We calculate the magnetic-field dependence (oscillations) of the Nernst coefficient in bismuth at low temperatures for the case when the magnetic field is directed along the trigonal axis of the crystal. In the calculations we take into account the scattering of the electrons and holes in bismuth on impurities and the dependence of this scattering on the magnetic field. The results of these calculations are compared with the experimental data recently published.
We report a theoretical calculation explaining the quantum Nernst effect observed experimentally in a bismuth single crystal. Generalizing the edge-current picture in two dimensions, we show that the peaks of the Nernst coefficient survive in three dimensions due to a van Hove singularity. We also evaluate the phonon-drag effect on the Nernst coefficient numerically. Our result agrees with the experimental result for a bismuth single crystal.
In elemental Bismuth, 10$^5$ atoms share a single itinerant electron. Therefore, a moderate magnetic field can confine electrons to the lowest Landau level. We report on the first study of metallic thermoelectricity in this regime. The main thermoelectric response is off-diagonal with an oscillating component several times larger than the non-oscillating background. When the first Landau level attains the Fermi Energy, both the Nernst and the Ettingshausen coefficients sharply peak, and the latter attains a temperature-independent maximum. A qualitative agreement with a theory invoking current…
Elemental bismuth provides a rare opportunity to explore the fate of a three-dimensional gas of highly mobile electrons confined to their lowest Landau level. Coulomb interaction, neglected in the band picture, is expected to become significant in this extreme quantum limit with poorly understood consequences. Here, we present a study of the angular-dependent Nernst effect in bismuth, which establishes the existence of ultraquantum field scales on top of its complex single-particle spectrum. Each time a Landau level crosses the Fermi level, the Nernst response sharply peaks. All such peaks are…
We calculate the phonon-drag contribution to the transverse (Nernst) thermoelectric power $S_{yx}$ in a bismuth single crystal subjected to a quantizing magnetic field. The calculated heights of the Nernst peaks originating from the hole Landau levels and their temperature dependence reproduce the right order of magnitude for those of the pronounced magneto-oscillations recently reported by Behnia et al [Phys.Rev.Lett. 98, 166602 (2007)]. A striking experimental finding that $S_{yx}$ is much larger than the longitudinal (Seebeck) thermoelectric power $S_{xx}$ can be naturally explained as the …
Nernst effect is the transverse electric field due to a longitudinal thermal gradient in presence of magnetic field. Its magnitude was recently studied up to 33 T in a bismuth single crystal with a magnetic field oriented along the trigonal axis . In addition to quantum oscillations of the Nernst response, a rich structure with several additional peaks was also detected when the magnetic field exceeded the quantum limit, which is of the order of 10 T in this configuration. The origin of these high-field Nernst anomalies is a subject of ongoing research. Very recently, Seradjeh, Wu, and Phillip…
Hetero-epitaxial films based on bismuth telluride with excess of Te were grown by hat wall technique at the surface of the mica (muscovite). Galvanomagnetic properties of the thin films were measured, and quantum oscillations of the magnetoresistance were found at the temperatures below 10 K in the magnetic field from 6 to 14 T. From analysis of magnetoresistance oscillations the main surface state parameters of the films were determined. Expementally obtained Landau level index shift and its temperature dependence are consistent with Berry phase specific for topological Dirac surface states. …
Metallic Bi2Te3 crystalline sheets with the room-temperature resistivity of above 10 mΩ cm were prepared and their magnetoresistive transport was measured in a field of up to 9 Tesla. The Shubnikov de Haas oscillations were identified from the secondly-derived magnetoresistance curves. While changing the angle between the field and normal axis of the sheets, we find that the oscillation periods present a cosine dependence on the angle. This indicates a two-dimensional transport due to the surface state. The work reveals a resolvable surface contribution to the overall conduction even in a meta…
In the presented report, the Nernst-Ettingshausen effect in layered conductors is investigated. Considering a Fermi surface (FS) consisting of a slightly corrugated cylinder and two corrugated planes distributed periodically in the momentum space, the thermoelectric effects are considered under general assumptions for the value of a magnetic breakdown probability. As a result of an external generalized force, the FS sheets in layered conductors with a multisheet FS appear to be so close that the charge carriers (as a result of magnetic breakdown) can move from one FS sheet to another. In addit…
In a Comment [arXiv:0910.1256], Behnia contends that the surface theory put forward in our recent Letter [Phys. Rev. Lett. 103, 136803 (2009); arXiv:0905.0689] as an alternative explanation of the anomalous peaks observed in Nernst measurement on a single-crystal bismuth sample at high (> 9 T) magnetic fields [Science 317, 1729 (2007); arXiv:0802.1993] is not consistent with the order of magnitude and shape of the anomalous peaks observed in the experiment. We explain in this Reply why this contention is not true.
Because of the long Fermi wavelength of itinerant electrons, the quantum limit of elemental bismuth (unlike most metals) can be attained with a moderate magnetic field. The quantized orbits of electrons shrink with increasing magnetic field. Beyond the quantum limit, the circumference of these orbits becomes shorter than the Fermi wavelength. We studied transport coefficients of a single crystal of bismuth up to 33 tesla, which is deep in this ultraquantum limit. The Nernst coefficient presents three unexpected maxima that are concomitant with quasi-plateaus in the Hall coefficient. The result…
We consider the Nernst-Ettingshausen (NE) effect in the presence of semiclassically strong magnetic fields for a quasi-two dimensional system with a parabolic or linear dispersion of carriers. We show that the occurring giant oscillations of the NE coefficient are coherent with the recent experimental observation in graphene, graphite and bismuth. In the 2D case we find the exact shape of these oscillations and show that their magnitude decreases/increases with enhancement of the Fermi energy for Dirac fermions/normal carriers. With a crossover to 3D spectrum the phase of oscillations shifts, …
Geometric oscillations of local Hall and Nernst effects in ballistic graphene at weak magnetic fields
Abrikosov Center for Theoretical Physics, MIPT（莫斯科物理技术学院阿布里科索夫理论物理中心）；Institute of Physics of DFRS of the RAS（俄罗斯科学院达吉斯坦共和国科学中心物理研究所）；School of Science, Westlake University（西湖大学理学院）
We study the manifestation of the Nernst effect in the Corbino disk subjected to the normal external magnetic field and to the radial temperature gradient. The Corbino geometry offers a precious opportunity for the direct measurement of the magnetization currents that are masked by kinetic contributions to the Nernst current in the conventional geometry. The magnetization currents, also referred to as the edge currents, are independent on the conductivity of the sample which is why they can be conveniently described within the thermodynamic approach. They can be related to the Landau thermodyn…
Electrons in a metal subject to magnetic field commonly exhibit oscillatory behavior as the field strength varies, with a period set by the area of quantized electronic orbits. Recent experiments on elemental bismuth have revealed oscillations for fields above 9 tesla that do not follow this simple dependence and have been interpreted as a signature of electron fractionalization in the bulk. We argue instead that a simple explanation in terms of the surface states of bismuth exists when additional features of the experiment are included. These surface electrons are known to have significant sp…
We report the observation of a dependence of the low temperature resistance of individual single-crystal bismuth nanowires on the Aharonov-Bohm phase of the magnetic flux threading the wire. 55 and 75-nm wires were investigated in magnetic fields of up to 14 T. For 55 nm nanowires, longitudinal magnetoresistance periods of 0.8 and 1.6 T that were observed at magnetic fields over 4 T are assigned to h/2e to h/e magnetic flux modulation. The same modes of oscillation were observed in 75-nm wires. The observed effects are consistent with models of the Bi surface where surface states give rise to …
We report measurements of the thermopower $S$ and Nernst signal $S_{yx}$ in graphene in a magnetic field $H$. Both quantities show strong quantum oscillations vs. the gate voltage $V_g$. Our measurements for Landau Levels of index $n\ne 0$ are in quantitative agreement with the edge-current model of Girvin and Jonson (GJ). The inferred off-diagonal thermoelectric conductivity $α_{yx}$ comes close to the quantum of Amps per Kelvin. At the Dirac point ($n=0$), however, the width of the peak in $α_{yx}$ is very narrow. We discuss features of the thermoelectric response at the Dirac point includin…
Photocurrent measurements provide a powerful means of studying the spatially resolved optoelectronic and electrical properties of a material or device. Generally speaking there are two classes of mechanism for photocurrent generation: those involving separation of electrons and holes, and thermoelectric effects driven by electron temperature gradients. Here we introduce a new member in the latter class: the photo-Nernst effect. In graphene devices in a perpendicular magnetic field we observe photocurrent generated uniformly along the free edges, with opposite sign at opposite edges. The signal…
We investigate proximity induced superconductivity in micrometer-long bismuth nanowires con- nected to superconducting electrodes with a high critical field. At low temperature we measure a supercurrent that persists in magnetic fields as high as the critical field of the electrodes (above 11 T). The critical current is also strongly modulated by the magnetic field. In certain samples we find regular, rapid SQUID-like periodic oscillations occurring up to high fields. Other samples ex- hibit less periodic but full modulations of the critical current on Tesla field scales, with field-caused ext…
We present a study of the angular dependence of the resistivity tensor up to 35 T in elemental bismuth complemented by torque magnetometry measurements in a similar configuration. For at least two particular field orientations a few degrees off the trigonal axis, the Hall resistivity was found to become field-independent within experimental resolution in a finite field window corresponding to a field which is roughly three times the frequency of quantum oscillations. The Hall plateaus rapidly vanish as the field is tilted off theses magic angles. We identify two distinct particularities of the…
Dirac and Weyl semimetals display a host of novel properties. In Cd$_3$As$_2$, the Dirac nodes lead to a protection mechanism that strongly suppresses backscattering in zero magnetic field, resulting in ultrahigh mobility ($\sim$ 10$^7$ cm$^2$ V$^{-1}$ s$^{-1}$). In applied magnetic field, an anomalous Nernst effect is predicted to arise from the Berry curvature associated with the Weyl nodes. We report observation of a large anomalous Nernst effect in Cd$_3$As$_2$. Both the anomalous Nernst signal and transport relaxation time $τ_{tr}$ begin to increase rapidly at $\sim$ 50 K. This suggests a…
We consider magnetic oscillations of resistivity of a clean hollow cylinder with fluctuating radius $R$, threaded by magnetic flux $Φ$. We demonstrate, that for very weak fluctuations $ΔR\ll p_F^{-1}$ the oscillations have a standard period $2Φ_0$ (where $Φ_0=πc\hbar/2e$), characteristic for oscillations in a clean system, while for $ΔR\gg p_F^{-1}$ they become $Φ_0$-periodic, which was expected only for dirty systems with $R\gg l$. The work is motivated by observation of predominantly $Φ_0$-periodic magnetic oscillations in very clean Bismuth wires.
Quantum oscillations of metallic systems at low temperatures is one of the key rules to experimentally access their electronic properties, such as energy spectrum, scattering mechanisms, geometry of Fermi surface and many other features. The importance of these knowledge is enormous, since from these a thorough understanding of anomalous Hall effect, thermopower and Nernst coefficients, just to name a few, is possible; and from those knowledge, a plenty of applications arise as emerging technologies. In this direction, the present contribution focus on a complete description of quantum capacit…
We have calculated the thermoelectric conductivity tensor $\varepsilon_{ij}$ and the thermal conductivity tensor $λ_{ij}$ of a unidirectional lateral superlattice (ULSL) ($i,j = x,y$, with the $x$-axis aligned to the principal axis of the ULSL), %, given as the first- and the second-order moments, employing based on the asymptotic analytic formulas of the electrical conductivity tensor $σ_{ij}$ in the literature valid at low magnetic fields where large numbers of Landau levels are occupied. With the resulting analytic expressions, we clarify the conditions for the Mott formula (Wiedemann-Franz…
We present a study of entropy transport in Bi2Se3 at low temperatures and high magnetic fields. In the zero-temperature limit, the magnitude of the Seebeck coefficient quantitatively tracks the Fermi temperature of the 3D Fermi surface at Γ-point as the carrier concentration changes by two orders of magnitude (10$^{17}$ to 10$^{19}$cm$^{-3}$). In high magnetic fields, the Nernst response displays giant quantum oscillations indicating that this feature is not exclusive to compensated semi-metals. A comprehensive analysis of the Landau Level spectrum firmly establishes a large $g$-factor in this…
We have investigated the spatial distribution of the electron temperature generated in a two-dimensional electron system (2DES) subjected to a perpendicular magnetic field. We measure thermoelectric voltages between Ohmic contacts located at the end of the voltage-probe arms of a Hall bar fabricated from a GaAs/AlGaAs 2DES wafer, immersed in the mixing chamber of a dilution refrigerator held at 20 mK. Magneto-oscillations due to the Landau quantization are examined for the thermoelectric voltages between the contact pairs straddling the main bar (arrangement to measure the transverse component…
The low magnetic field diffusion thermopower of a high mobility GaAs-heterostructure has been measured directly on an electrostatically defined micron-scale Hall-bar structure at low temperature (T = 1.6 K) in the low magnetic field regime (B < 1.2 T) where delocalized quantum Hall states do not influence the measurements. The sample design allowed the determination of the field dependence of the thermopower both parallel and perpendicular to the temperature gradient, denoted respectively by Sxx (longitudinal thermopower) and Syx (Nernst-Ettinghausen coefficient). The experimental data show cl…
Weyl semimetals (WSMs) have Weyl nodes where conduction and valence bands meet in the absence of inversion or time-reversal symmetry (TRS), or both. Interesting phenomena are expected in WSMs such as the chiral magnetic effect, anomalous Hall conductivity or Nernst effect, and unique quantum oscillations. The TRS-broken WSM phase can be driven from a topological Dirac semimetal (DSM) by magnetic field or magnetic dopants, considering that DSMs have degenerate Weyl nodes stabilized by rotational symmetry, i.e. Dirac nodes. Here we develop a Wannier-function-based tight-binding (WF-TB) model to …
