The Intrinsic Resistivity Of Several Kinds Of Metals

The intrinsic resistivity arisen from electron-phonon(e-ph)scattering is the key physical quantity to determine the electronic transport property of metals.Generally speaking,resistivity of metals is mainly contributed by the intrinsic resistivity at room temperature.Therefore,it has long been an important topic for both theoretical and experimental studies on intrinsic resistivity.Experiments show that the intrinsic resistivity p is proportional to temperature T at sufficiently high temperature.The onset of the linear p-T relationship is determined by two kinds of characteristic temperatures,i.e.the so-called Debye temperature(TD)and Bloch-Griineisen temperature(TBG).s According to conventional transport theory,TD can be understood as the temperature corresponding to the highest phonon frequency.So all phonon are fully activated at temperatures higher than TD with the average phonon number being about T/TD.When considering the e-ph scattering rate is proportional to the average phonon number,the linear ρ-T relation holds at temperatures higher or equal to TD.However,for some metallic materials with very small spherical Fermi surface,it is possible that the linear ρ-T relation emerges at a much lower temperature,i.e.TBG.This is because that the Fermi surface is so small that above TBG,the electrons around Fermi surface can be scattered to any other state near Fermi surface,including the back-scattering processes that contribute remarkably to the intrinsic resistivity.It means that all possible scattering processes are activated and the rising of temperature only means that the scattering rate increases linearly with the the increase of phonon number.Hence,the intrinsic resistivity is proportional to temperature.According to the analysis above,we can conclude that TBG depends on the size of Ferm surface.For metal with small Fermi surface,TBG is much smaller than TDNevertheless,many metals have much more complicated Fermi surfaces than spherical surfaces.Their Fermi surfaces are composed of many separated branches and these branches are from different valleys of the same bands or even different bands.In some cases,the Fermi surface is not closed in Brillouin zone and the so-called Umklapp processes are involved in electron-phonon scattering.In a word,the size and the shape of Fermi surface can not be described by a single parameter,such as TBG.As a result,Bloch-Griineisen temperature is not capable to predict the ρ-T relationship of these materials.In this case,the existence of linearρ-T relation and the onset linear temperature of metals with complicated Fermi surface is an important theoretical issue to be solved.The e-ph interaction not only determines the intrinsic resistivity of metals,but also other physical properties,such as the carrier mobility of semiconductors,the conventional superconductive transition temperature of metals,property of polaron,phonon involved optical process,and the correction of band structure,etc.The critical problem to investigate the e-ph interaction and related properties is to determine the analytical forms of e-ph interaction matrix elements.Based on this requirement,researchers proposed many analytical form of e-ph interaction matrix elements with empirical parameters,such as acoustic deformation poten-tial model,optical deformation potential model,piezoelectric potential model,and Frohlich polar-optical-phonon scattering model,etc.Nevertheless,the scope of these models is limited.Consequently,quantitative studies on specific mate-rials may not adopt these models.People cannot calculate the electron-phonon interaction matrix elements on the level of first-principles method until the density functional theory and density functional perturbation theory are put forward,but the cost is enormous.It is still a formidable work to perform an first-principles calculations on the intrinsic resistivity of a metallic material since it requires a fine Brillouin zone sampling for calculating e-ph scattering around the Fermi surface with high precision.It is indeed an unaffordable computation burden to perform an e-ph interaction investigation entirely on the level of the first-principles calcu-lations.In this context,we designed a viable solution to investigate the intrinsic resistivity of metals via the electron-phonon Wannier interpolation so that we can avoid the heavy burden of first-principles calculation and calculate the intrinsic resistivity of metals accurately.The brief introduction of our method is as follows.At first,we perform first-principles calculation on the electronic band structure,phononic dispersion and the electron-phonon interaction matrix elements in an uniform coarse grid of Brillouin zone.Then the Wannier interpolation method is employed to calculated these quantities in a very dense grid.The cost of interpolation is fairly cheap.Finally,the resistivity of metals are calculated via the Ziman formula based on the semi-classical Boltzmann transport theory.To deal with the integration with Dirac δ function in the Ziman formula,we generalized the tetrahedral integral method and wrote programs to calculate the intrinsic resistivity.By the method above,we choose borophene,beryllium,and trilayer graphene to investigate the characteristic of their intrinsic resistivity.Firstly,we investigated the anisotropic electrical resistivity limited by e-ph scattering of two two-dimension borophene allotropes,known as β12 and x3.Then we found the intrinsic resistivity of the two borophene allotropes shows notable anisotropy as the structure of them are anisotropy.For the case of β12,the resistiv-ity along y direction is 28%larger than that along x direction and such anisotropy depends on carrier doping sensitively.For the case of x3,the resistivity along y direction is 120%larger than x direction,while the anisotropy does not vary with respect to carrier doping prominently.When we calculated the temperature de-pendence of resistivity,we found that the intrinsic resistivity of the two structures is proportional to temperature at the high-temperature limit.However,at the low-temperature limit,the T4-law of intrinsic resistivity predicted by the Bloch-Gruneisen theory for 2D metals holds true only forβ12 but breaks down for x3.We found that two optical phonon modes,called ZOl and ZO2,along with an acoustic phonon mode,ZA,all of which describe out-plane atomic vibrations,play crucial roles in determining the intrinsic resistivity in the two materials.Besides,the inter-band scattering is always more important than the intra-band scattering inβ12;and in x3 Umklapp processes become the nontrivial scattering mechanism to limit the intrinsic resistivity at low-temperature.But both inter-band scattering and Umklapp processes are beyond the application of the Bloch-Gruneisen theo-ry.Therefore,we conclude that Bloch-Gruneisen theory is inapplicable to explain the temperature dependence of the intrinsic resistivity of the two borophenes and the conclusion drawn in a recent theoretical work based on the Bloch-Gruneisen theory is not appropriate.Then we investigated the e-ph scattering limited electrical resistivity of beryl-lium,a topological nodal-line semimetal,hoping to find the mechanism dominating the electronic transport property and the influence of topological states to its in-trinsic resistivity.First of all,our numerical results of the intrinsic resistivity of beryllium agree quantitatively to experimental data in a large temperature range and the intrinsic resistivity of Be starts to show linear temperature dependence at a very low critical temperature(200K).As a three dimension material,Be has a much complicated Fermi surface.Around each joint region between the elec-tron and hole pockets,the Fermi surface forms a pair of vertical facets(parallel to c-axis).Then,nesting effect between these Fermi surface segments can be re-alized by electron-phonon scattering.Such a Fermi surface nesting effect plays the dominant role to the intrinsic resistivity.In addition,it is also the underlying mechanism for linear temperature dependence of the intrinsic resistivity.In con-trast to the aforementioned Fermi surface nesting effect,the contribution of the topological nontrivial states near the nodal line to the intrinsic resistivity is less important.On the one hand,it is because that only a few of such states appear in the vicinity of Fermi surface due to the sizable dispersion of the nodal line.On the other hand,the back-scattering process between the nodal-line states are restrained so that these states are much less influential.In the end,we performed a calculation on the intrinsic resistivity of ABA and ABC,two trilayer graphene with different stacking sequence,to investigate the influence of layer stacking sequence on the intrinsics resistivity.The electronic band structures of the two trilayer graphene are distinctly different around the Fermi energy,while the phonon dispersions are almost the same.We calculated the resistivity of them with varied temperatures,finding that ABC structure has a much larger resistivity than ABA structure,which is due to the large difference of their density of states at the Fermi Energy.Besides,the resistivity of ABC struc-ture is proportional to the temperature above 500K,consistent with the general rule of resistivity of metals at high temperature limit,while the resistivity of A-BA structure does not obey this rule.When doped with electrons or holes,the resistivity of the two materials decreases rapidly.However,there are some differ-ence between their variation tendencies.The resistivity of ABA structure is kept constant after decreasing along with the increasing of doping concentration,while The resistivity of ABC structure vibrates before being constant.This facts can also be explained by the differences in the band structures.Finally,we analysed the contribution of the scattering mechanism to the resistivity of the two trilayer graphene and discovered that they are all dominated by the out-plane vibration phonon modes,and the intra-band scattering process is much more important than the inter-band scattering.