Bloch-Gruneisen Regime Research Articles

Phonon drag thermopower and energy loss rate in single and bilayer graphene due to piezoelectric surface acoustic phonons in Bloch-Gruneisen regime

In this report, we have both numerically and analytically calculated the phonon drag thermopower, Spz and, energy loss rate, Fpz due to piezoelectric surface acoustic phonons present in the underlying piezoelectric substrate at a distance, d apart from single and bilayer graphene as a function of carrier's temperature, concentration and phonon mean free path. The numerical results are then reduced to analytical results in the Bloch-Gruneisen regime and the corresponding power dependency of the quantities described above are obtained. We have also included the effect of Thomas-Fermi and Random Phase Approximation screening into account in these calculations. The obtained unscreened results are further reduced in much simpler form for a particular case having no distance between the graphene samples and the underlying substrate or at larger values of distances. The temperature dependency in Spz and Fpz for both SLG and BLG systems is found to be one order less than the in-plane acoustic phonons dependency, which is directly related to the matrix element of electron-in plane acoustic phonon interaction having a linear dependence on the phonon wave vector, q. Furthermore, a comparison of these calculations is made between the in-plane and piezoelectric surface acoustic phonon modes. One finds a crossover between these phonons modes depicting these are important in providing effective scattering channels for energy relaxation in typical experimental situations at different temperatures.

Electron single flexural phonon relaxation, energy loss and thermopower in single and bilayer graphene in the Bloch–Gruneisen regime

The flexural phonons serve as one of the important modes of interaction in graphene that can inhibit carrier mobility. For the estimation of scattering due to flexural phonons a two-phonon scattering process had been in place, as due to symmetry constraints out-of-plane deformations modulate electron hopping only in the second order. But recently it has been shown that electrostatic gating can break the planar mirror symmetry and activate single flexural phonon scattering processes (Gunst et al 2017 Phys. Rev. Lett. 118 046601). Motivated by this we perform single flexural phonon mechanism based analytical and numerical calculations of the electron phonon relaxation rate, energy loss rate and thermopower in single and bilayer graphene and obtain the power exponents of these quantities in the Bloch Gruneisen regime using the non-equilibrium Boltzmann transport equation. We find that the scattering by flexural phonons substantially changes the temperature dependencies from that observed due to in-plane phonons but the carrier concentration dependencies remain the same as of the in-plane phonons for all the three investigated quantities.

Chirality effect on electron phonon relaxation, energy loss, and thermopower in single and bilayer graphene in BG regime

We investigate the energy dependent electron-phonon relaxation rate, energy loss rate, and phonon drag thermopower in single layer graphene (SLG) and bilayer graphene (BLG) under the Bloch-Gruneisen (BG) regime through coupling to acoustic phonons interacting via the Deformation potential in the Boltzmann transport equation approach. We find that the consideration of the chiral nature of electrons alters the temperature dependencies in two-dimensional structures of SLG and BLG from that shown by other conventional 2DEG system. Our investigations indicate that the BG analytical results are valid for temperatures far below the BG limit (∼TBG/4) which is in conformity with a recent experimental investigation for SLG [C. B. McKitterick et al., Phys. Rev. B 93, 075410 (2016)]. For temperatures above this renewed limit (∼TBG/4), there is observed a suppression in energy loss rate and thermo power in SLG, but enhancement is observed in relaxation rate and thermopower in BLG, while a suppression in the energy loss rate is observed in BLG. This strong nonmonotonic temperature dependence in SLG has also been experimentally observed within the BG limit [Q. Ma et al., Phys. Rev. Lett. 112, 247401 (2014)].

+Cerenkov emission of terahertz acoustic phonons generated electrically from monolayers of transition metal dichalcogenides

We have studied theoretically the Cerenkov acoustic phonon emission from monolayers of transition metal dichalcogenides (TMDs) driven by an electric field. The features of emitted phonon spectrum P(ωq,θ) are investigated as a function of the phonon frequency ωq, electron concentration ns, electric field E, and phonon emission angle θ. Numerical evaluations are made for n-MoS2. Acoustic phonons in the terahertz (THz) frequency region are found to be generated when the Cerenkov emission condition is satisfied. Frequency dependence of P(ωq,θ) shows a maximum at a frequency ωqm, position which shifts to larger frequency for larger E and ns. We predict ωqm/ns1/2 and ωqm/E1/4 to be nearly constant. The angular dependence of P(ωq,θ) and the phonon intensity P(θ) show emission to be maximum along the direction of electric field, i.e., for θ = 0. Both P(ωq,θ) and P(θ) are found to be larger for larger E. In the Bloch-Gruneisen regime P(ωq,θ) and P(θ) ∼ ns−3/2 and P(θ) ∼ T4. The results are compared with those in the bulk and conventional two-dimensional semiconductors and graphene. It is suggested that monolayers of TMDs can find potential applications as a new source of electrical generation of acoustic phonons in the THz frequency region.

Phonon-drag thermopower in 3D Dirac semimetals

A theory of low-temperature phonon-drag thermopower Sg in three-dimensional (3D) Dirac semimetals has been developed considering screened electron–phonon deformation potential coupling. Numerical investigations of Sg, in the boundary scattering regime for phonons, are made in 3D Dirac semimetal Cd3As2, as a function of temperature T and electron concentration ne. Sg is found to increase rapidly for about T < 1 K and nearly levels off for higher T. It is also seen that Sg increases (decreases) with decreasing ne at lower (higher) T (<2 K). A screening effect is found to be very significant, strongly affecting T and ne dependence for about <1 K and becoming negligible at higher temperature. In the Bloch–Gruneisen (BG) regime the power laws Sg ~ T8 (T4) and Sg ~ with (without) screening are obtained. These laws with respect to T and ne are, respectively, characteristics of 3D phonons and Dirac 3D electrons. Comparison with diffusion thermopower Sd shows that Sg dominates (and is much greater than) Sd for about T > 0.2 K. Herring’s law Sgμp ~ T −1, relating phonon limited mobility μp and Sg in the BG regime, is shown to be valid in 3D Dirac semimetals. The results obtained here are compared with those in 3D semiconductors, low-dimensional semiconductor heterojunctions and graphene. We conclude that ne-dependent measurements, rather than T-dependent ones, provide a clearer signature of the 3D Dirac semimetal phase.

Electron–phonon interaction in a spin–orbit coupled quantum wire with a gap

Interaction between electron and acoustic phonon in an in-plane magnetic field induced gapped quantum wire with Rashba spin–orbit interaction is studied. We calculate acoustic phonon limited resistivity (ρ) and phonon-drag thermopower (Sg) due to two well-known mechanisms of electron–phonon interaction, namely, deformation potential (DP) and piezoelectric (PE) scattering. In the so-called Bloch–Gruneisen temperature limit, both ρ and Sg depend on temperature (T) in a power law fashion, i.e., ρ or . For resistivity, takes the value 5 and 3 due to DP and PE scattering, respectively. On the other hand, is 4 and 2 due to DP and PE scattering, respectively, for phonon-drag thermopower. Additionally, we find numerically that depends on Rashba parameter (α) and electron density (n). The dependence of on α becomes more prominent at lower density. We also study the variations of ρ and Sg with carrier density in the Bloch–Gruneisen regime. Through a numerical analysis, a similar power law dependence ρ or is established in which the effective exponent undergoes a smooth transition from a low density behavior to a high density behavior. At a higher density regime, matches excellently with the value obtained from theoretical arguments. Approximate analytical expressions for both resistivity and phonon-drag thermopower in the Bloch–Gruneisen regime are given.

Phonon-drag magnetothermopower in Rashba spin-split two-dimensional electron systems

We study the phonon-drag contribution to the thermoelectric power in a quasi-two-dimensional electron system confined in GaAs/AlGaAs heterostructure in the presence of both Rashba spin–orbit interaction and perpendicular magnetic field at very low temperature. It is observed that the peaks in the phonon-drag thermopower split into two when the Rashba spin–orbit coupling constant is strong. This splitting is a direct consequence of the Rashba spin–orbit interaction. We show the dependence of phonon-drag thermopower on both magnetic field and temperature numerically. A power-law dependence of phonon-drag magnetothermopower on the temperature in the Bloch–Gruneisen regime is found. We also extract the exponent of the temperature dependence of phonon-drag thermopower for different parameters like electron density, magnetic field, and the spin–orbit coupling constant.

Acoustic phonon limited mobility in two-dimensional semiconductors: Deformation potential and piezoelectric scattering in monolayer MoS2from first principles

We theoretically study the acoustic phonon limited mobility in n-doped two-dimensional MoS2 for temperatures T < 100 K and high carrier densities using the Boltzmann equation and first-principles calculations of the acoustic electron-phonon (el-ph) interaction. In combination with a continuum elastic model, analytic expressions and the coupling strengths for the deformation potential and piezoelectric interactions are established. We furthermore show that the deformation potential interaction has contributions from both normal and umklapp processes and that the latter contribution is only weakly affected by carrier screening. Consequently, the calculated mobilities show a transition from a high-temperature \mu T^{-1} behavior to a stronger \mu T^{-4} behavior in the low-temperature Bloch-Gruneisen regime characteristic of unscreened deformation potential scattering. Intrinsic mobilities in excess of 10^5 cm^2 V^{-1} s^{-1} are predicted at T < 10 K and high carrier densities (n > 10^{11} cm^{-2}). At 100 K, the mobility does not exceed ~7 x 10^3 cm2 V^{-1} s^{-1}. Our findings provide new and important understanding of the acoustic el-ph interaction and its screening by free carriers, and is of high relevance for the understanding of acoustic phonon limited mobilities in general.

Scattering mechanisms and diffusion thermopower in a bilayer graphene

Diffusion thermopower Sd is studied along with the scattering mechanisms and mobility μ by the Boltzmann transport equation technique in a bilayer graphene (BLG). Particular emphasis is made with respect to the sensitivity of Sd and μ to the possible scattering due to acoustic phonons (ap) and surface polar phonons (spp), charged impurity (ci) and short range disorder (sd). In the Bloch–Gruneisen regime, μap due to acoustic phonons shows T−4 behavior, a characteristic of 2D phonons, but with a small bump attributing to the chiral property of electrons. μ due to ci and sd are found to be almost independent of T. Numerical results of Sd, due to each of these mechanisms and the resultant, are presented as a function of temperature T and electron concentration ns for both supported and suspended BLG. The resultant Sd and that due to ap, ci and sd are found to be ∼T up to 100K and vary sublinearly above this temperature. However, the range of linear T behavior will be larger for larger ns.Sd due to sd is found to be dominant. A qualitative agreement is found with the experimental data. Sd∼ns−1 over the range of interest considered. Sd in suspended BLG is found to be greater than that in the supported one.

Electron-phonon interaction in a quantum wire in the Bloch-Gruneisen regime

We investigate the one-dimensional electron gas (1DEG) interaction with three-dimensional (3D) acoustic phonons in a quantum wire (QWR) by calculating the temperature $T$ and electron concentration ${n}_{l}$ dependence of phonon drag thermopower ${S}^{g}$ and energy loss rate $P$ in the Bloch-Gruneisen regime taking account of static screening. Electron scattering by acoustic phonons through piezoelectric field and deformation potential is considered. At very low temperature, ${S}^{g}$ and $P$ are dominated by the contribution due to piezoelectric scattering. The contribution due to deformation potential becomes significant at relatively higher temperature. The power laws for $T$ and ${n}_{l}$ dependences of ${S}^{g}$ and $P$ are obtained. The screening affects weakly the power of $T$ and significantly the power of ${n}_{l}$. The interesting feature is that ${S}^{g}$ is reduced by a factor of ${({\ensuremath{\nu}}_{s}∕{\ensuremath{\nu}}_{F})}^{2}$, where ${\ensuremath{\nu}}_{s}$ is the sound velocity and ${\ensuremath{\nu}}_{F}$ is the Fermi velocity, compared to that in two-dimensional electron gas (2DEG). A qualitative comparison is made between the calculated and experimentally observed energy loss rate in etched InGaAs QWRs. Herring's law ${S}^{g}{\ensuremath{\mu}}_{p}\ensuremath{\sim}{T}^{\ensuremath{-}1}$ (${\ensuremath{\mu}}_{p}$ is the phonon limited mobility) is validated in QWR. Our results are compared with those in 2DEG.

Effect of the Pekar mechanism on the phonon-drag thermopower inp-typeSi∕Si1−xGexheterostructures

We examine the effect of the Pekar mechanism on the coupling of two-dimensional (2D) holes with 3D acoustic phonons in $p$-type $\mathrm{Si}∕{\mathrm{Si}}_{1\ensuremath{-}x}{\mathrm{Ge}}_{x}$ heterostructures by using the phonon-drag thermopower, ${S}^{g}$, as a probe. The origin of the Pekar coupling is due to the variation of the dielectric permittivity under strain in nanostructures characterized by strong confining electric fields. The hole-phonon interaction via the Pekar mechanism resembles that of piezoelectric coupling in polar materials and dominates over the deformation potential coupling at low temperatures. It is found that ${S}^{g}$ shows a ${T}^{4}$ temperature dependence at low $T$ (Bloch-Gruneisen regime) which is in good agreement with the experimental data reported recently by Possanzini et al. [Phys. Rev. B 69, 195306 (2004)]. We perform detailed calculations of ${S}^{g}$ by considering both a screened Pekar coupling and a screened deformation potential coupling in $p$-type $\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ structures in the temperature range $0.25lTl6\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. The value of the deformation potential constant is taken to be $4\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, in good agreement with previous studies. The strength of the Pekar coupling is determined by the magnitude of the surface electric field ${E}_{s}$. We treat ${E}_{s}$ as an adjustable parameter in our calculations and we find good agreement with all the experimental data of ${S}^{g}$, available at the moment, by using a value for ${E}_{s}$ of the order of ${10}^{9}\phantom{\rule{0.3em}{0ex}}\mathrm{V}∕\mathrm{m}$. The possibility for the existence of such a large surface field in $\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ is discussed.

Effect of acoustic-phonon confinement on the phonon-drag thermopower of a two-dimensional electron gas in a semiconductor thin film

We study theoretically the effect of acoustic-phonon confinement on the phonon-drag thermopower ${S}^{g}$ of a two-dimensional (2D) electron gas in a quantum well embedded in a semiconductor thin film. The screened interaction of 2D electrons via the deformation and piezoelectric potentials in the film with stress-free surfaces is considered. Analytical expressions are obtained for ${S}^{g}$ (due to flexural, dilational, and shear modes) in the Bloch-Gruneisen regime where the nature of confinement is pronounced. Interestingly, ${S}^{g}$ dependence on temperature $T$ is the same for both deformation and piezoelectric potential mechanisms, which may be attributed to the similar phonon wave vector dependence of their interaction matrix elements at very low temperatures. We find ${S}^{g}\ensuremath{\sim}{T}^{3}$ for flexural modes and ${T}^{5}$ for dilatational and shear modes. ${S}^{g}$ due to flexural modes, which have quadratic dispersion and high density of states, is found to be dominant over contribution from other modes. ${S}^{g}$ in thin films is suppressed due to a smaller phonon mean free path compared to bulk phonons. Herring's formula relating ${S}^{g}$ to phonon limited mobility ${\ensuremath{\mu}}_{p}$ is discussed. We find ${S}^{g}{\ensuremath{\mu}}_{p}\ensuremath{\sim}{T}^{\ensuremath{-}1/2}$ for flexural modes and ${T}^{\ensuremath{-}1}$ for dilatational and shear modes showing the invalidity of the Herring's formula for flexural modes.

Direct measurements of energy relaxation time of electrons in AlGaAs/GaAs heterostructures under quasi-equilibrium conditions

For the first time, results are presented of a direct measurement of the energy relaxation time τ ε of 2D electrons in an AlGaAs/GaAs heterojunction at T = 1 and 5–20 K. A weak temperature dependence of τ ε for the T > 4K range and a linear temperature dependence of the reciprocal of τ ε for T < 4K have been observed. The linear dependence τ ε −1 ≈ T in the Bloch-Gruneisen regime is direct evidence of the predominance of the piezo-electric mechanism of electron-phonon interaction in non-elastic electron scattering processes. The values of τ ε in this regime are in very good agreement with the results of the Karpus theory. At higher temperatures, where the deformation-potential scattering becomes noticeable, a substantial disagreement between the experimental data and the theoretical results is observed.