arXiv subjects

E. J. Nicol

Publications and source records attributed to E. J. Nicol.

At least 19 recordsLinked to original sources

arXiv2023

Optical conductivity of tilted higher pseudospin Dirac-Weyl cones

We investigate the finite-frequency optical response of systems described at low energies by Dirac-Weyl Hamiltonians with higher pseudospin $\mathcal{S}$ values. In particular, we examine the situation where a tilting term is applied in the Hamiltonian, which results in tilting of the Dirac electronic band structure. We calculate and discuss the optical conductivity for the cases $\mathcal{S}=1$, $3/2$, and $2$, in both two and three dimensions in order to demonstrate the expected signatures in the optical response. We examine both undertilted (type I) and overtilted (type II) as well as the critically-tilted case (type III). Along with the well-known case of $\mathcal{S} =1/2$, a pattern emerges for any $\mathcal{S}$. We note that in situations with multiple nested cones, such as happens for $\mathcal{S}>1$, the possibility of having one cone being type I while the other is type II allows for more rich variations in the optical signature, which we will label as type IV behavior. We also comment on the presence of optical sum rules in the presence of tilting. Finally, we discuss tilting in the $α$-T$_3$ model in two dimensions, which is a hybrid of the $\mathcal{S}=1/2$ (honeycomb lattice) and $\mathcal{S}=1$ (dice or T$_3$ lattice) model with a variable Berry's phase. We contrast this model's conductivity with that of $\mathcal{S}=3/2$ and $\mathcal{S}=2$ as the resultant optical response has some similarities, although there are clear distinguishing features between the these cases.

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arXiv2021

Infrared imaging of samples in ultra high pressure diamond anvil cells

We describe an experimental platform that generates infrared images of micrometer-sized samples in the high pressure region of a diamond anvil cell. Using a 2.3 micron laser as a source of radiation, the system will be particularly useful in identifying hydride superconductors which exhibit an anomalous temperature dependence of reflectivity in the 2.3 micron region. Our system shows an intensity stability within one percent when the sample temperature is swept from 100 K to 300 K. The spatial stability is of the order of a few micrometers in the same temperature range.

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arXiv2020

Optical properties of superconducting pressurized LaH$_{10}$

Recently superconductivity has been discovered at around 200~K in a hydrogen sulfide system and around 260~K in a lanthanum hydride system, both under pressures of about 200 GPa. These record-breaking transition temperatures bring within reach the long-term goal of obtaining room temperature superconductivity. We have used first-principle calculations based on density functional theory (DFT) along with Migdal-Eliashberg theory to investigate the electron-phonon mechanism for superconductivity in the $Fm\bar{3}m$ phase proposed for the LaH$_{10}$ superconductor. We show that the very high transition temperature $T_c$ results from a highly optimized electron-phonon interaction that favors coupling to high frequency hydrogen phonons. Various superconducting properties are calculated, such as the energy gap, the isotope effect, the specific heat jump at $T_c$, the thermodynamic critical field and the temperature-dependent penetration depth. However, our main emphasis is on the finite frequency optical properties, measurement of which may allow for an independent determination of $T_c$ and also a confirmation of the mechanism for superconductivity.

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arXiv2019

Spectroscopic signatures of phonons in high pressure superconducting hydrides

The discovery of superconductivity at 203K in SH$_3$ is an important step toward higher values of $T_c$. Predictions based on state-of-the-art DFT for the electronic structure, including one preceding experimental confirmation, showed the mechanism to be the electron-phonon interaction. This was confirmed in optical spectroscopy measurements. For photon energies between $\sim 450$ and 600 meV in SH$_3$, the reflectance in the superconducting state is below that in its normal state. This difference decreases as $T$ approaches $T_c$. Decreasing absorption with increasing $T$ is opposite to what is expected in ordinary metals. Such an anomalous behavior can be traced back to the energy dependence of the superconducting density of states which is highly peaked at the energy gap value $Δ$ but decays back to the constant normal state value as energy is increased, on a scale of a few $Δ$, or by increasing $T$ towards $T=T_c$. The process of phonon-assisted optical absorption is encoded with a knowledge of the $T$-dependence of $Δ$, the order parameter of the superconducting state. Should the energy of the phonon involved be very large, of order 200 meV or more, this process offers the possibility of observing the closing of the superconducting order parameter with $T$ at correspondingly very large energies. The very recent experimental observation of a $T_c\simeq 250$ K in LaH$_{10}$ has further heightened interest in the hydrides. We compare the relevant phonon structure seen in optics with related features in the real and imaginary part of the frequency dependent gap, quasiparticle density of states, reflectance, absorption, and optical scattering rate. The phonon structures all carry information on the $T_c$ value and the $T$-dependence of the order parameter, and can be used to confirm that the mechanism involved in superconductivity is the electron-phonon interaction.

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arXiv2019

Signatures of merging Dirac points in optics and transport

We consider the optical and transport properties in a model two-dimensional Hamiltonian which describes the merging of two Dirac points. At low energy, in the presence of an energy gap parameter $Δ$, there are two distinct Dirac points with linear dispersion, these are connected by a saddle point at higher energy. As $Δ$ goes to zero, the two Dirac points merge and the resulting dispersion exhibits semi-Dirac behaviour which is quadratic in the $x$-direction ("nonrelativistic") and linear the $y$-direction ("relativistic").In the clean limit for each direction ($x,y$) the contribution of the intraband and interband optical transitions are both given by universal functions of photon energy $Ω$ and chemical potential $μ$ normalized to the energy gap. We provide analytic formulas for both small and large $Ω/2Δ$ and $μ/Δ$ limits. These define, respectively, Dirac and semi-Dirac-like regions. For $Ω/2Δ$ and $μ/Δ$ of order one, there are deviations from these asymptotic behaviors. Considering optics and also transport, such as dc conductivity, thermal conductivity and the Lorenz number, such deviations provide signatures of the evolution from the Dirac to the semi-Dirac regime as the gap $Δ$ is varied.

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arXiv2019

Optical properties of a semi-Dirac material

Within a Kubo formalism, we calculate the absorptive part of the dynamic longitudinal conductivity $σ(Ω)$ of a 2D semi-Dirac material. In the clean limit, we provide separate analytic formulas for intraband (Drude) and interband contributions for $σ(Ω)$ in both the relativistic and nonrelativistic directions. At finite doping, in the relativistic direction, a sumrule holds between the increase in optical spectral weight in the Drude component and that lost in the interband optical transitions. For the nonrelativistic direction, no such sumrule applies. Results are also presented when an energy gap opens in the energy dispersion. Numerical results due to finite residual scattering are provided and analytic results for the dc limit are derived. Energy dependence and possible anisotropy in the impurity scattering rate is considered. Throughout, we provide comparison of our results for $\sqrt{σ_{xx}σ_{yy}}$ with the corresponding results for graphene. A generalization of the 2D Hamiltonian to include powers of higher order than quadratic (nonrelativistic) and linear (relativistic) is considered. We also discuss the modifications introduced when an additional flat band is included via a semi-Dirac version of the $α$-${\cal T}_3$ model, for which an $α$ parameter tunes between the 2D semi-Dirac (graphene-like) limit and the semi-Dirac version of the dice or ${\cal T}_3$ lattice.

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arXiv2018

Detecting superconductivity in the high pressure hydrides and metallic hydrogen from optical properties

We present a new technique for measuring the critical temperature Tc in the high pressure, high Tc electron-phonon-driven superconducting hydrides. This technique does not require connecting leads to the sample. In the multiphonon region of the absorption spectrum, the reflectance mirrors the temperature variation of the superconducting order parameter. For an appropriately chosen value of photon energy of order twice the gap plus 1.5 times the maximum phonon energy, the temperature dependence of the reflectance varies much more rapidly below T=Tc than above. It increases with increasing temperature in the superconducting state while it decreases in the normal state. Examining the temperature dependence of the reflectance at a fixed photon energy, there is a cusp at T=Tc which provides a measurement of the critical temperature. We discuss these issues within the context of the recently observed metallic phase of hydrogen.

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arXiv2016

Spectroscopy of H$_3$S: evidence of a new energy scale for superconductivity

The discovery of a superconducting phase in sulfur hydride under high pressure with a critical temperature above 200 K has provided a new impetus to the search for even higher $T_c$. Theory predicted and experiment confirmed that the phase involved is H$_3$S with Im-3m crystal structure. The observation of a sharp drop in resistance to zero at $T_c$, its downward shift with magnetic field and a Meissner effect confirm superconductivity but the mechanism involved remains to be determined. Here, we provide a first optical spectroscopy study of this new superconductor. Experimental results for the optical reflectivity of H$_3$S, under high pressure of 150 GPa, for several temperatures and over the range 60 to 600 meV of photon energies, are compared with theoretical calculations based on Eliashberg theory using DFT results for the electron-phonon spectral density $α^2$F($Ω$). Two significant features stand out: some remarkably strong infrared active phonons at $\approx$ 160 meV and a band with a depressed reflectance in the superconducting state in the region from 450 meV to 600 meV. In this energy range, as predicted by theory, H$_3$S is found to become a better reflector with increasing temperature. This temperature evolution is traced to superconductivity originating from the electron-phonon interaction. The shape, magnitude, and energy dependence of this band at 150 K agrees with our calculations. This provides strong evidence of a conventional mechanism. However, the unusually strong optical phonon suggests a contribution of electronic degrees of freedom.

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arXiv2016

Magnetic properties of the $α$-$T_3$ model: magneto-optical conductivity and the Hofstadter butterfly

The $α$-$T_3$ model extrapolates between the pseudospin $S=1/2$ honeycomb lattice of graphene and the pseudospin $S=1$ dice lattice via parameter $α$. We present calculations of the magnetic properties of this hybrid pseudospin model, namely the absorptive magneto-optical conductivity and the Hofstadter butterfly spectra. In the magneto-optics curves, signatures of the hybrid system corollary a doublet structure present in the peaks, resulting from differing Landau level energies in the $K$ and $K^{\prime}$ valleys. In the Hofstadter spectra, we detail the evolution of the Hofstadter butterfly as it changes its periodicity by a factor of three as we vary between the two limiting cases of the $α$-$T_3$ model.

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arXiv2016

Optical and Transport Properties in 3D Dirac and Weyl Semimetals

Within a Kubo formalism, we study dc transport and ac optical properties of 3D Dirac and Weyl semimetals. Emphasis is placed on the approach to charge neutrality and on the differences between Dirac and Weyl materials. At charge neutrality, the zero-temperature limit of the dc conductivity is not universal and also depends on the residual scattering model employed. However, the Lorenz number L retains its usual value L_0. With increasing temperature, the Wiedemann-Franz law is violated. At high temperatures, L exhibits a new plateau at a value dependent on the details of the scattering rate. Such details can also appear in the optical conductivity, both in the Drude response and interband background. In the clean limit, the interband background is linear in photon energy and always extrapolates to the origin. This background can be shifted to the right through the introduction of a massless gap. In this case, the extrapolation can cut the axis at a finite photon energy as is observed in some experiments. It is also of interest to differentiate between the two types of Weyl semimetals: those with broken time-reversal symmetry and those with broken spatial-inversion symmetry. We show that, while the former will follow the same behaviour as the 3D Dirac semimetals, for the zero magnetic field properties discussed here, the latter type will show a double step in the optical conductivity at finite doping and a single absorption edge at charge neutrality. The Drude conductivity is always finite in this case, even at charge neutrality.

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arXiv2016

Hall quantization and optical conductivity evolution with variable Berry phase in $α$-$T_3$ model

The $α$-$T_3$ model is characterized by a variable Berry phase that changes continuously from $π$ to $0$. We take advantage of this property to highlight the effects of this underlying geometrical phase on a number of physical quantities. The Hall quantization of the two limiting cases is dramatically different - a relativistic series is associated with a Berry phase of $π$ and a non-relativistic series is associated with the other limit. We study the quantization of the Hall plateaux as they continuously evolve from a relativistic to a non-relativistic regime. Additionally, we describe two physical quantities that retain knowledge of the Berry phase, in the absence of a motion-inducing magnetic field. The variable Berry phase of the $α$-$T_3$ model allows us to explicitly describe the Berry phase dependence of the dynamical longitudinal optical conductivity and of the angular scattering probability.

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arXiv2015

Comparison of pressurized sulfur hydride with conventional superconductors

A recent report that sulfur hydride under pressure is an electron-phonon superconductor with a Tc of 190 K has been met with much excitement although it is yet to be confirmed. Based on several electron-phonon spectral density functions already available from density functional theory, we find that the electron-phonon spectrum is near optimum for Tc with a particularly large value of its characteristic phonon energy omega_ln which is due to the small hydrogen mass. We find that the thermodynamic universal BCS ratios are near those for Pb and Nb3Sn. We suggest that optical measurements could be a useful tool to establish the existence and nature of the superconductivity in this system. Conventional superconductors are in the impurity-dominated dirty limit. By contrast sulfur hydride will be in the clean limit because of its large energy gap scale. The AC optical conductivity will display distinct and separate signatures of the superconducting gap in the low-energy impurity-dominated range of the optical spectrum and additional phonon structures at higher energies where the clean limit applies.

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arXiv2012

Emergence of Plasmaronic Structure in the Near Field Optical Response of Graphene

The finite momentum optical response $σ({\boldsymbol{q}},ω)$ of graphene can be probed with the innovative technique of infrared nanoscopy where mid-infrared radiation is confined by an atomic force microscope cantilever tip. In contrast to conventional $q\sim 0$ optical absorption which primarily involves Dirac fermions with momentum near the Fermi momentum, $k\sim k_F$, for finite $q$, $σ({\boldsymbol{q}},ω)$ has the potential to provide information on many body renormalizations and collective phenomena which have been found at small $k< k_F$ near the Dirac point in electron-doped graphene. For electron-electron interactions, the low energy excitation spectrum characterizing the incoherent part of the quasiparticle spectral function of Dirac electrons with $k\sim k_F$ consists of a flat, small amplitude background which scales with chemical potential and Fermi momentum. However, probing of the states with $k$ near $k=0$ will reveal plasmarons, a collective state of a charge carrier and a plasmon. These collective modes in graphene have recently been seen in angle-resolved photoemission spectroscopy and here we describe how they manifest in near field optics.

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arXiv2012

Magneto-optical conductivity in graphene including electron-phonon coupling

We show how coupling to an Einstein phonon $ω_E$ affects the absorption peaks seen in the optical conductivity of graphene under a magnetic field $B$. The energies and widths of the various lines are shifted, and additional peaks arise in the spectrum. Some of these peaks are Holstein sidebands, resulting from the transfer of spectral weight in each Landau level (LL) into phonon-assisted peaks in the spectral function. Other additional absorption peaks result from transitions involving split LLs, which occur when a LL falls sufficiently close to a peak in the self-energy. We establish the selection rules for the additional transitions and characterize the additional absorption peaks. For finite chemical potential, spectral weight is asymmetrically distributed about the Dirac point; we discuss how this causes an asymmetry in the transitions due to left- and right-handed circularly polarized light and therefore oscillatory behavior in the imaginary part of the off-diagonal Hall conductivity. We also find that the semiclassical cyclotron resonance region is renormalized by an effective-mass factor but is not directly affected by the additional transitions. Last, we discuss how the additional transitions can manifest in broadened, rather than split, absorption peaks due to large scattering rates seen in experiment.

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arXiv2011

Effect of electron-phonon coupling on energy and density of states renormalizations of dynamically screened graphene

Electronic screening strongly renormalizes the linear bands which occur near the Dirac crossing in graphene. The single bare Dirac crossing is split into two individual Dirac-like points, which are separated in energy but still at zero momentum relative to the K-point. A diamond-like structure occurs in between as a result of the formation of plasmarons. In this work we explore the combined effect of electron-electron and electron-phonon coupling on the renormalized energy dispersion, the spectral function and on the electronic density of states. We find that distinct signatures of the plasmaron structure are observable in the density of states with the split Dirac points presenting themselves as minima with quadratic dependence on energy about such points. By examining the slopes of both the density of states and the renormalized dispersion near the Fermi level, we illustrate how one can separate $k$-dependent and $ω$-dependent renormalizations and suggest how this might allow for the isolation of the renormalization due to the electron-phonon interaction from that of the electron-electron interaction.

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arXiv2011

Effects of electron-phonon coupling on Landau levels in graphene

We calculate the density of states (DOS) in graphene for electrons coupled to a phonon in an external magnetic field. We find that coupling to an Einstein mode of frequency $ω_E$ not only shifts and broadens the Landau levels (LLs), but radically alters the DOS by introducing a new set of peaks at energies $E_n\pmω_E$, where $E_n$ is the energy of the $n$th LL. If one of these new peaks lies sufficiently close to a LL, it causes the LL to split in two; if the system contains an energy gap, a LL may be split in three. The new peaks occur outside the interval $(-ω_E,ω_E)$, leaving the LLs in that interval largely unaffected. If the chemical potential is greater than the phonon frequency, the zeroth LL lies outside the interval and can be split, eliminating its association with a single Dirac point. We find that coupling to an extended phonon distribution such as a Lorentzian or Debye spectrum does not qualitatively alter these results.

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arXiv2011

Electron pockets and pseudogap Dirac point in underdoped cuprate superconductors

We consider a model of the pseudogap specifically designed to describe the underdoped cuprates and which exhibits particle-hole asymmetry. The presence of electron pockets, besides the usual hole pockets, leads to the appearance of new vectors beyond the usual so-called octet model in the joint density of states (JDOS), which underlies the analysis of Fourier-transform scanning tunneling spectroscopy (FT-STS) data. These new vectors are associated with distinct patterns of large amplitude in the JDOS and are expected to occur primarily at positive bias. Likewise a pseudogap Dirac point occurs at positive bias and this point can be determined either through FT-STS or through extrapolation of data from the autocorrelation function of angle-resolved photoemission spectroscopy.

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arXiv2011

Phonon structures in the electronic density of states of graphene in magnetic field

Unlike in ordinary metals, in graphene, phonon structure can be seen in the quasiparticle electronic density of states, because the latter varies on the scale of the phonon energy. In a magnetic field, quantization into Landau levels creates even more significant variations. We calculate the density of states incorporating electron-phonon coupling in this case and find that the coupling has pronounced new effects: shifting and broadening of Landau levels, creation of new peaks, and splitting of any Landau levels falling near one of the new peaks. Comparing our calculations with a recent experiment, we find evidence for a phonon with energy similar to but somewhat greater than the optical $E_{2g}$ mode and a coupling corresponding to a mass enhancement parameter $λ\simeq 0.07$.

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