Pressure-Induced Superconducting and Topological Phase Transitions in the RuX2 (X=P,As,Sb) Family Compounds

CHEN Qun ,WU Jue-fei ,WANG Xiao-meng ,DING Chi ,HUANG Tian-heng ,LU Qing ,SUN Jian ,3

1.National Laboratory of Solid State Microstructures,School of Physics,Nanjing University,Nanjing 210093,China

2.School of Physical Science and Technology,ShanghaiTech University,393 Middle Huaxia Road,Shanghai 201210,China

3.Collaborative Innovation Center of Advanced Microstructures,Nanjing University,Nanjing 210093,China

Abstract: RuSb2,as a sister material of thermoelectric material FeSb2,has been extensively studied focusing on the comparisons with FeSb2,however,the properties of RuSb2 under pressure have not been surveyed thoroughly yet.In this work,we studied the properties of RuSb2 under pressure and explored the similarities and differences of crystal and electronic structures between the Ru-pnictides partners RuP2 and RuAs2.Using the crystal structures search method together with first-principles calculations,we found that this family undergoes a serial of structural phase transitions: (I) For RuSb2: Pnnm →I4/mcm →I4/mmm;(II)for RuP2: Pnnm →I41/amd →Cmcm;(III) for RuAs2: Pnnm →P-62m.The newly found five phases are all energetically and dynamically stable at high-pressure and exhibit metallic properties.The four high pressure phases of RuSb2 and RuP2 can be quenched to zero pressure.The superconducting transition temperatures of I4/mcm and I4/mmm phases of RuSb2 and I41/amd and Cmcm phase of RuP2 are predicted to be approximately 7.3 K,10.9 K,13.0 K,and 10.1 K at 0 GPa,respectively.In addition,the I4/mcm and I4/mmm phases of RuSb2 and the I41/amd phase of RuP2 exhibit non-trivial topological properties.Our studies illustrate that pressure is an effective way to tune the structural,electronic,and superconducting behavior of the Ru-pnictides compounds.

Key words: high-pressure;crystal structures search method;phase transition;superconductor;topological materials

I.INTRODUCTION

In recent decades,transition-metal pnictides have drawn lots of attention due to their unique physical properties,such as high-performance thermoelectricity in antimonides[1–6],unconventional superconductivity in arsenides[7–10],and topological properties in transition-metal oxypnictide[11–17].Among them,the thermoelectric properties of transition-metal antimonides have been studied intensively.For example,FeSb2was observed to show extremely large thermopower and thermal conductivity at low temperatures[18–21],which was predicted to be originated from strong electron-electron correlation[18,22,23]and the phonon-drag effect[21].In particular,the correlation strength is expected to increase with an increasing hybridization gap[23].

RuSb2,as a sister material of FeSb2,has been widely studied in recent decades.It is a narrow-gap semiconductor with an estimated gap of 0.26 eV,which is larger than that of FeSb2.Although RuSb2is originally introduced as one of the thermoelectric candidates[24],the recent discovery indicates that its thermoelectric power is relatively small,which is one order of magnitude less than that of FeSb2at low temperatures.Nevertheless,the distinct Seebeck patterns of RuSb2and FeSb2draw lots of attention,with a particular Seebeck peak located at about 10 K,whereas the Seebeck coefficient of other thermoelectric materials usually decreases monotonically with increasing temperature[19].Moreover,they have a different magnetic response[18].Despite RuSb2being often used as a reference to study the multiple disparities with FeSb2,focusing on the magnetic properties[25,26],not much work has been done on RuSb2itself.

As a fundamental thermodynamic parameter,pressure can be employed to control various properties of materials.It can cause a structural phase transition without contaminating materials with impurities[27–33],or synthesize new materials with exciting properties[34–41].Pressure-driven structural phase transitions often reshape electronic structures accompanied by exotic physical properties.For instance,pressure can induce superconductivity transition in topological materials[42,43],and change the superconducting transition temperature of unconventional superconductors[44,45].The transition-metal pnictides also exhibit novel properties under high pressure.For instance,FeSb2undergoes a phase transition from insulator to metal[46];the thermoelectric properties of CoSb3can be enhanced by pressure[47].

Among thermoelectric materials,RuSb2has received little attention and deserves further studies.Given the similarities in crystal and electronic structures in other Ru-pnictides partners RuP2and RuAs2,they can also be used for comparison.In this work,we employed pressure conditions up to 120 GPa to systematically study RuSb2,RuP2,and RuAs2compounds by crystal structure search method and firstprinciples calculations.We found that the RuX2(X=P,As,Sb)family compounds undergo a series of structural phase transitions.For RuSb2:Pnnm →I4/mcm →I4/mmm;for RuP2:Pnnm →I41/amd →Cmcm;and for RuAs2:Pnnm →P-62m.These newly predicted phases are energetically and dynamically stable at high pressure and even at ambient pressure.Then we studied their structural,electronic,and superconducting properties.When the spin-orbit coupling (SOC)effects were included,theI4/mcm,I4/mmmphases of RuSb2,and theI41/amdphase of RuP2are identified to be topologically non-trivial.Moreover,theI4/mcm,I4/mmmphases of RuSb2and theI41/amd,Cmcmphases of RuP2exhibit superconductivity at zero pressure.

II.COMPUTATIONAL METHOD

To search for the most stable structures at a given pressure for RuSb2,RuAs2,and RuP2,we used the machine-learning accelerated random structure searching code called Magus (machine learning and graph theory assisted universal structure searcher)[48,49]combined with theab-initiocalculations at 20,40,80,100,and 120 GPa respectively.The Magus code has successfully predicted many high-pressure structures in different systems[41,50–54].The maximum number of atoms in the simulation cell is up to 18.Structural optimizations and electronic structure calculations were carried out using the projector augmented wave (PAW)[55]method as implemented in the Viennaab-initiosimulation package (VASP)[56].We employed the generalized gradient approximation(GGA) exchange-correlation density functional parameterized by Perdew-Burke-Ernzerhof (PBE)[57].Electronic localization functions(ELF)calculated by VASP were displayed by Visualization for Electronic Structural Analysis (VESTA)[58].The valence electrons of the pseudopotentials are 4d105s25p1,5s25p3,3s23p3,and 4s24p3for Ru,Sb,P,and As,respectively.Structures were relaxed with high accuracy with the planewave basis energy cutoffof 400 eV.Brillouin zone was meshed using the Monkhorst-Pack method with a spacing of 2π×0.25nm-1(the equivalent grid forPnnm,I4/mcm,I4mmm,I41/amd,Cmcm,P-62mphase at the highest pressure are 9×12×10,19×19×6,16×16×8,9×12×10,12×12×14,and 14×14×10,respectively).In our atomic optimization,the tolerance of atomic force was set to 0.01 eV/nm.Phonon modes and frequencies of the structures were calculated by the VASP code,combining the PHONOPY code[59]which the phonon calculations were carried out with the finite-displacement method.We applied 2×2×2 supercells to calculate all structure’s force constants.Quantum Espresso (QE) code[60]is used to calculate the electron-phonon coupling (EPC) constants andTcusing an energy cutoffof 120 Ry.The PBE exchangecorrelation functional and norm-conserving pseudopotentials are used.We adopted 8×8×8k-point mesh for charge self-consistent calculation,16×16×16k-point mesh for electron-phonon coupling (EPC) linewidth integration,and 4×4×4q-point mesh for the dynamical matrix ofI4/mcmandI4/mmmphase.The surface states were obtained by constructing the maximally localized Wannier functions(MLWFs)[61]and using the surface Green function approach[62],as implemented in the WANNIERTOOLS package[63].

III.RESULTS AND DISCUSSIONS

A.Predicted structures under high pressure

We performed structure searches of the RuX2(X=P,As,Sb) family at 20,40,80,100,and 120 GPa,respectively.After tens of generations with more than three thousand structures are assessed at each pressure,we picked out five structures of the RuX2family,as shown in Fig.1(b)-(f).The predicted crystal structures of the RuX2family under high pressure are in Table A1.The enthalpy-pressure relations of the RuX2family are plotted in Fig.1(g)-(i).For RuSb2in Fig.1(g),the phase transition from thePnnmto the predictedI4/mcm(No.140),as shown in Fig.1(b)is predicted to occur around 30 GPa,followed by a phase transition to the predictedI4/mmm(No.139),as shown in Fig.1(c),at about 105 GPa.The calculated volume-pressure (V–P) curve shows that the volume collapse is about 5.6%and 1.2%at≈30 GPa and≈105 GPa,respectively.These are first-order phase transitions.Compared with thePnnmphase,the predicted tetragonal phaseI4/mcmandI4/mmmare stacked in Ru and Sb layers along thec-axis.The stacking sequence of theI4/mcmphase is Sb-Ru-Sb,while the sequence of theI4/mmmphase is Ru-Sb-Sb.

For RuAs2in Fig.1(h),thePnnmtransforms to the predictedP-62min Fig.1(d),at around 62 GPa,accompanying a volume collapse of about 4.8%.The predictedP-62mis hexagonal.Six As atoms surround Ru atoms and form hexagons.

For RuP2in Fig.1(i),it undergoes two phase transitions.The transition from thePnnmto the predictedI41/amd(No.141) in Fig.1(e) at around 49 GPa,following the transition to the predictedCmcm(No.63) in Fig.1(f) at around 118 GPa.The volume collapse is about 5.6 % and 1.2 % at≈49 GPa and≈118 GPa,respectively.The high-pressure phaseI41/amdis tetragonal andCmcmis orthorhombic.The P atoms are zig-zag patterns in both of the predicted structures.The zig-zag patterns form quasi-onedimensional chains along theaorc-axis in the predictedI41/amdin Fig.1(e),and the Ru atoms are interspersed among these chains.Moreover,the zig-zag patterns form P atom layers in the predictedCmcm,as shown in Fig.1(f),sandwiching the Ru atoms.

Then we calculated the phonon dispersions of these predicted structures both at high pressures and ambient pressure.The phonon dispersion of the predictedI4/mcmandI4/mmmof RuSb2and the predictedI41/amdandCmcmof RuP2have no imaginary curves within 120 GPa,suggesting their dynamical stability.But the predictedP-62mof RuAs2is stable above≈40 GPa.Our results indicate that the predicted structures of RuSb2and RuP2are likely to be synthesized after the pressure release.

B.Electronic structures

Fig.1.The ground Pnnm structure.The predicted structures (b) I4/mcm and (c) I4/mmm of RuSb2,(d) P-62m of RuAs2,(e)I41/amd and(f)Cmcm of RuP2.The silver,brown,purple,and green spheres denote Ru,Sb,P,and As atoms,respectively.The pressure-dependent enthalpy difference of (g) RuSb2 (relative to the I4/mcm),(h) RuAs2 (relative to the P-62m),and (i) RuP2 (relative to the I41/amd).The insets are the corresponding pressure-volume relation.

Fig.2.The band structure of (a) I4/mcm phase at 40 GPa with SOC,(b) I4/mcm phase at 0 GPa with SOC.The surface state of (c) I4/mcm phase at 0 GPa with SOC.The band structure of (d) I4/mmm phase at 110 GPa with SOC,(e) I4/mmm phase at 0 GPa with SOC.The surface state of (f) I4/mmm phase at 0 GPa with SOC.

As displayed in Fig.2,we calculated the band structure and the projected density of states(PDOS)of the predictedI4/mcmat 40 GPa [Fig.2(a)] and 0 GPa[Fig.2(b)]and the predictedI4/mmmat 110 GPa[Fig.2(d)] 0 GPa and [Fig.2(e)] for RuSb2with SOC.The band structures of the predictedI4/mcmandI4/mmmhave metallic features.The PDOS results indicate that the d electrons of the Ru atoms play a dominant role around the Fermi level.Comparing the PDOS at ambient pressure and high pressure of both predicted structures,the distribution of the d electrons is more localized around the Fermi level,such as the dominant peaks of the predictedI4/mcmandI4/mmmat ambient pressure,as shown in Fig.2(b)and (e).The increase of the DOS at the Fermi level may cause the elevation in superconductivity,which we will discuss in part III.C.

Besides,the band structures of both predicted phases at ambient pressure,as shown in Fig.2(b) and(e),open up the energy gap on the high symmetry path.Thus,we performed the calculation of the topological invariance with the help of MLWFs.The symmetry indicators for theI4/mcmphase areZ2w,i=1,2,3=1,Z2=1,Z4=1,Z8=1,and the symmetry indicators for theI4/mmmphase areZ2w,i=1,2,3=1,Z2=0,Z4=2,Z8=2,indicating that both phases are topologically non-trivial.For further confirmation,we calculated their surface states with SOC,as shown in Fig.2(c)forI4/mcmand Fig 2(f)forI4/mmm.The red regions indicate the projected bulk band structure.We projected the band structures alongk-pathfor theI4/mcm.Dirac-type surface states appear at the energy≈0.4 eV below the Fermi level around thepoint.Analogous to theI4/mcmphase,we projected the band structures along thek-point pathand find surface states around thepoint,confirming that theI4/mcmandI4/mmmphases are topologically non-trivial at ambient condition.

As for RuAs2,the band structures and the PDOS for the predictedP-62maround the transition point are in Fig.A1.It is topologically trivial and the d electrons of the Ru atoms make the main contribution around the Fermi level.

The band structure and the PDOS of theI41/amdphase at 50 GPa and 0 GPa with SOC are plotted in Fig.3(a) and (b).It has metallic features within 50 GPa,and the d electrons of the Ru atoms are dominant for the density of states around the Fermi level.The density of states near the Fermi level has a similar feature to the RuSb2phases.We also computed the symmetry indicators for theI41/amdphase at ambient conditions,with theZ2=1 andZ4=1.Then we projected the band structures along thek-point pathand observe the topologically protected gapless surface states in Fig.3(c).Dirac-type surface band appears at the energy of≈0.4 eV above the Fermi level atpoint,confirming that theI41/amdphase is topologically non-trivial.The band structures of the predicted phases without SOC are plotted in Fig.A3.

C.Superconductivity

To study the potential superconductivity of the predicted phases of the RuX2family,we performed the EPC calculations at different pressures.Phonon dispersions,phonon density of states (PDOS),the corresponding Eliashberg spectral functionα2F(ω) and the EPC parameterλare calculated.The superconducting transition temperatureTcwas estimated according to the Allen-Dynes modified McMillan formula:

with the typical Coulomb pseudopotentialμ*=0.1.The logarithmic averaged phonon frequencies (ωlog),and frequency-dependent EPCλ(ω) are obtained from the Eliashberg formalism.

Fig.4(a) and (b) are the EPC calculation results of the predictedI4/mcmof RuSb2at 0 GPa and 40 GPa,respectively.The EPC parameterλfor the predictedI4/mcmat 40 GPa is 0.56 andTc=3.4 K,while the EPC parameterλenhances to 0.94 andTc=7.3 K at ambient pressure.The Fermi surface of theI4/mcmphases at 0 GPa is shown in Fig.A4.We can see that several electron pockets and hole pockets are distributed along with the high symmetry points,where the electron pocket around the Γ point and the hole pocket wrapped around the M point are mainly composed of 4d electrons in Ru atoms.Although we can observe the phonon softening behavior around M and Z points,as shown in Fig.4(a),this part does not make enough contribution to the EPC parameterλ.As mentioned in part III.B,the DOS is more localized around the Fermi level for the predictedI4/mcmat ambient pressure.This may suggest that the electron distribution contributes more to the enhancement ofTcthan the softening behavior at M and Z points.

Fig.4(c) and (d) are the EPC results of the predictedI4/mmmof RuSb2at 0 GPa and 110 GPa,respectively.The EPC parameterλis 1.46 at 0 GPa and 0.32 at 110 GPa,and the correspondingTcis 10.9 K and 0.2 K,respectively.Its Fermi surface is plotted in Fig.A4.ForI4/mmmphase,there is an electron pocket around the Γ point and several electron pockets along the X-P path.Different from theI4/mcmphase,phonon softening behavior is along the Brillouin path,such as Γ-X and P-N.There is a sharp increase of EPC parameterλbetween 50 cm-1and 75 cm-1.Meanwhile,there is a localized peak in DOS at ambient pressure,while the DOS has little dispersion at 110 GPa,as shown in Fig.2(e) and (f).Therefore,the enhancement ofTcis the comprehensive interactions between phonons and electrons.Theλ,ωlog,andTcresults for the predicted structures of RuSb2are collected in Table I.

Fig.3.The band structure of I41/amd phase at (a) 40 GPa and at (b) 0 GPa with SOC.(c) The surface state of I41/amd phase at 0 GPa with SOC.

Fig.4.Phonon dispersion curves,Eliashberg spectral functions α2F(ω) together with the electron-phonon integral λ(ω)and phonon density of states (PHDOS) for I4/mcm phase at (a) 0 GPa,and (b) 40 GPa;and for I4/mmm phase at (c) 0 GPa,and (d) 110 GPa,respectively.

The EPC calculations of the predicted structures for RuP2are in Fig.5 and their Fermi surface results are in Fig.A4.There are several electron and hole pockets distributed along the high symmetry points.FortheI41/amdphase,there are several electron pockets around the Γ point,A point,and S point.These electron pockets are mainly composed ofdyz,dxz,anddx2-y2electrons in Ru atoms.The EPC results of the predictedI41/amdat 0 GPa and 40 GPa are plotted in Fig.5 (a) and (b).The EPC parameterλis elevated from 0.43 at 40 GPa to 0.87 at ambient pressure,and theTcis 2.4 K at 40 GPa and 13.0 K at ambient pressure.The phonon dispersions are analogous to the predictedI4/mmmof RuSb2.The phonon bands are in general softened along the Brillouin path,such as N-Γ-P,while the DOS around the Fermi level at ambient pressure is similar to that at 40 GPa,as shown in Fig.3(a) and (b).Hence,we propose that phonons make more contributions to EPC.

TABLE I.The electron-phonon coupling constant (λ),logarithmic average of phonon frequencies (ωlog),and estimated superconducting critical temperature (Tc) with the Coulomb potential (μ*) of 0.1 for I4/mcm and I4/mmm phase of RuSb2.

Fig.5.Phonon dispersion curves,Eliashberg spectral functions α2F(ω) together with the electron-phonon integral λ(ω)and phonon density of states(PHDOS)for the I41/amd phase of RuP2 at(a)0 GPa and(b)50 GPa,and the Cmcm phase of RuP2 at (c) 0 GPa and (d) 125 GPa,respectively.

As for the predictedCmcmphase,the EPC results at ambient pressure and 125 GPa are depicted in Fig.5(c)and(d).The EPC parameterλis 1.0 at ambient pressure and 0.38 at 125 GPa,withTc=10.1 K at ambient pressure andTc=1.2 K at 125 GPa.In analogous to the EPC results from the predictedI4/mcmof RuSb2,the softening behavior at ambient pressure along the Brillouin path Y-C and T-Y below 50 cm-1does not have enough contribution to the EPC parameter.Moreover,the DOS is more localized around the Fermi level at ambient pressure than 125 GPa,as shown in Fig.A2.We assume that the electron distribution contributes more to the enhancement ofTc.Theλ,ωlog,andTcresults for the predicted structures of RuP2are collected in Table II.

TABLE II.The electron-phonon coupling constant(λ),logarithmic average of phonon frequencies (ωlog),and estimated superconducting critical temperature (Tc) with the Coulomb potential(μ*)of 0.1 for I4/mcm and Cmcm phase of RuP2.

IV.CONCLUSIONS

In summary,using a machine-learning and graph theory accelerated crystal structure search package(Magus),we have investigated the pressure-induced phase transitions of the RuSb2family.It is found that the RuSb2family undergoes a series of transitions from the ambient (Pnnm) phase to several high-pressure phases: (I) For RuSb2,fromPnnmphase to a tetragonalI4/mcmphase,then to a tetragonalI4/mmmphase;(II) For RuP2,fromPnnmphase to a tetragonalI41/amdphase,then to a tetragonal Cmcm phase;(III) For RuAs2,fromPnnmphase to a tetragonalP-62mphase.Our calculations indicate that these phases are all stable at high-pressure.Except for the RuAs2,all these phases can be recovered to ambient pressure.TheI4/mcmandI4/mmmphases of RuSb2and theI41/amdphase of RuP2are predicted to be topological metals,taking into acount the SOC effect.The superconducting transition temperatureTcof RuSb2and RuP2shows a tendency of decreasing with increasing pressure.At 0 GPa,the maximumTcofI4/mcmandI4/mmmphases of RuSb2are 7.3 K and 10.9 K,and theTcof theI41/amdandCmcmphases of RuP2is 13.0 K and 10.1 K,respectively.We hope that this work will stimulate experimental efforts to realize them in the laboratory.

ACKNOWLEDGMENTS

J.S.gratefully acknowledges the financial support from the National Key R&D Program of China (grant nos.2022YFA1403201),the National Natural Science Foundation of China (grant no.12125404,11974162,and 11834006),and the Fundamental Research Funds for the Central Universities.The calculations were carried out using supercomputers at the High Performance Computing Center of Collaborative Innovation Center of Advanced Microstructures,the high-performance supercomputing center of Nanjing University.

APPENDIX

Fig.A1.RuAs2 P-62m band structure and PDOS at 65 GPa,phonon calculation at 65 GPa.

Fig.A2.RuP2 Cmcm band structure and PDOS at 0 GPa and 120 GPa.

TABLE A1.The crystal structure of the predicted RuX2 (X=Sb,As,P) family.