arXiv subjects

J. P. Carbotte

Publications and source records attributed to J. P. Carbotte.

At least 19 recordsLinked to original sources

arXiv2023

Hexagonal warping on optical conductivity of surface states in Topological Insulator Bi_{2}Te_{3}

ARPES studies of the protected surface states in the Topological Insulator $% Bi_{2}Te_{3}$ have revealed the existence of an important hexagonal warping term in its electronic band structure. This term distorts the shape of the Dirac cone from a circle at low energies to a snowflake shape at higher energies. We show that this implies important modifications of the interband optical transitions which no longer provide a constant universal background as seen in graphene. Rather the conductivity shows a quasilinear increase with a slightly concave upward bending as energy is increased. Its slope increases with increasing magnitude of the hexagonal distortion as does the magnitude of the jump at the interband onset. The energy dependence of the density of states is also modified and deviates downward from linear with increasing energy.

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arXiv2023

Longitudinal and spin/valley Hall optical conductivity in single layer $MoS_{2}$

A monolayer of $MoS_{2}$ has a non-centrosymmetric crystal structure, with spin polarized bands. It is a two valley semiconductor with direct gap falling in the visible range of the electromagnetic spectrum. Its optical properties are of particular interest in relation to valleytronics and possible device applications. We study the longitudinal and the transverse Hall dynamical conductivity which is decomposed into charge, spin and valley contributions. Circular polarized light associated with each of the two valleys separately is considered and results are filtered according to spin polarization. Temperature can greatly change the spin admixture seen in the frequency window where they are not closely in balance.

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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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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

Effect of chiral anomaly on the circular dichroism and Hall angle in doped and tilted Weyl semimetals

From the Kubo formula for transport in a tilted Weyl semimetal we calculate the absorptive part of the dynamic conductivity for both right and left handed circular polarized light. These depend on the real part of the longitudinal conductivity and the imaginary part of the transverse (Hall) conductivity. We include the effect of the chiral anomaly which pumps charge from negative to positive chirality node when the usual ${\bs E}\cdot{\bs B}$ term is included in the electrodynamics and obtain analytic expressions. To calculate the Hall angle we further provide expressions for the imaginary part of longitudinal and real part of the transverse conductivity and compare results with and without the pumping term. We also consider the case of a non centro symmetric Weyl semimetal in which the chiral nodes are displaced in energy by an amount $\pm \mathcal{Q}_0$. This leads to modification in dichroism and Hall angle which parallel the pumping case.

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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

Doping and tilting on optics in noncentrosymmetric multi-Weyl semimetals

Weyl semimetal (WSM) feature tilted Dirac cones and can be type I or II depending on the magnitude of the tilt parameter ($C$). The boundary between the two types is at $C=1$ where the cones are tipped and there is a Lifshitz transition. The topological charge of a WSM is one. In multi-Weyl it can be two or more depending on the value of the winding number $J$. We calculate the absorptive part of the AC optical conductivity both along the tilt direction ($σ_{zz}$) and perpendicular to it ($σ_{xx}$) as a function of the tilt ($C$) and chemical potential ($μ$). For zero tilt there is a discontinuous rise in both $σ_{xx}$ and $σ_{zz}$ at photon energy $Ω=2μ$ followed by the usual linear in $Ω$ law for $σ_{xx}$ at $J=1,2$ and $σ_{zz}$ at $J=1$. For $J=2$ and $σ_{zz}$ the interband background is constant rather than linear in $Ω$. For type I there is a readjustment of optical spectral weight as the tilt is increased. The absorption starts from zero at $2μ/(1+C)$ and then rises in a quasilinear fashion till it merge with the usual undoped untilted interband background at $2μ/(1-C)$. The discontinuous rise at twice the chemical potential of the untilted case is lost. For type II the interband background of the undoped untilted case is never recovered. For noncentrosymmetric materials the energies of a pair of opposite chirality Weyl nodes become shifted by $\pm Q_{0}$ and this leads to two separate absorption edges corresponding to the effective chemical potential of each of the two nodes at $2(μ+χQ_{0})$ depending on chirality $χ=\pm$. We provide analytic expressions for the conductivity in this case which depend only on the ratio $Q_{0}/μ$ and tilt when plotted against $Ω/μ$. The signature of finite energy shift $Q_{0}$ is more pronounced for $σ_{zz}$ and $J=2$ than for the other cases.

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arXiv2018

Imaginary part of Hall conductivity in tilted doped Weyl semimetal with both broken time reversal and inversion symmetry

We consider a Weyl semimetal (WSM) with finite doping and tilt within a continuum model Hamiltonian with both broken time reversal and inversion symmetry. We calculate the absorptive part of the anomalous AC Hall conductivity as a function of photon energy ($Ω$) for both type I and type II Weyl semimetal. For a given Weyl node, changing the sign of its chirality or of its tilt changes the sign of its contribution to the absorptive Hall conductivity with no change in magnitude. For a noncentrosymmetric system we find that there are ranges of photon energies for which only the positive or only the negative chirality node contributes to the imaginary (absorptive) part of the Hall conductivity. There are also other photon energies where both chirality contribute and there can be other ranges of $Ω$ where there is no absorption associated with the AC Hall conductivity in type I and regions where it is instead constant for type II. We comment on implications for the absorption of circular polarized light.

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arXiv2018

Anomalous DC Hall response in noncentrosymmetric tilted Weyl semimetals

Weyl nodes come in pairs of opposite chirality. For broken time reversal symmetry (TR) they are displaced in momentum space by $\bf{Q}$ and the anomalous DC Hall conductivity $σ_{xy}$ is proportional to $\bf{Q}$ at charge neutrality. For finite doping there are additive corrections to $σ_{xy}$ which depend on the chemical potential as well as on the tilt ($C$) of the Dirac cones and on their relative orientation. If inversion symmetry (I) is also broken the Weyl nodes are shifted in energy by an amount $Q_{0}$. This introduces further changes in $σ_{xy}$ and we provide simple analytic formulas for these modifications for both type I ($C<1$) and type II ($C>1$, overtilted) Weyl. For type I when the Weyl nodes have equal magnitude but oppositely directed tilts, the correction to $σ_{xy}$ is proportional to the chemical potential $μ$ and completely independent of the energy shift $Q_{0}$. When instead the tilts are parallel, the correction is linear in $Q_{0}$ and $μ$ drops out. For type II the corrections involve both $μ$ and $Q_{0}$, are nonlinear and also involve a momentum cut off. We discuss the implied changes to the Nernst coefficient and to the thermal Hall effect of a finite $Q_{0}$.

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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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arXiv2017

Optical response in Weyl semimetal in model with gapped Dirac phase

We study the optical properties of Weyl semimetal (WSM) in a model which features, in addition to the usual term describing isolated Dirac cones proportional to the Fermi velocity $v_{F}$, a gap term $m$ and a Zeeman spin-splitting term $b$ with broken time reversal symmetry. Transport is treated within Kubo formalism and particular attention is payed to the modifications that result from a finite $m$ and $b$. We consider how these modifications change when a finite residual scattering rate $Γ$ is included. For $Γ<m$ the A.C. conductivity as a function of photon energy $Ω$ continues to display the two quasilinear energy regions of the clean limit for $Ω$ below the onset of the second electronic band which is gapped at ($ m+b $). For $Γ$ of the order $m$ little trace of two distinct linear energy scales remain and the optical response has evolved towards that for $m=b=0$. Although some quantitative differences remain there are no qualitative differences. The magnitude of the D.C. conductivity $σ^{DC}(T=0)$ at zero temperature ($T=0$) and chemical potential ($μ=0$) is altered. While it remains proportional to $Γ$ it becomes inversely dependent on an effective Fermi velocity out of the Weyl nodes equal to $v_{F}^\ast=v_{F}\sqrt{b^2-m^2}/b$ which decreases strongly as the phase boundary between Weyl semimetal and gapped Dirac phase (GDSM) is approached at $b=m$. The leading term in the approach to $σ^{DC}(T=0)$ for finite $T/Γ$, $μ/Γ$ and $Ω/Γ$ is found to be quadratic. The coefficient of these corrections tracks closely the $b/m$ dependence of the $μ=T=Ω=0$ limit with differences largest near to the WSM-GDSM boundary.

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arXiv2017

Transport and optics at the node in a nodal loop semimetal

We use a Kubo formalism to calculate both A.C. conductivity and D.C. transport properties of a dirty nodal loop semimetal. The optical conductivity as a function of photon energy $Ω$, exhibits an extended flat background $σ^{BG}$ as in graphene provided the scattering rate $Γ$ is small as compared to the radius of the nodal ring $b$ (in energy units). Modifications to the constant background arise for $Ω\le Γ$ and the minimum D.C. conductivity $σ^{DC} $ which is approached as $Ω^2/Γ^2$ as $Ω\rightarrow0$, is found to be proportional to $\frac{\sqrt{Γ^2+b^2}}{v_{F}}$ with $v_{F}$ the Fermi velocity. For $b=0$ we recover the known three-dimensional point node Dirac result $σ^{DC}\sim \fracΓ{v_{F}}$ while for $b>Γ$, $σ^{DC}$ becomes independent of $Γ$ (universal) and the ratio $\frac{σ^{DC}}{σ^{BG}}=\frac{8}{π^2}$ where all reference to material parameters has dropped out. As $b$ is reduced and becomes of the order $Γ$, the flat background is lost as the optical response evolves towards that of a three-dimensional point node Dirac semimetal which is linear in $Ω$ for the clean limit. For finite $Γ$ there are modifications from linearity in the photon region $Ω\le Γ$. When the chemical potential $μ$ (temperature $T$) is nonzero the D.C. conductivity increases as $μ^2/Γ^2$($T^2/Γ^2$) for $μ/Γ$ $(T/Γ)\le 1$. For larger values of $μ>Γ$ away from the nodal region the conductivity shows a Drude like contribution about $Ω\approxeq 0$ which is followed by a dip in the Pauli blocked region $Ω\le 2μ$ after which it increases to merge with the flat background (two-dimensional graphene like) for $μ< b$ and to the quasilinear (three-dimensional point node Dirac) law for $μ> b$.

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arXiv2017

Absorption of circular polarized light in tilted Type-I and II Weyl semimetals

We calculate the A.C. optical response to circularly polarized light of a Weyl semimetal (WSM) with varying amounts of tilt of the Dirac cones. Both type-I and II (overtilted) WSM are considered in a continuum model with broken time reversal (TR) symmetry. The Weyl nodes appear in pairs of equal energies but of opposite momentum and chirality. For type-I the response of a particular node to right (RHP) and left (LHP) hand polarized light are distinct only in a limited range of photon energy $Ω$, $\frac{2}{1+C_{2}/v}<\fracΩμ<\frac{2}{1-C_{2}/v}$ with $μ$ the chemical potential and $C_{2}$ the tilt associated with the positive chirality node assuming the two nodes are oppositely tilted. For the over tilted case (type-II) the same lower bound applies but there is no upper bound. If the tilt is reversed the RHP and LHP response are also reversed. We present corresponding results for the Hall angle.

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arXiv2017

Optical Properties of Bogoliubov Quasiparticles

We calculated the optical conductivity $σ(T,Ω)$ of a gas of Bogoliubov quasiparticles (BQP) from their Green's function and the Kubo formula. We compare with corresponding normal state (N) and superconducting state (SC) results. The superconducting case includes the dynamic response of the condensate through additional contributions to the Kubo formula involving the Gor'kov anomalous Green's function. The differences in the optical scattering rate are largest just above the optical gap and become progressively smaller as the photon energy is increased or the temperature is raised. Our results are compared with those obtained using a recently advocated phenomenological procedure for eliminating the effect of the condensate [1]. The $δ$-function contribution at zero photon energy, proportional to the superfluid density, is dropped in the real part of the conductivity $[σ_1(T,Ω)]$ and its Kramers-Kronig transform is subtracted from the imaginary part $σ_2(T,Ω)$. This results in deviations from our BQP and superconducting state optical scattering rates even in the region where these have merged and are, in addition, close to the normal state result.

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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

Interband optical conductivity of the [001]-oriented Dirac semimetal Cd3As2

We measured the optical reflectivity of [001]-oriented $n$-doped Cd$_{3}$As$_{2}$ in a broad frequency range (50 - 22000 cm$^{-1}$) for temperatures from 10 to 300 K. The optical conductivity, $σ(ω) = σ_{1}(ω) + {\rm i}σ_{2}(ω)$, is isotropic within the (001) plane; its real part follows a power law, $σ_{1}(ω) \propto ω^{1.65}$, in a large interval from 2000 to 8000 cm$^{-1}$. This behavior is caused by interband transitions between two Dirac bands, which are effectively described by a sublinear dispersion relation, $E(k) \propto \lvert k \rvert ^{0.6}$. The momentum-averaged Fermi velocity of the carriers in these bands is energy dependent and ranges from $1.2 \times 10^{5}$ to $3 \times 10^{5}$ m/s, depending on the distance from the Dirac points. We detect a gaplike feature in $σ_{1}(ω)$ and associate it with the Fermi level positioned around $100$ meV above the Dirac points.

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arXiv2016

Optical Conductivity of Weyl Semimetals and Signatures of the Gapped Semimetal Phase Transition

The interband optical response of a three-dimensional Dirac cone is linear in photon energy ($Ω$). Here, we study the evolution of the interband response within a model Hamiltonian which contains Dirac, Weyl and gapped semimetal phases. In the pure Dirac case, a single linear dependence is observed, while in the Weyl phase, we find two quasilinear regions with different slopes. These regions are also distinct from the large-$Ω$ dependence. As the boundary between the Weyl (WSM) and gapped phases is approached, the slope of the low-$Ω$ response increases, while the photon-energy range over which it applies decreases. At the phase boundary, a square root behaviour is obtained which is followed by a gapped response in the gapped semimetal phase. The density of states parallels these behaviours with the linear law replaced by quadratic behaviour in the WSM phase and the square root dependence at the phase boundary changed to $|ω|^{3/2}$. The optical spectral weight under the intraband (Drude) response at low temperature ($T$) and/or small chemical potential ($μ$) is found to change from $T^2$ ($μ^2$) in the WSM phase to $T^{3/2}$ ($|μ|^{3/2}$) at the phase boundary.

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