We demonstrate a non-linear measurement scheme of the Shubnikov–de Haas effect based on Joule self-heating that builds on ideas of the $3\omega $-method used in thin films. While the temperature dependence of the resistance, *R*(*T*), of clean metals at low temperatures saturates, a significant temperature dependence, *dR*/*dT*, appears at high fields due to Landau quantization. We experimentally demonstrate this effect in the semi-metal CoSi, resolving well quantum oscillations at low magnetic fields in the non-linear channel, which appear as 3rd harmonics of the current drive frequency. To ensure the dominant self-heating originates in the crystal, not at the contacts, we fabricate crystalline microbars using focused ion beam machining. These oscillations in non-linear channel encode the ratio between the *dR*/*dT* and the thermal conductivity of the material, rendering it an interesting probe in situations of the broken Wiedemann–Franz law. Our results present a quantitative methodology that is particularly suited to investigate the electronic structure of micro- and nano-materials at intermediate temperatures.

Since the discovery of magneto resistance oscillations of single-crystal bismuth,^{1} the Shubnikov–de Haas (SdH) effect has been studied in a wide variety of metals.^{2} Contrary to magnetization oscillations, the de Haas–van Alphen effect (dHvA), SdH oscillations of a transport coefficient tend to be dominated by small Fermi surfaces due to the restricted phase-space of scattering. With the growing interest in topological band structure features, such quantum oscillations of small Fermi surfaces have recently shifted into attention.^{3–6} SdH is also ideally suited for mesoscopic samples as the total resistance increases with decreasing sample size, unlike the extensive magnetic moment that decreases. For example, measurements of dHvA oscillations on a two-dimensional electron gas (2DEG) are a metrological challenge,^{7–9} yet SdH oscillations are famously large in 2DEGs, leading to the discoveries of Quantum Hall states.

The de Haas-van Alphen effect is more widely used to determine the Fermi surface and quasiparticle effective masses of metals. In addition to the obvious preference to measure a thermodynamic quantity, the foremost advantage of dHvA is practical. The quantum coherent formation of Landau levels necessitates ultra-clean samples; yet, metals at such cleanliness are highly conductive, and hence, the voltage signals at low enough current values to minimize Joule heating are commonly small. dHvA measurements scale oppositely, as the signal size increases with increasing cleanliness due to the sharper modulation of the density of states in the Landau quantized regime.

This leaves an experimental conundrum. While the low sample resistance necessitates a high current to obtain measurable signals, the inevitable self-heating at high current levels suppresses quantum oscillations via thermal broadening. The common practice is to find a trade-off, to increase the current to the maximal level at which self-heating is not yet suppressing the oscillatory amplitudes. This sparked our curiosity: Self-heating encodes the energy dissipation landscape in a crystal and, hence, by itself should be sensitive to Landau quantization, too.

Here, we explore, rather than eliminate, Joule heating in strongly quantized metals and demonstrate that it enables a robust frequency and mass determination. This alternative measurement scheme may be generally applied to metallic crystals, yet here it is tailored to micrometer-sized samples. While the oscillatory component of the magnetoresistance is typically small, the effect of Landau quantization on the temperature-dependence of the resistance, *dR*/*dT*, is significant (Fig. 1). The resistance of metals at lowest temperatures is almost temperature independent once elastic defect scattering dominates. At Landau quantizing magnetic fields, a significant temperature dependence of *R*(*T*, *B*) is restored as the Fermi-Dirac distribution sharpens in the Landau quantized energy landscape upon cooling the sample. Thus, rather than resolving the magnetoresistance oscillations in *R*(*T*, *B*) on top of a large background, we aim to measure its slope, *dR*/*dT*, directly.

The strategy employed here builds on the $3\omega $ method, a successful approach to measure thermal properties of thin films.^{10–12} There, a micropatterned conductor is evaporated onto a thin-film of interest. With increasing applied current *I*, Joule heating raises the temperature of the conductor and thereby its resistance, $R(T+\Delta T(I))$, leading to non-linear current–voltage relations. The steady-state temperature increase, $\Delta T(I)$, naturally encodes the thermal conduction through the thin-film the conductor resides on, as it provides the dominant cooling path via heat diffusion. This non-linearity is conveniently measured via the generation of third-harmonic voltage, $V3\omega $, in a lock-in amplifier supplying a low-frequency ac-current. Interestingly, the conventional 3*ω* method naturally loses sensitivity in the low temperature limit^{11} for the very reason that enables our approach, the saturated temperature dependence of *R*(*T*) of metals at low temperatures. The key difference here to the standard $3\omega $ method is that the metallic sample itself acts as the heater and the thermometer. A thin crystalline bar in a standard 4-probe configuration is subjected to currents strong enough to change its temperature. The main cooling channel at lowest temperatures is its internal heat conductivity, draining the heat through the electric contacts [Fig. 1(a)].

Experimentally, we focus on quantum oscillations in Cobalt monosilicide (CoSi), which recently has attracted significant attention as it hosts multifold fermion excitations.^{13} Pronounced traditional SdH and thermoelectrical quantum oscillations have been reported in CoSi previously.^{14,15} Hence, this material provides an ideal testbed for this concept due to its simple quantum oscillation spectrum of only two, approximately angle-independent frequencies $F1\u223c$ 560 T and $F2\u223c$ 660 T with similar effective mass.^{14,15} These frequencies originate from Fermi surfaces around the *R* point, while the sheet around Γ has thus-far eluded observation. Indeed, the main idea of the paper is evident in the raw data (Fig. 2): At high magnetic fields, regular SdH oscillations are readily observed. The resistance under constant field corresponding to the valley/peak positions of the oscillations attains a strong temperature dependence, in contrast to the flat behavior of *R*(*T*) at zero field. Hence, detecting the slope, rather than the resistance itself, promises to be sensitive to the SdH effect.

To realize these transport bar geometries experimentally, we employ Focused Ion Beam (FIB) machining to carve an as-grown single crystal into a microstructured “U” shape structure [Fig. 2(a)]. The dimension of the bars is 78.5 × 2.7 × 5.7 *μ*m^{3}. The sample is mounted onto a sapphire substrate using two-component epoxy (Araldite brand) to minimize thermal coupling. Electrical connections are made using sputter-deposited thick Au films (200 nm), which also provide a robust thermal anchor to the sapphire acting as a thermal bath. An ac-current is applied to the device using the common current contacts, and two parallel bars can be measured simultaneously for reproducibility. The thermal situation under applied current was modeled by finite element simulations (COMSOL), which, not surprisingly, well reproduce the analytical results of a 1D heat conductor^{16} [Fig. 2(b)]. Here, the thermal boundary conditions were on the far side of the voltage terminals, which were set to the bath temperature, with all other surfaces thermally insulating. This corresponds to the approximation to neglect the heat leakage through the epoxy. This approximation appears well justified at low temperatures since the crystalline bars are decoupled from the substrate by a thick layer (>5 *μ*m) of epoxy, which, at low temperatures, is a good thermal insulator.^{17} The low resistance, Ohmic contacts furthermore speak against a significant contact barrier for electronic heat conduction. Good agreement of our data with this thermal profile as shown in Fig. 2(b) gives some confidence in these approximations; yet, they are not critical for the main result of the non-linear SdH observation.

Throughout this study, the devices are operated at a low frequency (73 Hz) to ensure that the corresponding duration period (13.7 ms) is well above the thermal time constant of the microstructure (∼0.4 *μ*s, estimated as $\tau \u223c\rho CpL2\pi 2\lambda $,^{16} where *ρ*, C_{p}, *L*, and *λ* are the mass density, specific heat, length, and thermal conductivity, respectively). In this adiabatic limit, the temperature instantaneously follows the current such that phase shifts due to the small heat capacity are absent. It will be interesting to explore in the future if high-frequency operation will further allow to reliably estimate the specific heat of this device; yet, the present devices are optimized for dc (<100 Hz) measurements.

A self-heating-induced temperature increase $\Delta T=Tsample\u2212Tbath$ by an ac-current of amplitude *I*_{0} (RMS) induces a third harmonic voltage as

(see the supplementary material for full derivation). Here, *L* and *S* are the length and the cross-sectional area of the microbar, respectively. *λ* denotes the thermal conductivity and $R\u2032=dR(B,T)/dT|T=Tbath$ the temperature derivative of the resistance. This approximation holds as long as $\Delta T$ is small enough to warrant a linear expansion, $R(T)\u2248R(Tbath)+R\u2032\Delta T$. This condition is straightforward to validate experimentally as any higher order corrections lead to the appearance of 5th or higher harmonics, which we, indeed, observe experimentally when overdriving the devices. Throughout the experiment, the currents are adjusted, such that no 5th harmonic is measurable above the noise floor.

Indeed, the experimental data confirm these intuitive considerations [Fig. 2(c)]. While the magnetoresistance oscillations contribute a small oscillatory signal to a large background on the $Rosc(B)/R(B)\u223c10\u22124$ level, the third harmonic is completely dominated by the quantum oscillation signal. Importantly, these curves are not simply related by differentiation, as we measure the slope in temperature, not field. One remarkable feature is the absence of a background in the non-linear data [Fig. 2(c), no background subtracted]. In contrast to *R*(*B*), which is dominated by the magnetoresistance, *R*(*T*) in the absence of Landau quantization remains rather flat irrespective of the value of the field in the low-temperature limit, hence the absent (weak) background. Thus, despite benefiting from the increased sensitivity of a differential technique, we do not amplify the magnetoresistive background but only obtain large oscillatory signals.

An interesting property of this measurement mode is the somewhat counter-intuitive scaling of the oscillatory component of $V3\omega $ on the applied transport current (Fig. 3). In a standard SdH measurement, the ideal current level is a trade-off between the $V\omega $ signal size and the unavoidable and detrimental self-heating. Over-driving the sample into the self-heating regime increases the overall background signal, $V\omega $, yet reduces the quantum oscillation amplitude in it. However, as the dynamic amplitude suppression during the *ac* power cycle is the source of the non-linear $V3\omega $ signal, the oscillatory amplitude grows with increasing self-heating. While the zero-field background increases proportional to $I03$ as for any non-linear conductor, also the quantum oscillation amplitudes grow with increasing current (Fig. 3). Naturally, this increase in signal does not carry on indefinitely but is terminated when $\Delta T$ grows beyond the bounds of the linearized resistance model, which occurs at a temperature scale given by the cyclotron effective mass $m*$. Figure 3(c) here is to show that the currents we use are in the linear regime at zero field. For the magnetic field dependence of $V3\omega $ shown next, we always check a current sweep at the magnetic field of the highest landau level first to make sure the current we use is not over-driving.

Next, we turn to quantitatively formalize this model. The thermal damping of the linear SdH oscillation is captured by the *R _{T}* term in the Lifshitz–Kosevich (LK) formalism:

^{2}$RT=\alpha Xsinh(\alpha X)$. The non-linear SdH oscillation is described by its derivative: $RT\u2032=\alpha X(\alpha Xcoth(\alpha X)\u22121)Tsinh(\alpha X)$, where $\alpha =2\pi 2kBe\u210f$ and $X=m*TB$.

Despite their simplicity, these considerations on $RT\u2032$ capture our experimental observations very well (Fig. 4). First, regular SdH oscillations can be measured at lowest current levels without any self-heating, and indeed, the temperature dependence of the oscillations is well described by the usual Lifshitz–Kosevich form of *R _{T}*. By increasing the current to optimize the self-heating while avoiding any 5th harmonic, the non-linear SdH can be recorded. A comparison of the data to theory requires the

*a priori*unknown thermal conductivity,

*λ*. Here, we approximate

*λ*from the Wiedemann–Franz law as $\lambda =L0T\sigma $, where

*L*

_{0}denotes the Lorentz number and

*σ*is the electrical conductivity, which is directly obtained from the linear measurements. Using this approximation allows us to obtain the oscillatory term $RT\u2032$, which well follows the predictions of the Lifshitz-Kosevich theory. This agreement confirms that

*R*is the dominant source of temperature dependence of the resistance. For both curves, the cyclotron mass is the only free fitting parameter. We remark that a small discontinuity at 1.5 K is due to different thermal situations in the two different cryostats: a dilution refrigerator below 1.5 K and a He bath cryostat above 1.5 K. While present naturally in both datasets, such issues become visually more apparent due to the increased temperature sensitivity of the non-linear method. To estimate the effective mass more accurately, we fit the data measured in the two cryostats individually. Table I lists the fitting results from linear and non-linear SdH measured in these two cryostats, respectively. These masses match well with previously reported effective masses of electrons in CoSi,

_{T}^{18,19}yet do not exactly overlap. One natural origin of this small deviation would be the different thermal situations. The presence of exchange gas in a bath cryostat provides a second channel of thermal coupling, contrary to our assumption of thermal relaxation only within the material.

. | He bath cryostat . | Dilution refrigerator . | ||
---|---|---|---|---|

Methods . | F_{1} (m_{e})
. | F_{2} (m_{e})
. | F_{1} (m_{e})
. | F_{2} (m_{e})
. |

Resistance | 0.89 ± 0.046 | 0.94 ± 0.062 | 0.70 ± 0.014 | 0.74 ± 0.014 |

Derivative of resistance | 0.98 ± 0.024 | 1.06 ± 0.070 | 0.91 ± 0.025 | 0.94 ± 0.027 |

. | He bath cryostat . | Dilution refrigerator . | ||
---|---|---|---|---|

Methods . | F_{1} (m_{e})
. | F_{2} (m_{e})
. | F_{1} (m_{e})
. | F_{2} (m_{e})
. |

Resistance | 0.89 ± 0.046 | 0.94 ± 0.062 | 0.70 ± 0.014 | 0.74 ± 0.014 |

Derivative of resistance | 0.98 ± 0.024 | 1.06 ± 0.070 | 0.91 ± 0.025 | 0.94 ± 0.027 |

In conclusion, we describe a differential method based on self-heating of Landau-quantized metals to detect quantum oscillations in resistance in microscopic or patterned samples. The appearance of $RT\u2032$ instead of *R _{T}* in quantum oscillations is a natural consequence of thermal gradients that non-linear SdH shares with thermal transport and thermopower measurements.

^{20}Indeed, exploring non-linear SdH was inspired by magnetothermopower oscillations of macroscopic CoSi crystals,

^{14}which are in quantitative agreement with our data. While it is interesting to see this phenomenon manifests in the data, it remains to discuss if and where this approach holds practical advantages over the linear SdH effect. The first natural consideration concerns the amplitude of oscillations and its temperature dependence. While

*R*is maximal at zero temperature, its derivative $RT\u2032$ peaks at finite temperature. Therefore, quantum oscillations may be easier to resolve in the non-linear channel at elevated temperatures. This is particularly relevant in situations of heavy quasiparticles, which necessitates ultra-low temperatures to resolve the Landau levels—a challenge while avoiding self-heating. However, this large effective mass greatly enhances $RT\u2032$; it would be interesting to investigate the enhanced sensitivity for such heavy electrons.

_{T}Even in situations when the non-linear oscillatory amplitude itself may not be greatly enhanced over the linear one, the strong suppression of the background signal is an asset of this technique. In conventional, linear SdH measurements, subtracting a polynomial background is a common practice; yet, this inevitably compromises the frequency resolution on a scale given by the frequency cutoff of the background fitting function. This impacts the frequency resolution, may generate spurious low-frequency peaks in the spectrum, and strongly impacts any phase analysis based on tracing out Landau Level positions from minima/maxima of the data. Interestingly, all these points are of importance in topological semi-metals when small Fermi surfaces are studied.

Naturally, one wonders if the thermal conductivity can be measured with this technique at low temperature and high magnetic field, and the Wiedemann–Franz law be checked rather than used as an input. The determination of the thermal conductivity in micrometer-shaped quantum materials would certainly yield new insights on the current distribution and the evolution of correlated states in them. However, making quantitative statements about the thermal conductivity will require improvements on the detailed control of the thermal environment of the sample. This includes testing the assumption of negligible thermal coupling of the bar to the substrate, and further reducing the thermal boundary resistance at the contacts to avoid spurious heat-buildup. It will be interesting to apply this methodology to a variety of materials and, thereby, explore the different regions of materials space in which non-linear SdH provides new and orthogonal insights to its linear cousin.

See the supplementary material for the details about crystal growth, device fabrication, third harmonic voltage calculation, and transport measurement.

This project was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (“MiTopMat”—Grant Agreement No. 715730). K.M. and C.F. acknowledge the financial support from the European Research Council (ERC) Advanced Grant No. 742068 “TOP-MAT,” European Union's Horizon 2020 research and innovation program (Grant Nos. 824123 and 766566), and Deutsche Forschungsgemeinschaft (DFG) through SFB 1143. This project received funding by the Swiss National Science Foundation (Grant No. 176789). Additionally, K.M. acknowledges Max Plank Society for the funding support under the Max Plank–India partner group project.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts of interest to disclose.

## DATA AVAILABILITY

All raw data underpinning this publication are openly available in Zenodo at https://zenodo.org/deposit/5512433, Ref. 21.

## References

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