thickness (d) of the sample becomes an integer multiple of helix pitches. In ballistic samples, where the mean free path limited by disorder is significantly longer than d, this condition leads to oscillations in conductivity, as observed in numerous experiments during the second half of the last century^{8,9,10,11,12,13,14,15,16}. More recently, two studies have been devoted to Sondheimer oscillations in the Weyl semi-metal WP₂¹⁷ and in graphite¹⁸.

Fig. 1: Sondheimer oscillations and their extraction in two thin cadmium crystals.

a Experimental setup. Electrons in a metallic solid conduct electricity in the presence of a magnetic field perpendicular to the current direction. Magnetoresistivity and Hall signals were measured in longitudinal and transverse configurations. b Front view of electron trajectories. c Field dependence of longitudinal resistivity (ρ_xx) in the 12.6 μm sample. Red curve: experimental data; black line: non-oscillatory background. Inset: Oscillatory component after background subtraction. d Hall resistivity (ρ_xy) of the 12.6 μm sample. e Field dependence of the longitudinal resistivity (ρ_xx) in the 25 μm sample. Red curve: experimental data; black line: non-oscillatory background. Inset: Oscillatory component after background subtraction. f Hall resistivity (ρ_xy) of the 25 μm sample.

The periodicity of the oscillations can be quantified thanks to a simple expression put forward by Harrison¹⁹ for velocity:

$${v}_{z}=\frac{\hslash }{2\pi {m}^{* }}\frac{\partial A}{\partial {k}_{z}}$$

(1)

Here, v_z and k_z represent the Fermi velocity and the wavevector along the magnetic field, respectively. A is the cross-sectional area of the Fermi surface in the momentum space perpendicular to the magnetic field. The time for an electron to travel across the sample thickness is τ₁ = d/v_z. One revolution of the helix takes a time equal to ${\tau }_{2}=\frac{2\pi {m}^{\star }}{eB}$. The ratio τ₁/τ₂ increases linearly with magnetic field and periodically reaches an integer. At these specific fields, the thickness equals an integer multiple of the helical pitch. The period, ΔB, is therefore given by refs. ^3,7,8,17,20:

$$\Delta Bd=\frac{\hslash }{e}\frac{\partial A}{\partial {k}_{z}}.$$

(2)

In general, ∂A/∂k_z is not constant but varies with k_z. Therefore, a well-defined periodicity implies a singular value prevailing over the others^3,7,21,22. In a spherical or an ellipsoidal Fermi surface, this singularity occurs at the end point of the Fermi surface along the orientation of the magnetic field. A distinct type of singularity occurs when ${\partial }^{2}A/\partial {k}_{z}^{2}=0$ at an inflection point on the Fermi surface. The field dependence of the amplitude of oscillations is expected to depend on the type of singularity. In the case of an end point, the amplitude should fall off as ∝ B⁻⁴³. This is what has been reported in cadmium and zinc^{8,11,12,13,23,24,25,26,27}. In the case of an inflection point, one expects a field dependence ∝ B^−2.5 21. This behavior has been observed in aluminum^9,10. Hurd⁷ has proposed that other possible exponents for the power law decay can arise for other types of ∂A/∂k_z singularity.

We present here a study of longitudinal and Hall resistivity of cadmium single crystals with varying thicknesses (d), ranging from 12.6 to 475 μm. Employing focused ion beam (FIB) technology²⁸ to etch the samples, we ensured planar and parallel surfaces. Our extensive study on five different samples with a forty-fold variation in thickness has led to solid conclusions regarding the period and the amplitude of oscillations and their evolution with magnetic field and thickness. The period of oscillations, ΔB, agrees with what was reported by Grenier et al.⁸ for a thicker sample. In contrast, the amplitude of oscillations in thin samples, follows δσ ∝ B^−2.5e^−B/3ΔB. Such a field dependence was never proposed or observed before. Only when the samples become sufficiently thick, and the magnetic field sufficiently high, the field dependence becomes σ ∝ B⁻⁴ as previously reported. The crossover between the two regimes of field dependence occurs at  ~ 10 oscillations.

Section results and discussion presents our extensive data sets leading to an empirical determination of the evolution of the oscillation amplitude as a function of magnetic field and thickness. We show that for all crystals, the best fit is given by $\delta \sigma \propto {B}^{-2.5}{e}^{-B/{B}_{0}}$. Over a factor of forty variation in thickness, the amplitude scales ∝ d⁻². We also show that the temperature dependence of the oscillations is such that they vanish when the thermal energy exceeds the spacing between the Landau levels, implying a role played by the latter, which are not invoked in the semiclassical scenario.

Then, we contrast this distinct dependence on magnetic field and thickness with what is expected in the semiclassical models. We show that in our case, the data can be expressed as the oscillations of a dimensionless conductivity as a function of a dimensionless period. We argue that the specific electronic structure of Cd²⁹ is crucial for this scalability. This conjecture is supported by a comparison between Cd and Cu. We then argue that the degeneracy of ∂A/∂k_z over a significant portion of the Fermi surface leads to an interplay between Landau levels and spatial confinement. The competition between two distinct discretizations of the energy spectrum, imposed by Landau quantization and z-axis confinement, generates oscillations whose period set by the commensurability between these two quantum degeneracies. The exponential term, unexpected in any of the available semiclassical pictures, can be tracked as the consequence of tunneling across two neighboring Landau levels. This scenario provides a basis for the expression empirically derived in section results and discussion.

Results and discussion

Samples

The crystal growth and the fabrication of the samples using FIB are discussed in detail in the supplement (Note 1)³⁰. Table 1 lists the samples used in this study. Their thickness was varied by a factor of 40. In the thickest samples, the zero-field resistivity at 2 K was found to be as low as  ~ 1.5 nΩ ⋅ cm, implying a mean free path as long as ~ 80 μm. This corresponds to the mean free path set by the distance between defects. In samples thinner than 80 μm, the residual resistivity increases because of the ballistic limit on the mean-free-path. In this regime, the mean-free-path extracted from the zero-field conductivity is comparable to the geometric average of thickness and width of the sample.

Table 1 Samples used in this study

Extracting the oscillating component of the conductivity

Figure 1c and 1d show the data for our thinnest samples, with a thickness of 12.6μm. As seen in Fig.1c, the longitudinal resistivity is dominated by a large background. Cadmium is a compensated metal with a finite overlap between the conduction and the valence bands and an equal density of holes and electrons. Like other elemental semi-metals, such as bismuth³¹ and antimony³², it hosts a large orbital magnetoresistance, which, thanks to perfect compensation, does not saturate even in the high field limit (μB ≫ 1, where μ is the electronic mobility)³³. On top of this background, one can detect oscillations. The inset shows the oscillating signal extracted by using a quadratic fit to the monotonic background. Two oscillations and the beginning of the third one are clearly visible.

Figure 1d shows the Hall resistivity of the same sample. Here, one can see that field-periodic oscillations dominate the response, which is not surprising given the almost perfect compensation between the density of electrons and holes. The slope of the black solid line implies a finite non-oscillating Hall resistivity, presumably due to a slightly larger mobility of electrons compared to holes.

Figure 1e, f show similar data for a twice thicker (d = 25 μm) sample. There are twice more oscillations in the same field window and the frequency of oscillations become twice faster. Meanwhile, the amplitude of oscillations is also reduced. The shape of oscillations is close to sinusoidal, but the evolution of the amplitude with magnetic field is complex.

Raw data for three thicker samples are presented in supplementary note 3³⁰. These data confirm the features seen in the thinnest samples.

Period of oscillations

Figure 2 displays the evolution of the signal in two Cd single crystals with d ranging from 12.6 to 475 μm. Note the scales of both the vertical and horizontal axes vary across the panels. The absolute amplitude of oscillations and their period both enhance with thinning. Since oscillations are squeezed in thicker samples, the field axis is limited to a lower bound to make them visible. More than one frequency may be present in the thickest (d = 475μm) sample. Here, we will focus on the main one.

Fig. 2: Thickness dependence of oscillation period and Fermi surface topology.

a, b The oscillatory component of Sondheimer oscillations (SO) in the longitudinal and transverse resistivity for the 12.6 μm sample, respectively. As the field increases, the amplitude of oscillation grows. c, d The oscillatory component of SO in the longitudinal and transverse resistivity for the 475 μm sample, respectively. e Oscillation period, ΔB, as a function of thickness, d. The solid line represents a linear variation with a slope corresponding to ΔB ⋅ d = 58 T ⋅ μm. The data point reported by Grenier et al.⁸ for a 1.02 mm thick sample is also included. Inserting this ΔBd in Equation (2) yields $\frac{\partial A}{\partial {k}_{z}}=8.76$ Å⁻¹. f The computed cross section of A for the `lens-shaped' electron-like pocket as a function of k_z shown near the southern pole²⁹. This pocket, as shown in the inset, is not elliptical. The plot of A vs. k_z is perfectly linear, and its slope yields $\frac{\partial A}{\partial {k}_{z}}=\pm 8.73$ Å⁻¹. This corresponds within a margin less than a percent to the experimentally derived $\frac{\partial A}{\partial {k}_{z}}$. The linear out-of-plane dispersion of this band, which hosts semi-Dirac fermions, is the origin of the constant $\frac{\partial A}{\partial {k}_{z}}$ over a k_z interval of at least 0.014 Å⁻¹.

The evolution of the main period as a function of the thickness is shown in Fig. 2e. One can see that our data for 10μm  < d < 500μm are in excellent agreement with what was found by Grenier et al.⁸ for a sample with d = 1020μm. The solid black line represents a linear relation between thickness and frequency. The product of period and thickness is constant (ΔB ⋅ d = 0.058 T ⋅ mm) even when d varies by two orders of magnitude. Inserting this in Equation (2) leads to $\frac{\partial A}{\partial {k}_{z}}$ = 8.76 Å⁻¹. This is to be compared with what is calculated by ab initio theory $\frac{\partial A}{\partial {k}_{z}}$ = 8.73 Å^−1 29. The agreement is remarkable.

The semi-Dirac nature of the band giving rise to the electron-like pocket plays an important role. This pocket is not an ellipsoid^8,29 (See the inset in Fig. 2f). In an ellipsoid, the extremal $\frac{\partial A}{\partial {k}_{z}}$ at each pole (or apex) is equal to $\frac{\partial A}{\partial {k}_{z}}=2\pi \frac{{k}_{Fr}^{2}}{{k}_{Fz}}$ = 12.6 Å⁻¹, significantly different from the experimentally measured value (The computed pocket dimensions are k_Fr = 0.742Å⁻¹ and k_Fz = 0.2755Å⁻¹). Moreover, in contrast to an ellipsoid, a large portion of the Fermi surface near the poles shares an identical $\frac{\partial A}{\partial {k}_{z}}$. This can be seen in Fig. 2f, which shows A as a perfectly linear function of k_z near a pole (${k}_{z}^{max}=-0.2755$ Å⁻¹). This will play a crucial role in our discussion in section results and discussion.

Amplitude of oscillations

To compare the amplitude of oscillations with theoretical expectations, we inverted the measured resistivity tensor to quantify the conductivity tensor. Figure 3 displays δσ_xxB⁴ and δσ_xyB⁴ as a function of magnetic field in four samples. By plotting the data in this way one can see that the postulated ∝ B⁻⁴ field dependence does not hold and σ_ijB⁴ oscillation peaks do not keep a fixed amplitude.

Fig. 3: Failure of the B⁻⁴ field dependence.

a, b Multiplying the longitudinal and transverse conductivity of the 12.6 μm sample by B⁴ does not yield oscillations with a constant amplitude. c, d Same (a) and (b), but for the 25 μm sample. e, f Same (a) and (b), but for the 97.5 μm sample. A change in regime is observed above 7.1 T. g, h For the 200 μm sample, the amplitude of δσ_xxB⁴ oscillations becomes constant above 2.6 T.

To empirically explore possible fits to the field dependence of the amplitude of conductivity oscillations, we multiplied the longitudinal conductivity by various powers of the magnetic field. Figure 4 shows the data for the 97.5 μm sample. Plots of δσ_xxB^α (with α = 2, 2.5 and 3) are shown. None of them are satisfactory. Upon closer examination, we found that δσ_xxB^α becomes non-monotonic when α > 2.5, but is monotonic when α < 2.5. Moreover, when α = 2.5, δσ_xxB^2.5 displays an exponential decay (see Fig. 4c, indicated by the red dashed line). This led us to try $\propto {B}^{-2.5}{e}^{-B/{B}_{0}}$ (see Fig. 4d) with B₀ = 1.8 T. Required for dimensional consistency, B₀ is a parameter indispensable for ensuring an almost perfect sinusoidal behavior. The deviation from this behavior seen in high fields (Fig. 4d) is also meaningful.

Fig. 4: Identification of the best-fit scaling: $\propto {B}^{-2.5}\,{e}^{-B/{B}_{0}}$.

a The longitudinal conductivity multiplied by B². The envelope of the oscillation amplitude decreases monotonically. b The longitudinal conductivity multiplied by B³. The oscillation amplitude is non-constant and its dependence on field is non-monotonic. c The longitudinal conductivity multiplied by B^2.5, showing an exponential decay (indicated by a red dashed line). d Multiplying by ${B}^{2.5}\,{e}^{B/{B}_{0}}$ leads to a flat evolution of oscillation amplitude with magnetic field. e The longitudinal conductivity of the 12.6 μm sample multiplied by ${B}^{2.5}{e}^{B/{B}_{0}}$. Sinusoidal oscillations are observed with a period of 5.14 T. Here, the B₀ equals 15.2 T. The dashed line indicates the sine fitting. f The same analysis for Hall conductivity yields a similar period B₀ approximately 16.3 T. g, h Application of the same method for the 25 μm sample reveals sinusoidal oscillations with a period of approximately 2.42 T. B₀ is 6.8 T and 7.2 T for the longitudinal and transverse conductivity.

This fitting function, $\propto {B}^{-2.5}{e}^{-B/{B}_{0}}$, was applied to the data obtained from other samples (d < 100μm). As illustrated in Fig. 4, in samples with a thickness of 12.6 (Fig. 4e, f) and 25 (Fig. 4g, h) μm, oscillations of conductivity (both longitudinal and transverse) are $\propto {B}^{-2.5}{e}^{-B/{B}_{0}}$ up to the highest magnetic field (B < 14 T). The few oscillations are almost perfectly sinusoidal with no deviation occurring at high magnetic fields. In the 200 and 475 μm samples, conductivities follow $\propto {B}^{-2.5}{e}^{-B/{B}_{0}}$ below a threshold field, which we call B₁. Above B₁, both conductivities follow ∝ B⁻⁴ (see the supplementary note 4³⁰).

Figure 5a shows the thickness dependence of B₀. We find B₀ ∝ d⁻¹. As mentioned above, the period of oscillations is also proportional to d⁻¹. As a result, the ratio of B₀ to ΔB is constant, as revealed in Fig. 5b. Even more striking is the fact that in five samples studied, the ratio is close to three: $\frac{{B}_{0}}{\Delta B} \sim 3$. Thus, B₀ is not an additional parameter, but is simply three times ΔB. To summarize our results, the field dependence of the (peak-to-peak) amplitude of oscillations has two distinct regimes:

$$\delta {\sigma }^{pp}\propto \left\{\begin{array}{lr}{B}^{-2.5}{e}^{-B/3\Delta B}, & B < {B}_{1}.\\ {B}^{-4},\hfill & B > {B}_{1}.\end{array}\right.$$

(3)

Let us now turn our attention to the significance of B₁.

Fig. 5: Thickness dependence of B₀ and B₁.

a B₀ is inversely proportional to the thickness. b In all samples, B₀ is about 3 times ΔB. c B₁ in different samples. It decreases with increasing d. The data for d = 1.02 mm is taken from Ref. ⁸. The error bar originates from the uncertainty in the extraction of the threshold field, B₁³⁰. d The ratio of B₁/ΔB is close to 10 in all samples. e) A comparison of g(x) = x⁻⁴ and f(x) = exp( − x/3) ⋅ x^−2.5. The two functions cross each other at x^⋆ ≃ 10.5.

Boundary between the two regimes

As the sample thickens, the threshold field B₁ decreases. This ensures consistency between our data and what was reported in the previous study of Sondheimer oscillations in cadmium, which was performed on a 1.02 mm thick sample⁸ and concluded that the amplitude of oscillations follows B⁻⁴. Interestingly, the authors found a deviation from this behavior in their data below  ~ 0.45 T (See Fig.13 in ref. ⁸), in agreement with our findings.

As seen in Fig. 5c, which shows the thickness dependence of B₁, the latter is proportional to the inverse of the thickness. The plot includes a data point for d = 1.02 mm⁸ with B₁  ~ 0.45 T. Figure 5d shows that for all samples, $\frac{{B}_{1}}{\Delta B}\simeq 10$. For more details on the boundary between the two regimes, see the supplementary note 4³⁰.

Insight into this is offered by considering the two functions g and f of x = B/ΔB. With g(x) = x⁻⁴ and f(x) = exp( − x/3)x^−2.5, one can see that they cross each other at x^⋆ ≃ 10.5. In other words, f > g when x < x^⋆ and f < g when x > x^⋆. This means that the B₁ = 10ΔB is not an independent parameter, but a marker of domination between two competing field dependencies. ΔB is the fundamental field scale.

Thus, ${\sigma }_{ij}\propto {B}^{-2.5}{e}^{-B/3\Delta B}\cos (B/\Delta B)$ field dependence, which holds in thinner samples, coexists with the ${\sigma }_{ij}\propto {B}^{-4}\cos (B/\Delta B)$ field dependence. The latter dominates in thicker samples and high magnetic field, just because it decays slower at higher fields.

Let us now turn our attention to the thickness dependence.

Thickness dependence and the scaling of the amplitude

Figure 6a is a plot of the product of δσ^pp and B^2.5 as a function of oscillation number, confirming the relevance of the exponential term. Figure 6b shows how the peak-to-peak amplitude of the oscillations, δσ^pp, evolves with thickness. The product of δσ^pp and ${B}^{2.5}{e}^{B/{B}_{0}}$ in each sample is plotted as a function of its thickness, d. It is clear that the best fit is close to d⁻². As we will discuss in the next section, neither this thickness dependence nor this field dependence is expected in the available semiclassical theories. As seen in Fig. 6c, d, this field and thickness dependence leads to a scaling of the amplitude of the oscillations (both longitudinal and transverse in all cadmium single crystals used in this study). All curves collapse on top of each other.

Fig. 6: Thickness dependence and scaling of the amplitude.

a The peak-to-peak amplitude of the oscillations, $\delta {\sigma }_{xx}^{pp}{B}^{2.5}$, evolves with oscillation number and exhibits an exponential decay. b The peak-to-peak amplitude of the oscillations, $\delta {\sigma }_{xx}^{pp}{B}^{2.5}{e}^{B/{B}_{0}}$, scales with thickness d. The best fit is close to d⁻². c Oscillations of normalized conductivity as a function of the product of magnetic field and thickness (Bd). Curves for all samples fall on top of each other. The vertical axis represents conductivity divided by the conductance quantum (${G}_{0}=\frac{2{e}^{2}}{h}$) and multiplied by ${\ell }_{B}^{-5}{d}^{2}{e}^{B/{B}_{0}}$. This normalized quantity has dimensions of [L⁻⁴]. d The same scaling applied to the Hall conductivity data. All curves collapse, with normalized amplitude twice that of the longitudinal component.

Thus, not only the period but also the amplitude of the first ten oscillations is determined by the thickness and Fermi surface geometry. This scaling implies that the amplitude can be written as:

$$\delta {\sigma }^{pp}={G}_{0}{d}^{-2}{e}^{-B/3\Delta B}{\ell }_{B}^{5}{k}_{s}^{4}$$

(4)

Here, ${\ell }_{B}=\sqrt{\frac{\hslash }{eB}}$ is the magnetic length. The parameter k_s is a momentum space length scale, determined by the Fermi surface geometry.

Temperature dependence of the oscillations

Only the temperature dependence of resistivity in the 25 μm sample was carefully studied. As shown in Fig. 7, magneto-morphic oscillations in both longitudinal and Hall resistivity weaken with increasing temperature. The evolution of $\delta {\sigma }_{xy}{B}^{2.5}{e}^{B/{B}_{0}}$ with temperature is shown in Fig. 7c. These oscillations remain sinusoidal across all measured temperatures. As seen in Fig. 7d, the amplitude of the oscillations decreases linearly with increasing temperature and vanishes when T ≃ 8.5 K.

Fig. 7: Temperature dependence of oscillations in 25 μm sample.

a Longitudinal resistivity ρ_xx as a function of magnetic field at different temperatures. b Transverse resistivity ρ_xy as a function of magnetic field at different temperatures. c Transverse conductivity δσ_xy multiplied by ${B}^{2.5}{e}^{B/{B}_{0}}$, showing the flattened evolution of the oscillation amplitude with magnetic field. The dashed line indicates the sine fitting. d The amplitude of the third oscillation marked by the black line in (c) as a function of temperature.

Given that the effective mass of carriers is not far from the bare electron mass, in a magnetic field of  ~ 10 T, this is a temperature at which the thermal energy exceeds the spacing between Landau levels.

Our extensive data analysis led to an empirical expression for oscillating conductivity (4). In this section, we begin by showing that this expression deviates from the available semiclassical theories. A comparison with copper indicates that specific features in the band structure of cadmium are essential. Then, we will show that, within a quantum picture, the oscillations arise because sweeping the magnetic field drives two distinct discretizations into commensurability at regular intervals of magnetic field, and the exponential term in equation (4) can be traced to the mesoscopic degeneracy of $\frac{\partial A}{\partial {k}_{z}}$. Finally, we reformulate the empirical equation (4) and render its connection to the Fermi surface geometry transparent.

Comparison with semiclassical predictions

Abrikosov ²² considered the field dependence of Sondheimer oscillations in two distinct cases. In the first case, the Fermi surface has an inflection point (at which $\frac{{\partial }^{2}A}{\partial {k}_{z}^{2}}=0$). He found that in this case, the amplitude of oscillations decays with increasing magnetic field (following B^−2.5) and thickness (following d^−3/2). In other words:

$$\delta {\sigma }_{ip}^{pp}\propto {\ell }_{B}^{5}{d}^{-3/2}$$

(5)

This result was originally derived by Gurevich²¹. In addition, Abrikosov considered the case of an end point on a featureless Fermi surface and found a result similar to what was found by Chambers³. In the latter case, the amplitude of oscillations was found to be:

$$\delta {\sigma }_{ep}^{pp}\propto {\ell }_{B}^{8}{d}^{-3}$$

(6)

As shown in the supplementary Note 4³⁰, our high-field oscillations do not display a scaling as remarkable as that seen for low-field oscillations. However, their thickness and field dependence are roughly compatible with Eq. (6). This is understandable given that there is indeed an end point in the Fermi surface of cadmium. As we will see below, the field dependence in copper is similar to what is expected by Eq. (5).

Our low field empirical expression Eq. (4), however, differs from both Equations (5) and (6). The most striking difference is the presence of e^−B/3ΔB; the thickness dependence also differs. There are other semiclassical scenarios, in which the exponent of the power law is different from -2.5 or -4 depending on Fermi surface details⁷. However, none of these semiclassical scenarios account for the exponential term, which is reminiscent of the canonical quantum tunneling expression e^−s/ℏ, where s represents an action³⁴.

Contrasting cadmium with copper

To determine whether the scaling discovered in cadmium is general, we used FIB to fabricate thin copper crystals in a manner identical to that employed for cadmium. As discussed in the supplement (note 5)³⁰, we detected Sondheimer oscillations in both longitudinal and Hall conductivities after subtracting the monotonic background. The oscillation periods found in our crystals agree with those reported by Sakamoto¹⁵. The product of the oscillation period and thickness (ΔB ⋅ d = 14.7 × 10⁻⁶ T ⋅ m) combined with Equation (2) implies $\frac{\partial A}{\partial {k}_{z}}$ = 1.28 Å⁻¹. Sakamoto showed that this value closely matches an extremum in the belly orbit of the well-established Fermi surface of copper¹⁵. The optimal scaling for the amplitude was found to be ∝  B^−2.5, leading to a nearly constant amplitude for δσB^2.5 over the first few oscillations. As discussed above, an exponent of 2.5 is indeed what is expected for an inflection point singularity of $\frac{\partial A}{\partial {k}_{z}}$^7,21,22.

The remarkable difference between the quality of the experimental data in copper and in cadmium is partially an issue of materials science. However, there are fundamental distinctions. There is no exponential term in the field dependence of δσ^pp. The thickness dependence is much weaker and, finally, the oscillation amplitude of Hall conductivity is less than half that of the longitudinal conductivity. All these features are in striking contrast with those observed in cadmium, despite the fact that in both cases, the period of oscillations multiplied by thickness is equal to what is expected by equation (2) and a singular value of $\frac{\partial A}{\partial {k}_{z}}$.

This leads us to conclude that the specific electronic band structure in Cd is crucial for the emergence of the empirical scaling identified in this study. In contrast to copper, cadmium is a compensated metal with equal volumes of hole-like and electron-like Fermi surface pockets. This may play a role in locking the amplitude of δσ_xy to  ~ 2δσ_xx. More importantly, in cadmium, $\frac{\partial A}{\partial {k}_{z}}$ is constant over a large portion of the Fermi surface. In copper, its singularity is sharply localized and corresponds to a single inflection point in momentum space.

Interplay between two distinct energy quantizations

Spatial confinement is known to generate discrete energy levels. Quantum effects are expected when the thickness approaches the Fermi wavelength (d ≈ λ_F). However, our case is far from this limit. In a finite magnetic field, energy levels become discrete. The semiclassical picture breaks down at high magnetic fields (when ℓ_B ≈ λ_F). This is not our case either.

However, when well-defined $\frac{\partial A}{\partial {k}_{z}}$ is shared by a significant number of electronic states, a bridge builds up between these two distinct discretizations. A magnetic field truncates the Fermi surface into Landau tubes oriented along the magnetic field (Fig. 8a). Each tube corresponds to states sharing an identical Landau level index, n. Two neighboring Landau levels, n and n + 1, are separated by an energy of ℏω_c, where ${\omega }_{c}=\frac{eB}{{m}^{\star }}$ is the cyclotron energy and m^⋆ is the effective mass⁶.

Fig. 8: Landau tubes and Fermi surface steps.

a A magnetic field truncates the Fermi surface into Landau tubes. Neighboring Landau levels, indexed n and n + 1, are separated by an energy interval, which depends on magnetic field (ℏω_c). b Confinement of electrons in real space modifies the smooth Fermi surface and induces steps in k-space (red line). Their height along k_z is ${k}_{d}=\frac{\pi }{d}$. The energy distance between two neighboring levels, indexed l and l + 1, depends on the thickness d, (it is $\frac{{\hslash }^{2}}{2\pi {m}^{\star }}{k}_{d}\partial A/\partial {k}_{z}$). c Sketch of quantization in the (energy E, k_z) plane: Vertical lines refer to the discretization of k_z introduced by spatial confinement. Oblique lines represent Landau levels and their dispersion. Their intersections are marked as circles (black when occupied and empty when unoccupied). These correspond to eigen-states common to both components of the global Hamiltonian. l can take negative and positive values, but n is exclusively positive. Thus introduces a factor of 2 in the comparison between the two sets of quantum numbers. When the ratio of the two degeneracies per unit square in k-space is an even number (left panel), all highest occupied states have an identical energy and this matches the location of the chemical potential. When the ratio of the two is an odd number (right panel), the incommensurability between the two sets of discretization becomes extreme and the chemical potential is farthest away from allowed states. Alternation between these two extreme situations is periodic in magnetic field.

A finite thickness of d along the z direction in real space implies that the wave-vector along this direction ceases to be continuous. It can only be an integer times $\frac{\pi }{d}$. Such a constraint sculpts k_z steps on the Fermi surface with a height of ${k}_{d}=\frac{\pi }{d}$, as illustrated in Fig. 8b. Note that k_d ℓ ≥ 1 and therefore, k_d is well-defined. This reorganization redistributes the in-plane and the out-of-plane components of the kinetic energy. In an infinite sample, k_r = 0 states have no in-plane kinetic energy and the share of in-plane (out-of-plane) kinetic energy increases (decreases) smoothly with increasing k_r. In contrast, in a finite sample, k_d steps on the Fermi surface are concomitant with abrupt transfers between the in-plane and out-of-plane components equal to:

$$\delta E=\frac{{\hslash }^{2}}{2\pi {m}^{\star }}{k}_{d}\frac{\partial A}{\partial {k}_{z}}$$

(7)

Thus, at zero magnetic field, the out-of-plane and the in-plane kinetic energies are discrete and there are δE steps. Now, a finite magnetic field opens a cyclotron gap equal to $\hslash {\omega }_{c}=\hslash \frac{eB}{{m}^{\star }}$ in the in-plane energy spectrum. The interplay between these two distinct energy quantizations, one set by thickness and another by the magnetic field, can provide a straightforward account of oscillations.

Let us consider the degeneracy of states for each type of quantization. The degeneracy of Landau levels per unit area is simply ${D}_{LL}=\frac{eB}{h}$. As for z-axis confinement, the flat rings carved on the Fermi surface are each a set of degenerate states. In each ring, indexed l, radial wave-vectors (which are good quantum numbers at zero field) are distributed within a finite window (k_Fr,l+1 < k_Fr < k_Fr,l). The area of each ring is equal to:

$$\pi ({k}_{Fr,l+1}^{2}-{k}_{Fr,l}^{2})={k}_{d}\frac{\partial A}{\partial {k}_{z}}$$

(8)

Each level l has an out-of-plane Fermi energy of E_F⊥,l = δE∣l∣ and a degeneracy of ${D}_{z}={(2\pi )}^{-2}{k}_{d}\frac{\partial A}{\partial {k}_{z}}$ per unit area.

Thus, each discretization generates a set of degenerate states whose energy is identical, but is distinguished either by their radial wave-vector (in the case of confinement) or by their angular momentum (in case of Landau quantization). Let us now introduce a dimensionless quantity, Ω:

$$\Omega \equiv d{\ell }_{B}^{-2}{\left(\frac{\partial A}{\partial {k}_{z}}\right)}^{-1}$$

(9)

One can see that Ω, which is proportional to the magnetic field is also set by the ratio of the two degeneracies:

$$\frac{{D}_{LL}}{{D}_{z}}\equiv 2\Omega$$

(10)

The D_LL/D_z ratio, which quantifies the number of flat rings in each Landau level, is akin to an inverted filling factor. Recall the filling factor in a two-dimensional electron gas, $\nu =\frac{h{n}_{2d}}{eB}$, which is proportional to the inverse of the magnetic field³⁵. In our case, the number of rings in each Landau level increases linearly with magnetic field. In both cases, oscillations arise because ν and Ω become integers at regular intervals (either of the magnetic field or the inverse of it).

Thus, as the magnetic field is swept, the inverted filling factor Ω steadily increases. At specific magnetic fields, it becomes an integer. This situation is depicted on the left-hand side of Fig. 8c. Each discretization is represented by a set of solid lines and states allowed by their combination by empty circles. Commensurability between D_LL and D_z allows occupation of allowed states up to the chemical potential. However, this is not the case in an arbitrary magnetic field. The right hand of Fig. 8c sketches a situation in which Ω is a half-integer and the chemical potential resides at midway between allowed states. Since dimensionless Ω periodically becomes an integer with sweeping magnetic field, this picture implies the existence of oscillations that are periodic in B, with a period identical to the semiclassical one. Indeed, equation (9) combined with equation (2) yields:

$$\Omega =\frac{B}{\Delta B}$$

(11)

Thus, in regular intervals of increasing magnetic field, Ω becomes an integer and the two quantizations become commensurate. In other words, the boundary between occupied and unoccupied states of the two sets of discrete levels match each other. In contrast, at halfway between two successive integer values of Ω, the chemical potential is located between the energy levels allowed by the two discretizations. This leads to a competition between the two energy spacings. When the magnetic field is large or the sample is thick, ℏω_c > δE and the cyclotron gap dominates. Conversely, when the field is weak or the sample is thin, what dominates is the discontinuity in the out-of-plane kinetic energy imposed by spatial confinement.

Cadmium is a compensated metal. Charge conservation implies that the densities of mobile electrons and mobile holes remain identical in a stoichiometric crystal. However, the chemical potential and the absolute values of carrier density can both shift if it lowers energy. This feature has been documented in detail in the extreme quantum limit of bismuth subject to a strong magnetic field³⁶.

In the case depicted in the left side of Fig. 8c, the occupied and unoccupied states are separated by a cyclotron gap. Tunneling is expected to occur between two neighboring Landau levels. Equation (4) includes a $\exp (-B/{B}_{0})$ term, which indicates quantum tunneling. We saw that B₀ = 2.9 ± 0.4ΔB (See Fig. 5b). Assuming B₀ = πΔB, an assumption compatible with our experimental accuracy, the exponential term can be rewritten as $\exp (-\Omega /\pi )$. Quantum tunneling refers to a process with the probability of $\exp (-s/\hslash )$, where s has the dimensions of an action³⁴. In our case:

$$s=eB{k}_{d}^{-1}{\left(\frac{\partial A}{\partial {k}_{z}}\right)}^{-1}$$

(12)

There is a deep connection between quantum tunneling and the uncertainty principle³⁷. Despite conservation of energy, a particle can tunnel across an energy barrier in real space with a height larger than its kinetic energy. This is because one cannot know both the position and the momentum with infinite accuracy. In the Wentzel-Kramers-Brillouin (WKB) approximation, the tunneling action for a particle with momentum p is given by³⁷:

$$s={\int }_{\!\!\!\!{x}_{1}}^{{x}_{2}}p({x}^{{\prime} })d{x}^{{\prime} }$$

(13)

Comparing Equations (12) and (13), one can see that in our case, the action, instead of being a product of momentum and position, is the product of the charge of electron, e, and a magnetic flux, $B{k}_{d}^{-1}{(\frac{\partial A}{\partial {k}_{z}})}^{-1}$³⁰. Let us recall that Noether’s theorem links the conservation of electric charge to the gauge symmetry. In quantum mechanics, the electric charge and magnetic flux are conjugate variables. An intuitive account of Equation (12) is provided by noting that an electron which does not belong with a Landau level is only allowed because the product of the uncertainties in electric charge and magnetic flux is bounded by the Planck constant. Thus, there is an analogy between such an electron tunneling across two neighboring Landau levels and a particle crossing an energy barrier.

The quantum origin of the empirical scaling

Given the considerations detailed above, we can rewrite Equation (4) as:

$$\delta {\sigma }^{pp}=\alpha {G}_{0}{k}_{Fz}{k}_{Fr}{\ell }_{B}{\Omega }^{-2}{e}^{-\Omega /\pi }$$

(14)

Here, α is a dimensionless fitting parameter and k_Fz, k_Fr are the Fermi radii.

Figure 9, confronts our experimental data with equation (14). For all samples, the normalized δσ is plotted against the normalized magnetic field. Both axes are dimensionless. The horizontal axis is Ω, the magnetic field normalized by the [thickness-dependent] period. The vertical axis is δσ_ii or δσ_ij normalized by the conductance quantum and the product of k_Fzk_Frℓ_BΩ⁻²e^−Ω/π. For each sample, Ω has been extracted from Equation (9) and the thickness of the sample. The experimental data is compared with a sinusoidal function whose amplitude is set by Eq. (14) given the computed²⁹ values of k_Fz and k_Fr. Only α is a free parameter. The slight frequency mismatch may be attributed to a difference between nominal and effective thickness. α is sample dependent, but remains close to unity. Save for the first oscillation period, the agreement between our empirical equation (14) and the data is remarkable. Of course, such an empirical agreement is not a rigorous explanation. In the absence of the latter, a challenge to condensed matter quantum theory, let us restrict ourselves to a set of remarks.

Fig. 9: Equation (14) and experimental data.

a Oscillations of dimensionless conductivity as a function of dimensionless period index. The vertical axis is the oscillating conductivity divided by the product of ${G}_{0}=\frac{2{e}^{2}}{h}$ and k_Frk_Fzℓ_BΩ⁻²e^−Ω/π. The horizontal axis represents $\Omega ={\ell }_{B}^{-2}{\left(\frac{\partial A}{\partial {k}_{z}}\right)}^{-1}d$. The solid blue lines represents αsin(Ω − ϕ). Save for the first oscillation, in all samples, the data matches the sine fit, with α ≈ 1 and ϕ ≈ 0.15π. b Same for Hall conductivity. A similar fit matches the data with an α, which is twice larger and a ϕ, which is shifted by  − π/2.

Equation (14) is reminiscent of a three-dimensional Drude conductivity expressed as Landauer transmission ($\sigma =\alpha {G}_{0}{k}_{F}^{2}\ell$). A naive interpretation would be the following: as the magnetic field is swept, a fraction ($\propto {\Omega }^{-2}{e}^{-\frac{\Omega }{\pi }}$) of states at Fermi level oscillate between localization and conduction. When they conduct their mean free path, is  ≈ ℓ_B.

The exponential term in Equation (14), which refers to tunneling across neighboring Landau levels, is to be contrasted with another case of tunneling, namely magnetic breakdown⁶. There, tunneling occurs across cyclotron orbits separated by an energy barrier. We also note that tunneling between Landau levels across an n − p junction of Weyl semi-metals was framed³⁸ as a manifestation of Landau-Zener tunneling in a two-level system³⁹. A similar framework was proposed for the case of magnetic breakdown⁴⁰. The two cases give rise to an exponential term with magnetic field either in the numerator or the denominator of the exponent. The relevance of these ideas to our experimental context remains an open question.

A comparison of oscillations in the transverse and in the longitudinal channels reveals puzzles that are yet to be understood and constitute a challenge to theory. In all five samples studied, there is a π/2 phase shift between the two sets of oscillations. Moreover, the transverse amplitude is roughly twice the longitudinal amplitude (see Fig. 9 and the supplementary note 6³⁰ for more details).

Conclusion

In summary, we showed that in thin single crystals of cadmium hosting ballistic electrons, Sondheimer (or magnetomorphic) oscillations of conductivity display a field dependence never encountered before. Their amplitude is set by the quantum of conductance and a length scale depending only on the thickness, the magnetic length, and the Fermi surface geometry. We argued that to explain this experimental observation, one needs to go beyond the available semiclassical theories and take into account the interplay between Landau tubes and the steps sculpted on the Fermi surface by spatial confinement. This led us to an empirical scaling in which the key parameter is a dimensionless parameter set by a combination of length scales which set by the Fermi surface geometry, the thickness and the magnetic length. A rigorous account of our experimental finding is missing and remains a challenge to theory.

Methods

Samples

Cd single crystals were grown using the vapor-phase transport method. A quartz tube containing appropriate amounts of starting material, Cd (99.9997%), was kept at the growth temperature for two weeks in a two-zone furnace. Shiny plate-like crystals were produced under a temperature gradient of 300 -200°C.

Measurements

All transport experiments were performed in a commercial measurement system (Quantum Design PPMS). Electric transport responses were measured by a standard four-probe method using a current source (Keithley6221) and a DC-nanovoltmeter (Keithley2182A).