Electronic structural critique of interesting thermal and optical properties of C 17 Ge germagraphene
DDOI:
Electronic structural critique of interesting thermaland optical properties of C Ge germagraphene.
Sujoy Datta, a , b Debnarayan Jana a DATE:
In this communication, we report a theoretical attempt to understand the involvement of electronicstructure in determination of optical and thermal properties of C Ge germagraphene, a buckledtwo dimensional material. The structure is found to be a direct bandgap semiconductor with lowcarrier effective mass. Our study has revealed the effect of spin-orbit coupling on the band struc-ture and in appearance of spin Hall current in the material. A selectively high blue to ultravioletlight absorption and a refractive index comparable to flint glass open up the possible applicabilityof this material for optical devices. From electronic structural point of view, we investigate thereason behind its moderately high Seebeck coefficient and power factor comparable to traditionalthermoelectric materials. Besides its narrow bandgap, relatively smaller work function of C Ge( . eV ) than graphene ( . eV ) and germanene ( . eV ) assures more easily removal ofelectron from the surface. This material is turned out to be an excellent alternative for futuristicsemiconductor application from optical to thermal device regime. The discovery of graphene and related compounds revolution-ized modern semiconductor industry. Subsequently, other 2D ma-terials like silicene , black-phosphorene , and borophene weresynthesized experimentally.Though graphene exhibits extraordinary thermal, mechanicaland electrical properties, its zero band-gap imposes a severe con-straint on its practical applicability. As a result, opening up theband-gap of graphene and other planar zero-gap materials hasbeen considered to be the top-most priority in semiconductor en-gineering. It has been seen earlier that introducing defects ingraphene or graphene-nanotubes has a large influence on theirelectronic properties .Experimental work on such materials have been prolific. Theaim of this communication is to use the latest theoretical ap-proaches, together with what we shall argue to be better modifi-cations, and explain these experimental results. Dependable the-oretical predictions will then provide the experimentalist a muchbetter handle in choosing their desired materials out of a plethoraof possibilities.Silicon and Germanium being the same group IV materialas Carbon, are always the first choice for doping graphene.Several theoretical studies on silicon doped graphene, i.e.,siligraphene have been carried out by varying the rela-tive concentrations of Silicon and Carbon. Siligraphene with Si : C = showed superior sunlight absorbance . Wang etal. found theoretically that siligraphene had wonderful Li-ionstorage capacity . Dong et al. have reported that the stablesiligraphene structure (SiC ) exhibited a direct band-gap and su-perior light absorbance making it a promising donor material foroptical-devices . † corresponding author: [email protected] a Department of Physics, University of Calcutta, Kolkata 700009, India. b Department of Physics, Lady Brabourne College, Kolkata 700017, India.
Theoretically, Ge doping was seen to tune the band-gap but these studies never explored the situation of buckled struc-ture. As Ge is much heavier atom than C, doping with Ge is sup-posed to result in off-the-plain buckling and it is never easy todope graphene by such a heavier Ge atom. Very recently Tripathi et al. successfully implanted Ge on graphene and found a buck-led structure when a Ge atom replaces a C atom. However, whenGe replaces a C-C bond, it can be accommodated in the plane. Hu et al. found buckled C Ge to be stable and reported lithium ionabsorption in this buckled structure . However, for functional-ization of the germagraphene, electronic structural study shouldbe done more extensively. Here, in this systematic study, we tryto explore the electronic, optical and thermal properties of thegermagraphene structure. The basic calculations are carried out in plane wave based tech-niques used in the Quantum Espresso (QE) code . Slab ge-ometry for two dimensional system is simulated by introduc-ing 12 Å vacuum on either side of the sheet. The structure isenergetically optimized first using variable cell structural relax-ation technique. Projected augmented wave (PAW) basis of theQuantum Espresso (QE) is utilised using Perdew-Burke-Ernzerhof(PBE) exchange-correlation potentials . Charge-densities andenergies for each calculation were converged to 10 − Ry. withthe maximum force of 0.001 Ry./atom. × × k-point meshesare used and force convergence and pressure thresholds are setas 0.0001 Ry./au and 0.0000 Kbar, respectively. The self consis-tent field (SCF) calculation on the relaxed structure is done using × × dense k-point mesh for PBE and × × k and q pointmeshes for HSE calculation.For calculation of spin Hall conductivity (SHC) and trans-port properties we have used Wannier90 package for construct-ing the maximally localized Wannier functions (MLWF). For con-struction of MLWF relatively coarse k-ponts as applied in SCF1 a r X i v : . [ c ond - m a t . m t r l - s c i ] M a r alculation is enough. However, for SHC and transport calcu-lations much dense k-points is needed . That is why utilisingthe Wannier functional way is always beneficial . A fine sampleof × × and × × k-points are used for SHC andthermal property calculations, respectively. Wannier interpolationtechnique is also applied for the HSE band plotting.BoltzWann module of Wannier90 is a powerful tool for calcu-lating transport properties of material at moderate computationalcost . The high accuracy in the Brillouin zone integrals can beachieved through Wannierization. In thermal properties calcu-lations we have used adaptive interband smearing and constantrelaxation-time of f s .For optical property predictions, we calculate the complex di-electric tensor using random phase approximation (RPA). ε αβ ( ω ) = + π e Ω N k m ∑ n , n (cid:48) ∑ k (cid:104) u k , n (cid:48) | ˆ p α | u k , n (cid:105)(cid:104) u k , n | ˆ p † β | u k , n (cid:48) (cid:105) ( E k , n (cid:48) − E k , n ) (cid:20) f ( E k , n ) E k , n (cid:48) − E k , n + ( ¯ h ω + i ¯ h Γ ) + f ( E k , n ) E k , n (cid:48) − E k , n − ( ¯ h ω + i ¯ h Γ ) (cid:21) (1)Here, Γ is the inter-smearing term tending to zero. Sinceno excited-state can have infinite lifetime, we have introducedsmall positive Γ in order to produce an intrinsic broadening toall excited states. The imaginary part of the dielectric function ε ( i ) αβ has been calculated first and the real part ε ( r ) αβ is found us-ing the Kramers-Kronig relation. Optical-conductivity, refractiveindex and absorption-coefficients were calculated using real andimaginary parts of dielectric functions. Dielectric tensor: ε αβ = ε ( r ) αβ + i ε ( i ) αβ (2)Optical Conductivity: Re [ σ αβ ( ω )] = ω π ε ( i ) αβ ( ω ) (3)Complex Refractive Index: µ αα = n + αα + in − αα (4)Absorption Coefficient: A αα ( ω ) = ω n − αα ( ω ) c (5)where, n ± αα ( ω ) = (cid:115) | ε αα ( ω ) | ± ε ( r ) αα ( ω ) A 3 × Ge structureis shown as shaded region in Fig. 1.The geometrically relaxed structure shows buckling. Our cal-culations complement the experimental finding of buckled ger-magraphene structure as well as the theoretical result of Hu. et al. who also showed the dynamical stability of the structure.Being a heavier atom, Ge tends to distort the planar graphenestructure more than that of siligraphene structures . The in-plane lattice constant is 7.541 Å and the Ge atom at 0.917 Å off- Fig. 1 × Ge. a and b are the primitive vec-tors. The shaded region in grey shows the primitive unit cell and theshaded region in pink is used for elastic properties calculation. Optimizedbond lengths labelled by 1, 2, 3 are 1.411Å, 1.466Å, 1.450 Å, while theC-Ge bond-length is 1.863Å. C-Ge-C, C-C-Ge and C-C-C bond anglesare 97.834 o , 115.987 o and 122.968 o . the-plane. The separation between layers is kept at 12 Å to nullifythe intra-layer interaction, prerequisite for 2D calculations. Wecan clearly visualize a rhombic formation of Ge atoms connectedby blue lines in Fig.1. C-Ge bond length is 1.863 Å, whereas,the C-C bonds denoted by 1, 2, 3 have lengths of 1.411, 1.466,1.450 Å respectively. So, all the hexagonal rings with only carbonatoms are not regular-hexagons after Ge doping. C-Ge-C, C-C-Geand C-C-C bond angles are 97.834 o , 115.987 o and 122.968 o , re-spectively. The buckled structure suggests a deviation from thepure sp hybridization which is a signature of planar structure. Elastic constants:
In Fig. 1 the shaded region in pink (dashed line) is used asthe cell for the energy of state (EOS) calculation. Uniaxial strainis produced along either in direction (x) or in direction (y). Forbiaxial stress condition, same percentage of strain is producedalong both direction. The total energy is fitted using the equa-tion : u = C δ x + C δ y + C δ x δ y + C δ xy (6)where, u is the the strain energy per unit area of two dimensionalstructure. In standard Vigot notation δ x and δ y are the uniaxial Fig. 2
Strain energy with respect to uniaxial (blue-dashed) and biaxial(brown-solid) strain. ig. 3 (left)The band structures and density of states for C Ge using PBE. The p z projected partial DOS of Ge and C are shown in blue and red,whereas, the total DOS is in grey. The Γ − Γ direct band gap is . eV between HOMO originating from C (in red) and LUMO from Ge (in blue). (right)Charge density plots for the states near Fermi energy. in plane strain and δ xy is the sheering strain. As the structure issymmetrical along x and y, so, C = C .In Fig.2 the EOS plot is depicted. The elastic constantsare found from the fitting curve as C = . GPa-nm and C = . GPa-nm. These satisfy the Born criteria for mechan-ical stability of 2D materials, C × C (cid:105) C . The in plane Young’smodulus as derived from the formula Y = C − C C is 353.949 GPa-nm which is higher than that of graphene (335 GPa-nm) andabout three times of MoS (123 GPa-nm) . The Poission’s ratiodefined as υ = C C is 0.043. The band structure and densities of states (DOS) for C Ge areplotted in Fig.3 . The valence band maxima (VBM) is taken asthe reference zero of energy axis. From band structure plot itis evident that C Ge is a direct band-gap semiconductor. Theestimated band-gap using PBE is . eV at Γ point.Near the Fermi energy (E F ), there is one band approachingfrom below, the highest occupied (HO) band. In conduction bandregion, there are two lower lying bands, one is almost flat, thelowest unoccupied (LU) and another is sharp dipping (LU+1). Inthe vicinity of Γ , HO and LU + bands show almost linear dis-persion if moved along Γ ( , , ) to K ( / , / , ) as well as along Γ ( , , ) to M ( / , , ) . So, similar E-k relation is found for thesetwo bands, both along k b and k b , where b and b are the re-ciprocal lattice vectors in k x k y plane for hexagonal system. Suchdispersion isotropy is also present in LU but with much smallercurvature which will affect in the mobility of carriers. However,the dispersive nature of HO − and LU + bands along Γ − K and Γ − M paths are not the same. A HO and HO − crossover is seenat about the midway of Γ − M . At Γ there is a band degeneracy inconduction band as LU and LU + bands overlap.The total density of state (DOS/atom/state) is plotted in grey.Approaching E F from the occupied states below, the total DOS(TDOS) shows a gradual fall, whereas, there is a sharp peak at thelowest unoccupied state. To understand the origin of the bands,we plot the p z projected DOS (pDOS) of C and Ge in red andblue, respectively. The mixing of Ge and C states are evidentfrom pDOS plots. The sharp pDOS peaks of both Ge and C just near CBM suggests the dominant existence of Ge-p z state in LU.The flatness of LU has given rise to this sharp peak, i.e., higherdensity of states. In valence band (VB) region, the Ge energystates are more available near the E F than at lower energy. Exceptthe sharp peak in LU, the C pDOS rises gradually in both higherand lower energy direction with respect to E F . So, leaving thepeak, this nature is a reflection of linear dispersive systems, smallDOS around E F . This nature of carbon pDOS and the dominantexistence of Ge states in the vicinity of Fermi energy, both in CBand VB confirm the role of Ge in opening the gap between the HOand LU.The logical conclusion on the origin of the bands near Fermienergy finds a stronger base from the charge density plots in Fig.3. We have plotted the charge densities for HO, LU as well asHO − and LU + states. These plots are helpful to identify ac-tive sites in C Ge. The HO and LU states dominated by Ge- p z states whereas, HO − and LU + states are composed of C-Gebonding states with smaller contribution from Ge. As none of theHO, HO − , LU, LU + sates are localized, the efficiency of photo-absorption is expected. The prevalent existence of Ge- p chargedensities in HO and LU reassert the role of Ge in band-gap open-ing. Carrier Mobility:
Due to the degeneracy of bands at CBM, one flat and a sharpband, it would be interesting to calculate the carrier mobility. Mo-bility is the defining factor of the dynamical activity of the chargecareers in any semiconductor. Higher mobility means speedycharge transfer and low energy loss, yielding better performanceand durability of an electronic device.Mobility is inversely proportional to the effective mass (m ∗ )of carrier hole or electron. From the band-structure, m ∗ can becalculated as m ∗ = ¯ h / d Edk near the band-edges, for electron atCBM and at VBM for hole.The calculated hole effective mass is m ∗ h = − . m where m is electron rest mass. The electron effective mass has two dif-ferent values due to the band degeneracy at CBM (at Γ ). The elec-tron effective mass originating from LU, m ∗ e ( L ) = . m is less han th that of m ∗ for the LU + band, m ∗ e ( L + ) = . m .Such low light-electron and hole effective mass indicate very highcarrier mobility in germagraphene, which is much higher thanmany 2D materials and reported so far .As we find C Ge as a narrow gap semiconductor, so, its ap-plicability should be compared with the range of II-VI narrow gapsemiconductors. For example, both the light-electron and holeeffective masses of this germagraphene are almost th to thoseof HgCdTe ( m ∗ ≈ . m ) . So, C Ge readily can be consid-ered as a better option for the optoelectronic devices like infrareddetector, photovoltaic devices etc., those now built by alloyingsemimetals with II-VI semiconductors . Effect of Intrinsic Spin-Orbit Coupling:
For those systems where space inversion symmetry holds,time reversal symmetry would produce double degeneracy of thebands. Graphene, like many other two-dimensional (2D) mate-rials, preserves inversion symmetry. So, even in the presence ofintrinsic spin-orbit coupling (SOC), the Bloch states remain dou-bly spin degenerate at K where the Dirac point originally formed,however, a gap opens up . Application of transverse externalelectric field can lift the spin degeneracy by the Rashba effect.Adatoms having stronger spin-orbit coupling can be doped toopen up the band-gap of graphene . The system we are dealinghere can be otherwise viewed as a periodically doped grapheneand having a band gap as well, even in the absence of SOC. So,in presence of intrinsic SOC it is interesting to explore the bandstructure of C Ge.
Fig. 4
Band structure of C Ge with spin-orbit coupling (SOC). Differentregions of interest are enlarged.
Band structure of C Ge in presence of intrinsic SOC is shownin Fig.4. Though the plot looks almost same as that without SOC(Fig.3), a close look at several region reveals the difference asrepresented in the insets. The Γ point, being the centre of recip-rocal space, is the most symmetrical point preserving both timereversal and inversion symmetries. So, no spin-orbit spiting isexpected at Γ and the regions denoted by 1 and 2 endorse that.However, the band degeneracy of LU and LU + is now removedat Γ point by a small value of . meV (Fig.4-1). This has an effecton the effective mass of electron as removal of band degeneracynow confirms this value m ∗ e ( L ) = . m .The HO − band has considerable C and Ge state mixing andas Ge atom is much heavier than C, spin-orbit coupling effect should be more evident in that band. In the vicinity of Γ point, thesplitting is observable and at the Brilouin zone edge K ( / , / , ) the splitting is as large as . meV (Fig.4-4). The degeneratebands in occupied region, below − . eV , are now separated by . meV as shown in sub-figure (2).It was never expected that this C Ge structure would displaylarge SOC effect, still, the band structure exhibits ample SOCfootprint in removing band degeneracies. The spin degeneracyremoval, however small it be, can generate spin Hall current indevices and we focus on that in the next section.
Spin Hall Effect:
Intrinsic SOC is an interesting feature of materials as it gen-erates an electric field within such substance, even in absence ofany external perturbation. Due to the removal of band degener-acy in intrinsic spin-orbit coupled materials, this transverse elec-tric field triggers spin current. This phenomena is known as spinHall effect. For different semiconductors and metals this featureis prolific . The ab-initio calculations have produced excellentresults but these type of direct calculations are excessively ex-pensive as almost number of k-pints are needed for simplesemiconducting systems like Ge, GaAs, etc. . So, for larger sys-tems with lesser symmetries direct ab-initio calculations becomealmost impossible. Qiao et al. successfully found an alternativeway of calculation through Wannier interpolation and could re-produce the ab-initio results effectively and further applied onlayered materials .In presence of electric field along β direction, the response ofspin ( σ γ ) current along the α direction can be calculated usinglinear response theory and the Kubo formula for spin Hall con-ductivity (SHC) tensor is given by: σ γαβ = − e ¯ hV N k ∑ k , n f n k Ω γ n , αβ ( k ) (7)where, V is the volume of the unit cell, N k is the number of k-points in Brilouin zone and Ω γ n , αβ ( k ) is the Berry curvature liketerm expressed as: Ω γ n , αβ ( k ) = − ¯ h ∑ m (cid:54) = n Im [ (cid:104) n k |{ ˆ σ γ , ˆ v α }| m k (cid:105)(cid:104) m k | ˆ v β | n k (cid:105) ]( E n , k − E m , k ) − ( ¯ h ω + i η ) (8)The term in curly bracket is coming from the spin current, ˆ j γα = { ˆ σ γ , ˆ v α } and ˆ v α is the velocity operator. The matrix elementsassociated with these two quantities are calculated using Wannierinterpolation techniques . In dc-SHC calculation, setting ω = blows up the term in Eq.3.2. To avoid that inter-band smearinghas been introduced through an adaptive technique .For two dimensional systems, only two components of Hallconductivity tensor is important, σ zxy and σ zyx . For symmetricalsystems, like we are dealing here, σ zxy = − σ zyx . Though for dc-SHC, the imaginary part is always zero, for ac-SHC a variation ofboth real and imaginary parts of σ zxy is expected.In Fig. 5, we have plotted the variation of real and imag-inary parts of ac spin Hall conductivity as a function of en- ig. 5 Variation of real and imaginary parts of ac-SHC σ zxy with energy. ergy. As ω → , Im [ σ zxy ] → which is obvious for systems withband gap. The real part of SHC approaches a constant value of − . ( ¯ h / e )( Ω m ) − for zero energy, whereas, the imaginary parttends to zero. A sharp peak is located around the bandgap value . eV and after that they are almost flat till . eV . Thenthere are random variations though not much sign change is ob-served. The real part changes sign ∼ . eV where a small peakof imaginary part appears. This peak is not as sharp as observedat . eV . The highest value of Re [ σ zxy ] is . ( ¯ h / e )( Ω m ) − and Im [ σ zxy ] is . ( ¯ h / e )( Ω m ) − . These values are not as largeas metallic or semimetallic systems , where SOC effect is muchprominent. The C- p and Ge- p electrons are not expected toexhibit large spin Hall effect, however, this result is interestingenough to establish the effect of small doping of same group ma-terial in pristine graphene by displaying measurable SHC. Optical properties of materials are directly related to their elec-tronic structure. While for metallic systems both inter-band andintra-band transitions play role in optical properties, in semicon-ductors the optical property is mostly governed by inter-bandjump. The absorption of a photon by any semiconducting sys-tem leads to the transition of an electron from VB to CB. So, theminimum amount of energy required to form the charge carri-ers should be greater than or equal to its band-gap. The absorp-tion of a photon with energy similar to the bandgap produces ahole in the valence band and an electron in the conduction band.Photons with energy greater than bandgap initiates transition tohigher level of conduction band. So, for optical property calcula-tions always plenty of empty bands in CB are taken into consider-ation. In VB region, the number of electrons participating in theinter-band transitions can be calculated using the expression ofeffective electron number (n e f f ) given as: n e f f ( E m ) = mNe h (cid:90) E m E . ε ( i ) αα ( E ) dE ; (9)Here, m, e and N are electron mass, charge and density. Asaturation of the n e f f with photon energy reflects the unavailabil-ity of electrons lower than that value for production of hole byjump towards CB. Our calculated value has shown that electronslying below eV from VBM is not contributing in the opticaltransitions. So, all the plots are focused in the − eV range.In Fig.6, we have depicted the real and imaginary parts of di- agonal element of dielectric tensor ε ( r / i ) αβ and electron energy lossspectrum (EELS) as a function of energy. There are two signif-icantly different polarization of electric fields to be considered,one is parallel to the plane of the 2D material (denoted by ε (cid:107) )and perpendicular to that plane (denoted by ε ⊥ ). The two or-thogonal directions (say, x and y) in the plane are symmetric, so, ε (cid:107) = ε xx = ε yy .Both the real and imaginary parts of parallel dielectric con-stants ε ( r / i ) (cid:107) show rapid fluctuations in lower energy region. Inaddition, ε ( i ) (cid:107) changes sign four times in this energy region. ε ( r / i ) (cid:107) becomes almost flat only at high energy region ( > eV ). Fig. 6
Variation of real and imaginary part of diagonal element of dielec-tric tensor ε ( r / i ) αβ for (a) parallel polarization, (b) perpendicular polarizationof electric field; (c) Electron energy loss spectrum (EELS). The plasmafrequency is shown by circular pointer. The real part of dielectric constant crosses the zero axis fromnegative to positive at five different energies, two below eV andthree in the − eV range. The plasma frequency is defined asthe frequency when ε ( r ) shows positive slope while crossing thezero axis and ε ( i ) (cid:104) . The fourth zero crossing energy . eV sat-isfies both the conditions and identified as the plasma frequency(encircled in Fig.6) for parallelly polarized electric field. This canbe readily verified by the peak in EELS. In most of the semicon-ductors the plasma frequency lies deep into ultraviolet region andgermagraphene is not an exception.We find relatively flat ε ( r / i ) ⊥ than its perpendicular counter-part. The values are smaller as well. The imaginary part shows asharp peak around ∼ . eV . Unlike ε ( r ) (cid:107) , ε ( r ) ⊥ becomes negativeto positive only once, at . eV which is the plasma frequencyfor perpendicular polarization. ig. 7 Graphical (a) absorption spectra, (b) refractive index, and (c) op-tical conductivity as a function of photon energy for parallel and perpen-dicular polarizations of electric field.
Electron energy loss spectrum (EELS) is a properties whichcan be verified through experiments. In Fig.6(c) the plot of EELSis presented. While the EELS for parallel polarization takes a bellshape centring around . eV , the perpendicular counterpart ex-hibit significant growth only after eV . There is a crossoverof these two at ∼ . eV and thereafter, the perpendicular partis dominant. The sharp peak of EELS ⊥ is at the correspondingplasma frequency . eV as found from dielectric constant plot. Absorption coefficient, refractive index and optical conductiv-ity:
Transition of electron from VB to CB is a result of absorptionof photonic energy. So, absorption coefficient preserves the sig-nature of not only band gap but also the full characteristics ofbands. Near the Fermi energy, such correspondence can be read-ily established.In Fig. 7(a) & (b), we plot absorption coefficient as a functionof photon energy. To study the one to one correspondence be-tween the electronic structure and photonic absorption we haveto look back to Fig.3. The very first small peak in the absorptioncoefficient (A) for parallel polarization is at ∼ . eV correspondsto the band gap of the material. The second peak ∼ . eV is be-tween VBM and the small hump of LU band in Γ − K direction,the third peak around . eV is between the flat region of HO − around Γ and CBM. In visible region, the absorption coefficient isshowing a gradual increase first, then at around . eV starts torise stiffly. The jump from around . eV to . eV is attributed from HO to LU + transitions as well as from the HO − to LU + transitions around the midway of Γ − K and Γ − M path. This re-gion corresponds to blue-violet colour region. Blue to ultravioletlight absorption is particularly useful for safety of different opticaldevices through coating by those high absorbing materials.The absorption coefficient for perpendicular polarization ( A ⊥ )is negligible with respect to the parallel counterpart till eV . Afterthat it is dominant as observed for dielectric constant.In the application for coating on devices another importantoptical property is the refractive index. The real refractive in-dex of germagraphene structure is depicted as function of photonenergy in Fig. 7(b). In the visible range ( . − . eV ), the refrac-tive index for parallel polarized electric field varies from . to . having an average ≈ . . For comparison, we quote the re-fractive index of flint glass as . − . , so, this germagraphenematerial has similar average refractive index to glass for parallelelectric field. For perpendicular field, the variation is in . − . range.There is a direct connection of optical absorption spectrawith the imaginary part of the refractive index through the Eq.5.Therefore, the nature of refractive index plots are expected toreplicate the peaks and valleys of absorption spectra.The optical conductivity in Fig. 7(c) shows similar characterto absorption coefficient which is logically evident. As more elec-trons move to CB through photon absorption, the carrier concen-tration increases resulting in higher optical conductivity. The sig-nature of semiconductors, a gap at the beginning of optical con-ductivity is clearly visible. The first peak is around the band-gapvalue of . eV . The peak is sharper than that of the absorp-tion coefficient because the absorption of one photon creates twocarriers, one electron and one hole. In materials electrons carry both charge and heat. The temper-ature gradient of any material produces an electric field whichopposes the natural diffusion of electrons. So, the electronic con-ductivity becomes dependent on temperature. The opposing elec-tric field produces the so called Seebeck voltage. The Seebeckcoefficient and electronic conductivity depends on this Seebeckvoltage. As a gradient dependent quantity, the Seebeck voltageis directly correlated with the grade of asymmetry of electronicdistribution around Fermi energy.A narrow band gap is one of the primary requirement to iden-tify thermoelectric materials for further extensive study, high car-rier mobility (low effective mass) is another. Low bandgap helpsto pump electrons for conduction even with the help of small ther-mal excitation with tendency of increasing the electrical conduc-tivity. Besides, direct nature of gap is an additive benefit. Further,low effective mass ensures the high mobility of the carriers, thus,increases the electrical conductivity. However, the decisive fac-tors for searching a promising thermoelectric material consist ofa variety of different interconnected properties. From electronicpoint of view, electrical conductivity ( σ ), Seebeck coefficient (S)or the power factor ( σ S ) are the key parameters to dictate thethermoelectric performance. ig. 8 Thermal properties of C Ge: (a) Electrical conductivity ( σ ), (b)Seebeck Coefficient (S) and (c) power factor ( σ S ) as a function of chem-ical potential. In Fig.8 we have plotted electrical conductivity, Seebeck coef-ficient and power factor at different temperature (100, 300, 500,700 K) for C Ge as functions of chemical potential ( µ ) as mea-sured from E F . The zero of chemical potential represents thepristine condition and positive (negative) value indicates the elec-tron (hole) doping. In pristine condition, the conductivity riseswith temperature which is a sign of the semiconducting nature ofC Ge.We have seen in Fig.3 that the DOS has much higher peakaround CBM than around VBM. This means more electron con-centration in CB edge than hole concentration at VB edge. Hence,asymmetric plot about the zero chemical potential is justified andmore electrical conductivity at n-side is a result of higher carrierconcentration near CBM.The electrical conductivity for K has sharper variationswith comparison to the other temperature plots. Conductivity atroom temperature ( K ) rises gradually, attains its highest value . × ( Ω m ) − at µ = . and then shows a dip at ∼ µ = . before staring to rise again. Similar variation is seen for K ,however at K this dip is absent.At room temperature (300 K) the highest value of Seebeckcoefficient is − µ V / K at µ = . eV for n-type doping and µ V / K at µ = − . eV for p-type doping. These values aremuch higher than the value for pristine graphene, graphene basedheterostructure , and comparable to Bi Te . A comparisonwith other thermoelectric materials is presented in Table. 1 .High Seebeck effect can be mapped with the rapid variance of DOS. High energy gradient of DOS ( dDOSdE ) is always beneficial forthermoelectric performance . Under doping, if the Fermi levelmoves to high DOS peak region, a large asymmetry of thermo-electric power is developed between the hot electrons of energyhigher than E F and cold electrons of energy below Fermi level.Higher anisotropy means higher S, as we can verify for our sam-ple, where the sharp peaks in CB (near CBM) is responsible forhighest peaks in Seebeck coefficients. Lower anisotropy of DOS inVB is the reason of lower peak values in µ < region than µ > .As temperature rises, the increase in Joule heating brings downthe value of Seebeck coefficient. So, in the figure, we see lessSeebeck coefficient for higher temperature.The power factor of any material is defined as PF = σ S . Tounderstand the electronic structural effect on power factor wehave to look back to the carrier effective masses. Higher effec-tive mass is favourable for Seebeck coefficient, whereas it has anegative effect on electrical conductivity. In PF, the second or-der dependence on S overwhelms the unfavourable effect on σ .C Ge has an unusual semiconductor with a very flat LU and verysharp HO band. This in turn initiates the large anisotropy be-tween hole and electron effective masses. So, the power factorfollows that anisotropy and for positive µ the PF peaks are signif-icantly higher than negative µ . Even the peak for K outplaysthe peak of K for n-type doping. For µ < , higher temperaturepossesses higher maxima.The power factor at K is maximum ( . mW m − K − ) near µ = . eV due to significant increase of the electrical conduc-tivity at the high electron doping range. The value of power factoris in between two orthodox thermoelectric materials, Bi Te andSb Te and better than some other proposed materials .Moreover, the excellent value of S establishes germagraphene asa potential candidate for thermoelectric application. The work-function ( Φ ) of any material is defined as the energyrequired to remove an electron from the surface of the mate-rial. The highest energy of electron in material is its Fermi en-ergy ( E F ), and the energy at the point just outside the surfaceis originally the vacuum energy ( E vacuum ). Work function is thedifference of these two: Φ = E vacuum − E F (10)In intrinsic semiconductors E F is the mid point energy of VBMand CBM, so, in absolute scale, the orientation of the bands canbe found accordingly.It is well-known that the PBE functional is not adequateenough to determine the work function correctly, whereas,Heyd − Scuseria − Ernzerhof (HSE) screened hybrid func-tional has shown great success for semiconductors. The failureof Local Density Approximation (LDA) and the PBE based Gen-eralized Gradient Approximation (PBE-GGA) in determining theband gap exactly is also evident . HSE functional can solve thatproblem in most of the cases.We have done the HSE calculation for the same optimizedgermagraphene unit cell and same energy and charge density cut- ffective Mass ( m )Material Hole Electron Seebeck Coefficient( µ V / K ) Power Factor( mW m − K − ) Work Function(eV)C Ge -0.00292 heavy:0.59528light:0.00291 309 1.56 4.361 Graphene – – 31 ∼ Germanene – – – – 4.682
GeAs -0.20 0.15 400 0.55 – MoS -0.56 0.55 500-550 WSe -0.42 0.46 450-500 Bi Te -0.024 0.178 323.0 ∼ – Sb Te -0.054 0.045 115.6 ∼ Table 1
Effective mass, Seebeck coefficient, power factor and work function of C Ge at K compared with other 2D (graphene, germanene, MoS ,GeAs , WSe ) and 3D (Bi Te , Sb Te ) materials. The results of present work are in shaded cells. off. The k and q point grids are taken as × × and OptimizedNorm-Conserving Vanderbilt (ONCV) pseudopotentials are used.The HSE calculated band-gap is . eV which is ≈ higherthan PBE calculated value ( . eV ). Fig. 9 (left) HSE calculated work function and band-gap of C Ge.(right) Electrostatic potential plot for graphene (red), germanene (blue)and C Ge (black) for calculation of corresponding work functions ( Φ ) In Fig.9 we plot the electrostatic potentials of germagraphenein comparison with graphene and germanene. The vacuum levelis different for different systems and dependent on the softwarepackages, so, for comparison this vacuum level is always stan-dardized to the zero energy. Using Eq.3.5 the work functionsof these three materials are calculated. Our predicted value ofgraphene monolayer ( . ) is at par with experimental result . eV and better than the PBE estimated value ≈ . eV .The Φ of germanene is . eV similar to the already reportedvalue of . eV . The HSE calculated work function of C Geis . eV which is lower than both of these. Lower work func-tion is beneficial for removal of electron from material surface.The Φ is of particular interest in the field of photo-catalytic de-vices through heterojunction formation. Being a material of lowerwork function than graphene, it may be considered as an alterna-tive to those heterojunction devices where graphene is used now. We show that the stable buckled C Ge structure is a direct bandgap semiconductor having a PBE calculated band gap of . eV .The band degeneracy at the conduction band minima is left whenspin orbit coupling is considered. This spin dependent electronic property leaves its footprint on spin hall conduction with highestvalue of conductivity as . ( ¯ h / e )( Ω m ) − . Though this value isnot large but effective enough to showcase the effect of anisotropyinduced by Ge on the spin dependent electronic property. Wealso find an exceptionally small hole and light-electron effectivemass. These are preferable for optoelectronic devices like in-frared detector, photovoltaic devices etc. where high mobility is aperquisite criteria.The optical and thermal properties of C Ge are also been ex-plored. This material has exhibited significant blue to ultravioletlight absorption. So, we predict if devices are coated by this ma-terial then they can avoid the harm from lower UV region. Therefractive index comparable to flint glass is an additional advan-tage of this material in optoelectronic world.The narrow band gap of germagraphene makes it a poten-tial candidate for thermoelectric applications. C Ge has shownrelatively larger value of Seebeck coefficient and power factorcomparable to traditional thermoelectric materials such as Bi Te and Sb Te at room temperature. Thus, C Ge affirms brilliantthermoelectric performance. In a sum up, we can conclude thatthe efficiency of germagraphene in futuristic applications is quitepromising.
Acknowledgement
The authors acknowledge the contribution of
Late Prof. Abhi-jit Mookerjee in forming the problem at the initial stage. SDwants to thank
Dr. Prashant Singh of Ames Lab., USA and
Dr.Chhanda Basu Chaudhuri of Lady Brabourne College, India fortheir suggestions.
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