- Email: [email protected]
Prerequisites for chiral charge order
Physica B 407 (2012) 1779–1782
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Prerequisites for chiral charge order Jasper van Wezel n Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
a r t i c l e i n f o
abstract
Available online 9 January 2012
The chiral charge density wave state which was recently discovered in TiSe2 can be understood as a combination of orbital and charge order. Here, we discuss the prerequisite material properties for this type of chiral charge order to emerge. We find that although both the lattice and orbital structure constrain the set of candidate materials, there remains a class of materials in which chiral charge order is expected to emerge. & 2012 Elsevier B.V. All rights reserved.
Keywords: Chirality Charge order Orbital order
1. Introduction The charge ordered state of the layered dichalcogenide 1T–TiSe2 has been suggested to have chiral symmetry [1–3]. The three charge density wave components that make up the charge order have different intensities, and the propagation direction of the dominant component rotates as one descends through consecutive atomic layers, thus creating a helical charge ordered pattern. Such spiral phases are well-known in spin ordered materials, where a rotation of the magnetization vector along an axial propagation vector trivially yields a chiral state. In charge ordered materials on the other hand, the order parameter is a scalar quantity, and TiSe2 is the first known material in which chirality emerges as a property of the ordered state. It was recently shown that the helical symmetry of TiSe2 can be understood as the consequence of the combined presence of charge and orbital order [3]. Here, we describe the prerequisite material properties for chirality to emerge in such a combined orbital ordered and charge ordered state. The constraints on the electronic structure and the symmetries of the lattice presented here indicate that although not all three-component charge density waves can be chiral, there remains a broad class of materials in which chiral charge order may be expected to emerge.
2. Chiral charge and orbital order in TiSe2 Titanium–diselenide is a quasi-two dimensional, layered material. Its band structure has a small (positive or negative) indirect gap of the order of 150 meV or less [4,5]. The predominantly Se-4p valence band is located at the center of the first n
Tel.: þ1 630 252 4764; fax: þ1 630 252 7777. E-mail address: [email protected]
0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2012.01.028
Brillouin zone and the Ti-3d conduction band forms electron pockets at the zone boundaries (see Fig. 1). Although the nesting between the central hole pocket and the outer electron pockets is poor at best [7], TiSe2 is observed to undergo a phase transition at 202 K in which a commensurate 2 2 2 charge density wave forms, accompanied by a periodic lattice distortion. The mechanism underlying the formation of this charge order is still heavily debated. One hypothesis is that a variant of the Jahn–Teller effect is driving the transition [7–11], in which a commensurate spatial reconstruction of the lattice lowers the energy of both the conduction and the valence bands close to the Fermi surface by facilitating partial charge transfer between neighboring Se-4p and Ti-3d orbitals [8,9]. The main competing scenario is that the transition is driven by exciton formation [5,12–14], and possibly condensation [15]. The exciton formation is made possible by the paucity of charge carriers in the system and the correspondingly poorly screened Coulomb interaction. With sufficient electron–hole coupling between the valence and conduction bands, the system is unstable to the formation of excitons and deforms with a periodicity governed by the wave vector connecting them. Both scenarios are supported by numerical as well as experimental results, but since neither can fully explain all the observed effects, it has been suggested that a cooperative mechanism, combining both excitons and Jahn–Teller phonons, may be responsible for the charge density modulations in TiSe2 [16–18]. 2.1. Orbital structure Regardless of the precise mechanism underlying the formation of the ordered state, it is clear that the three components of the charge density wave in TiSe2 are associated with the transfer of electronic weight from the central Se-4p pocket to the three Ti-3d pockets on the edges of the first Brillouin zone, as indicated in the left of Fig. 1. The orbital character of each of the Ti-3d pockets can
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J. van Wezel / Physica B 407 (2012) 1779–1782
sector, the electron–phonon coupling will in general no longer be isotropic, and the displacement wave may acquire a transversal component [3]. In the case of TiSe2, the orbitals involved in each of the components of the charge density wave in fact force the ionic lattice displacements to be purely transversal, so that its polarization vector E^ is perpendicular to the propagation vector q (as shown in the right of Fig. 1). 2.2. Relative phases
Fig. 1. (Left) A schematic representation of constant energy maps near EF forming electron and hole pockets in the first Brillouin zone of TiSe2 (adapted from Ref. [6]). The arrow indicates the direction of electronic weight transfer associated with one of the three components of the charge density wave. (Right) The position in the lattice structure of the orbitals involved in the formation of the charge density wave component along propagation vector q with polarization vector e ? q.
Having found both the propagation vectors and the polarization vectors of the displacement waves in TiSe2, the only remaining parameters determining the lattice structure in the low temperature phase are the relative phase differences jj of the three components u ¼ u0
3 X
E^ j sinðqj x þ jj Þ:
ð1Þ
j¼1
These relative phases cannot be determined from the band structure alone. They depend on interactions between the charge density wave components, which are mediated by the direct Coulomb interaction as well as through the lattice deformations. Both of these effects can be included in the general form of the Ginzburg–Landau expansion of the free energy [21] Z 2 2 2 F¼ dxfadr2 þ bdr3 þ cdr4 þd½9c1 c2 9 þ 9c2 c3 9 þ 9c3 c1 9 g: ð2Þ
Fig. 2. (Left) The weight of each orbital component in the conduction band of a 6-orbital tight binding fit to the band structure of TiSe2 [18]. (Right) A schematic representation of the dominant orbital contribution to each of the electron and hole pockets involved in the formation of the charge ordered state. Notice that the orbital character of the central hole pocket varies as a function of direction in reciprocal space.
Here drðxÞ is the modulation of the average charge density, which can be written as a sum of three complex components cj ¼ c0 eiqj x þ jj . To simplify the expression, we assume that all components have the same intensity c0 , and are aligned with the preferred propagation vectors qj . The cross terms in the last line of Eq. (2) signify the competition over the Fermi surface of the central Se-4p pocket, which is shared by all components. For the evaluation of the integrals it is essential to take into account the Umklapp processes associated with the presence of a discrete lattice. This can be done by forcing the spatially varying part of the coefficients to reflect the symmetry of the lattice, as well as the internal structure of the unit cell [21] X X X a ¼ a0 þ a1 eiGi x þ ga1 eiGi x eiGi RSe1 þ ga1 eiGi x eiGi RSe2 þ i
i
i
ð3Þ be calculated using a tight-binding fit to the band structure [18]. As indicated in Fig. 2, each pocket is dominated by only a single Ti orbital. Likewise, the central hole pocket is found to be dominated by just two out of the three possible orientations of Se-4p orbitals. The electronic weight transfer that accompanies the formation of the charge density wave in TiSe2 is thus restricted to a specific orbital sector for each of its three components. The orbital character of the different components does not have a direct effect on the modulation of the electronic charge density, but it does have profound consequences for the periodic rearrangement of ionic lattice sites, which accompanies the formation of a charge density wave in any real material, due to the omnipresent electron–phonon coupling [19,20]. The electronic density modulation by itself may be characterized completely by its propagation vector: dr ¼ A cosðq xÞ. The displacement wave describing the lattice deformations on the other hand, requires a polarization vector in addition to the propagation vector: u ¼ u0 E^ sinðq xÞ. If the electron–phonon coupling is completely isotropic, the ionic displacements simply follow the gradient in electronic density, and the displacement wave is always purely longitudinal. However, if the charge density modulations occur within a specific orbital
Here Gi are the shortest reciprocal lattice vectors and RSe1;2 denote the positons of the two Se atoms within the unit cell. The factor g reflects the difference between the electron–phonon couplings on the Ti and Se sites. Higher order terms in the expansion include longer reciprocal lattice vectors. Evaluating the expression of Eq. (2) taking into account the Umklapp terms, and minimizing the result with respect to the relative phases of the charge density wave components, finally yields the solution
j1 ¼
p 2
,
j2 ¼ j3 , j3 ¼ 7
" # 1 3c2 c0 2a1 ð1gÞ cos1 : 2 2 6c2 c
ð4Þ
0
Inserting these solutions into the expression of Eq. (1) for the total ionic displacement pattern, it is easily seen that they represent precisely the chiral phase observed in TiSe2, with the sign of j3 determining the handedness [3,22]. Because each component of the displacement wave corresponds to the transfer of electronic
J. van Wezel / Physica B 407 (2012) 1779–1782
weight within a specific orbital sector, the phase differences in Eq. (4) necessarily describe the formation of an orbital ordered as well as a charge ordered state [3].
3. Requirements for chirality The chirality of the charge ordered state in TiSe2 emerges from the combination of three differently polarized charge density wave components with nonzero relative phase differences. To see whether this mechanism may also occur in other charge ordered materials, we consider the role of each of the basic building blocks in establishing the chiral crystal structure. A chiral structure is defined by a simultaneous rotation and translation: the dominant displacement direction rotates as it is translated along an axial propagation vector. For the charge order to give rise to chirality, both a sense of rotation and of translation thus need to be identified in the periodic lattice distortions accompanying the charge density wave X u ¼ u0 E^ j sinðqj ½x þ dxj Þ: ð5Þ j
Writing the relative phases jj as spatial shifts of the displacement patterns shows how a sense of translation can be accomplished in the overall pattern. The common component of all spatial shifts defines an overall propagation vector for the displacement pattern. The first requirement for chirality to be allowed, is thus the existence of a common component of the spatial shift differences X dxj dxj þ 1 a 0: ð6Þ j
Notice that the definition qj dxj ¼ jj only defines the components of the shifts parallel to the vectors qj . To identify the other components of the spatial shifts of the displacement pattern, it is necessary to take into account the detailed lattice structure. In TiSe2 for example, a shift of the charge density modulation along a propagation vector qj moves the maximum displacement from the Ti atom in the center of the layer to one of the Se atoms at either the top or the bottom of the layer. The translation of the displacement wave can thus acquire a component perpendicular to the atomic layers, even though all propagation vectors qj lie wholly within the atomic plane. This is made possible by the fact that the Se atoms in the 1T–TiSe2 lattice structure are not arranged symmetrically with respect to the central Ti plane. To identify a sense of rotation in the displacement pattern of Eq. (5), the polarizations of the different charge density wave components may be used. For any rotation to be possible, one needs at least three different components. Moreover, to define a chiral pattern one needs at least three different components which do not coincide with the axial translation vector
E^ i a E^j for i a jEf1; 2,3g, E^ i a
X
dxj dxj þ 1
for iEf1; 2,3g:
ð7Þ
j
Because the polarizations of the displacement waves arise from the occupation of specific orbital sectors, the requirement of having differently polarized components with nonzero relative phase shifts always implies the presence of orbital order in the distorted lattice structure. Chiral charge order of the type discussed here therefore cannot occur without the simultaneous presence of orbital order. In the case of TiSe2, the orbital order consists of the uneven occupation of different Ti-3d or Se-4p orbitals in each atomic layer [3]. With the requirements of Eqs. (6) and (7) satisfied, the only remaining obstacle to the emergence of the chiral charge ordered
1781
state in any real material is the energy required to establish the lattice displacements. From the Ginzburg–Landau expression of Eq. (2) it is clear that in order for the chiral state to be energetically favorable as compared to a non-chiral charge density wave, the presence of Umklapp terms is essential. The Umklapp terms can only contribute to the free energy if the charge density wave is commensurate with the lattice structure. For the chiral state to be energetically favorable then, the propagation vectors qj defined by the band structure should either be commensurate, or very close to commensurate.
4. Conclusions The recently discovered chiral state in the layered dichalcogenide 1T–TiSe2 is the result of the simultaneous formation of orbital and charge density wave order. The three components of the charge density wave occupy three distinct orbital sectors, giving rise to three distinct polarization directions for their associated displacement waves. Superposing the components with non-zero relative phase differences then yields the observed chiral pattern. In general, the prerequisites for finding this type of chiral order are: the presence of at least three components in the charge density wave; the existence of an axial propagation vector, defined by the differences between the spatial shifts of the displacement wave components in response to the relative phase differences of the charge density wave components; and finally, a sense of rotation around the propagation vector, defined by the polarization directions of the different displacement wave components. In addition, the chiral state needs to be energetically favored over other types of charge order, which suggests that the charge density wave vectors should be nearly commensurate with the crystal lattice. Both the definition of the axial propagation vector, and the presence of distinct polarizations places constraints on the possible lattice structures in which the type of combined chiral orbital and charge order discussed here can occur. In TiSe2 both the absence of a mirror plane in the Ti layer and the spatial orientation of the Ti-3d orbitals are crucial in establishing the chiral pattern. Many other charge density wave materials, even within the dichalcogenide family, lack these prerequisite characteristics. One noteworthy example of a material that does meet all requirements is 1T–TiTe2. If it can be induced to undergo a charge density wave transition under doping, its charge ordered state is expected to be chiral.
Acknowledgments This work was supported by the US DOE, Office of Science, under Contract No. DE-AC02-06CH11357. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
J. Ishioka, et al., Phys. Rev. Lett. 105 (2010) 176401. J. van Wezel, P. Littlewood, Physics 3 (2010) 87. J. van Wezel, Europhys. Lett. 96 (2011) 67011. F. diSalvo, J. Waszczak, Phys. Rev. B 17 (1978) 3801. A. Zunger, A.J. Freeman, Phys. Rev. B 17 (1978) 1839. C. Monney, et al., Europhys. Lett. 92 (2010) 47003. K. Rossnagel, et al., Phys. Rev. B 65 (2002) 235101. M.H. Whangbo, et al., J. Am. Chem. Soc. 114 (1992) 9587. H.P. Hughes, J. Phys. C 10 (1977) L319. K. Motizuki, et al., Solid State Commun. 40 (1981) 995. N. Suzuki, et al., J. Phys. Soc. Jpn. 54 (1985) 4668. G. Li, W. Hu, et al., Phys. Rev. Lett. 99 (2007) 027404. J. Wilson, Solid State Commun. 22 (1997) 551. H. Cercellier, et al., Phys. Rev. Lett. 99 (2007) 146403.
1782
[15] [16] [17] [18] [19]
J. van Wezel / Physica B 407 (2012) 1779–1782
C. Monney, et al., Phys. Rev. B 79 (2009) 45116. J. van Wezel, et al., Phys. Rev. B 81 (2010) 165109. J. van Wezel, et al., Europhys. Lett. 89 (2010) 47004. J. van Wezel, et al., Phys. Status Solidi 247 (2010) 592. R.E. Peierls, Quantum Theory of Solids, Oxford University Press, Oxford, 1955.
¨ [20] G. Gruner, Density Waves in Solids, Addison-Wesley, Reading, 1994. [21] W.L. McMillan, Phys. Rev. B 12 (1975) 1187. [22] If the argument of the inverse cosine is less than 1, the lowest energy solution instead becomes the non-chiral triple-q mode given by j1 ¼ j2 ¼ j3 ¼ p=2.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Prerequisites for chiral charge order Jasper van Wezel n Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA
a r t i c l e i n f o
abstract
Available online 9 January 2012
The chiral charge density wave state which was recently discovered in TiSe2 can be understood as a combination of orbital and charge order. Here, we discuss the prerequisite material properties for this type of chiral charge order to emerge. We find that although both the lattice and orbital structure constrain the set of candidate materials, there remains a class of materials in which chiral charge order is expected to emerge. & 2012 Elsevier B.V. All rights reserved.
Keywords: Chirality Charge order Orbital order
1. Introduction The charge ordered state of the layered dichalcogenide 1T–TiSe2 has been suggested to have chiral symmetry [1–3]. The three charge density wave components that make up the charge order have different intensities, and the propagation direction of the dominant component rotates as one descends through consecutive atomic layers, thus creating a helical charge ordered pattern. Such spiral phases are well-known in spin ordered materials, where a rotation of the magnetization vector along an axial propagation vector trivially yields a chiral state. In charge ordered materials on the other hand, the order parameter is a scalar quantity, and TiSe2 is the first known material in which chirality emerges as a property of the ordered state. It was recently shown that the helical symmetry of TiSe2 can be understood as the consequence of the combined presence of charge and orbital order [3]. Here, we describe the prerequisite material properties for chirality to emerge in such a combined orbital ordered and charge ordered state. The constraints on the electronic structure and the symmetries of the lattice presented here indicate that although not all three-component charge density waves can be chiral, there remains a broad class of materials in which chiral charge order may be expected to emerge.
2. Chiral charge and orbital order in TiSe2 Titanium–diselenide is a quasi-two dimensional, layered material. Its band structure has a small (positive or negative) indirect gap of the order of 150 meV or less [4,5]. The predominantly Se-4p valence band is located at the center of the first n
Tel.: þ1 630 252 4764; fax: þ1 630 252 7777. E-mail address: [email protected]
0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2012.01.028
Brillouin zone and the Ti-3d conduction band forms electron pockets at the zone boundaries (see Fig. 1). Although the nesting between the central hole pocket and the outer electron pockets is poor at best [7], TiSe2 is observed to undergo a phase transition at 202 K in which a commensurate 2 2 2 charge density wave forms, accompanied by a periodic lattice distortion. The mechanism underlying the formation of this charge order is still heavily debated. One hypothesis is that a variant of the Jahn–Teller effect is driving the transition [7–11], in which a commensurate spatial reconstruction of the lattice lowers the energy of both the conduction and the valence bands close to the Fermi surface by facilitating partial charge transfer between neighboring Se-4p and Ti-3d orbitals [8,9]. The main competing scenario is that the transition is driven by exciton formation [5,12–14], and possibly condensation [15]. The exciton formation is made possible by the paucity of charge carriers in the system and the correspondingly poorly screened Coulomb interaction. With sufficient electron–hole coupling between the valence and conduction bands, the system is unstable to the formation of excitons and deforms with a periodicity governed by the wave vector connecting them. Both scenarios are supported by numerical as well as experimental results, but since neither can fully explain all the observed effects, it has been suggested that a cooperative mechanism, combining both excitons and Jahn–Teller phonons, may be responsible for the charge density modulations in TiSe2 [16–18]. 2.1. Orbital structure Regardless of the precise mechanism underlying the formation of the ordered state, it is clear that the three components of the charge density wave in TiSe2 are associated with the transfer of electronic weight from the central Se-4p pocket to the three Ti-3d pockets on the edges of the first Brillouin zone, as indicated in the left of Fig. 1. The orbital character of each of the Ti-3d pockets can
1780
J. van Wezel / Physica B 407 (2012) 1779–1782
sector, the electron–phonon coupling will in general no longer be isotropic, and the displacement wave may acquire a transversal component [3]. In the case of TiSe2, the orbitals involved in each of the components of the charge density wave in fact force the ionic lattice displacements to be purely transversal, so that its polarization vector E^ is perpendicular to the propagation vector q (as shown in the right of Fig. 1). 2.2. Relative phases
Fig. 1. (Left) A schematic representation of constant energy maps near EF forming electron and hole pockets in the first Brillouin zone of TiSe2 (adapted from Ref. [6]). The arrow indicates the direction of electronic weight transfer associated with one of the three components of the charge density wave. (Right) The position in the lattice structure of the orbitals involved in the formation of the charge density wave component along propagation vector q with polarization vector e ? q.
Having found both the propagation vectors and the polarization vectors of the displacement waves in TiSe2, the only remaining parameters determining the lattice structure in the low temperature phase are the relative phase differences jj of the three components u ¼ u0
3 X
E^ j sinðqj x þ jj Þ:
ð1Þ
j¼1
These relative phases cannot be determined from the band structure alone. They depend on interactions between the charge density wave components, which are mediated by the direct Coulomb interaction as well as through the lattice deformations. Both of these effects can be included in the general form of the Ginzburg–Landau expansion of the free energy [21] Z 2 2 2 F¼ dxfadr2 þ bdr3 þ cdr4 þd½9c1 c2 9 þ 9c2 c3 9 þ 9c3 c1 9 g: ð2Þ
Fig. 2. (Left) The weight of each orbital component in the conduction band of a 6-orbital tight binding fit to the band structure of TiSe2 [18]. (Right) A schematic representation of the dominant orbital contribution to each of the electron and hole pockets involved in the formation of the charge ordered state. Notice that the orbital character of the central hole pocket varies as a function of direction in reciprocal space.
Here drðxÞ is the modulation of the average charge density, which can be written as a sum of three complex components cj ¼ c0 eiqj x þ jj . To simplify the expression, we assume that all components have the same intensity c0 , and are aligned with the preferred propagation vectors qj . The cross terms in the last line of Eq. (2) signify the competition over the Fermi surface of the central Se-4p pocket, which is shared by all components. For the evaluation of the integrals it is essential to take into account the Umklapp processes associated with the presence of a discrete lattice. This can be done by forcing the spatially varying part of the coefficients to reflect the symmetry of the lattice, as well as the internal structure of the unit cell [21] X X X a ¼ a0 þ a1 eiGi x þ ga1 eiGi x eiGi RSe1 þ ga1 eiGi x eiGi RSe2 þ i
i
i
ð3Þ be calculated using a tight-binding fit to the band structure [18]. As indicated in Fig. 2, each pocket is dominated by only a single Ti orbital. Likewise, the central hole pocket is found to be dominated by just two out of the three possible orientations of Se-4p orbitals. The electronic weight transfer that accompanies the formation of the charge density wave in TiSe2 is thus restricted to a specific orbital sector for each of its three components. The orbital character of the different components does not have a direct effect on the modulation of the electronic charge density, but it does have profound consequences for the periodic rearrangement of ionic lattice sites, which accompanies the formation of a charge density wave in any real material, due to the omnipresent electron–phonon coupling [19,20]. The electronic density modulation by itself may be characterized completely by its propagation vector: dr ¼ A cosðq xÞ. The displacement wave describing the lattice deformations on the other hand, requires a polarization vector in addition to the propagation vector: u ¼ u0 E^ sinðq xÞ. If the electron–phonon coupling is completely isotropic, the ionic displacements simply follow the gradient in electronic density, and the displacement wave is always purely longitudinal. However, if the charge density modulations occur within a specific orbital
Here Gi are the shortest reciprocal lattice vectors and RSe1;2 denote the positons of the two Se atoms within the unit cell. The factor g reflects the difference between the electron–phonon couplings on the Ti and Se sites. Higher order terms in the expansion include longer reciprocal lattice vectors. Evaluating the expression of Eq. (2) taking into account the Umklapp terms, and minimizing the result with respect to the relative phases of the charge density wave components, finally yields the solution
j1 ¼
p 2
,
j2 ¼ j3 , j3 ¼ 7
" # 1 3c2 c0 2a1 ð1gÞ cos1 : 2 2 6c2 c
ð4Þ
0
Inserting these solutions into the expression of Eq. (1) for the total ionic displacement pattern, it is easily seen that they represent precisely the chiral phase observed in TiSe2, with the sign of j3 determining the handedness [3,22]. Because each component of the displacement wave corresponds to the transfer of electronic
J. van Wezel / Physica B 407 (2012) 1779–1782
weight within a specific orbital sector, the phase differences in Eq. (4) necessarily describe the formation of an orbital ordered as well as a charge ordered state [3].
3. Requirements for chirality The chirality of the charge ordered state in TiSe2 emerges from the combination of three differently polarized charge density wave components with nonzero relative phase differences. To see whether this mechanism may also occur in other charge ordered materials, we consider the role of each of the basic building blocks in establishing the chiral crystal structure. A chiral structure is defined by a simultaneous rotation and translation: the dominant displacement direction rotates as it is translated along an axial propagation vector. For the charge order to give rise to chirality, both a sense of rotation and of translation thus need to be identified in the periodic lattice distortions accompanying the charge density wave X u ¼ u0 E^ j sinðqj ½x þ dxj Þ: ð5Þ j
Writing the relative phases jj as spatial shifts of the displacement patterns shows how a sense of translation can be accomplished in the overall pattern. The common component of all spatial shifts defines an overall propagation vector for the displacement pattern. The first requirement for chirality to be allowed, is thus the existence of a common component of the spatial shift differences X dxj dxj þ 1 a 0: ð6Þ j
Notice that the definition qj dxj ¼ jj only defines the components of the shifts parallel to the vectors qj . To identify the other components of the spatial shifts of the displacement pattern, it is necessary to take into account the detailed lattice structure. In TiSe2 for example, a shift of the charge density modulation along a propagation vector qj moves the maximum displacement from the Ti atom in the center of the layer to one of the Se atoms at either the top or the bottom of the layer. The translation of the displacement wave can thus acquire a component perpendicular to the atomic layers, even though all propagation vectors qj lie wholly within the atomic plane. This is made possible by the fact that the Se atoms in the 1T–TiSe2 lattice structure are not arranged symmetrically with respect to the central Ti plane. To identify a sense of rotation in the displacement pattern of Eq. (5), the polarizations of the different charge density wave components may be used. For any rotation to be possible, one needs at least three different components. Moreover, to define a chiral pattern one needs at least three different components which do not coincide with the axial translation vector
E^ i a E^j for i a jEf1; 2,3g, E^ i a
X
dxj dxj þ 1
for iEf1; 2,3g:
ð7Þ
j
Because the polarizations of the displacement waves arise from the occupation of specific orbital sectors, the requirement of having differently polarized components with nonzero relative phase shifts always implies the presence of orbital order in the distorted lattice structure. Chiral charge order of the type discussed here therefore cannot occur without the simultaneous presence of orbital order. In the case of TiSe2, the orbital order consists of the uneven occupation of different Ti-3d or Se-4p orbitals in each atomic layer [3]. With the requirements of Eqs. (6) and (7) satisfied, the only remaining obstacle to the emergence of the chiral charge ordered
1781
state in any real material is the energy required to establish the lattice displacements. From the Ginzburg–Landau expression of Eq. (2) it is clear that in order for the chiral state to be energetically favorable as compared to a non-chiral charge density wave, the presence of Umklapp terms is essential. The Umklapp terms can only contribute to the free energy if the charge density wave is commensurate with the lattice structure. For the chiral state to be energetically favorable then, the propagation vectors qj defined by the band structure should either be commensurate, or very close to commensurate.
4. Conclusions The recently discovered chiral state in the layered dichalcogenide 1T–TiSe2 is the result of the simultaneous formation of orbital and charge density wave order. The three components of the charge density wave occupy three distinct orbital sectors, giving rise to three distinct polarization directions for their associated displacement waves. Superposing the components with non-zero relative phase differences then yields the observed chiral pattern. In general, the prerequisites for finding this type of chiral order are: the presence of at least three components in the charge density wave; the existence of an axial propagation vector, defined by the differences between the spatial shifts of the displacement wave components in response to the relative phase differences of the charge density wave components; and finally, a sense of rotation around the propagation vector, defined by the polarization directions of the different displacement wave components. In addition, the chiral state needs to be energetically favored over other types of charge order, which suggests that the charge density wave vectors should be nearly commensurate with the crystal lattice. Both the definition of the axial propagation vector, and the presence of distinct polarizations places constraints on the possible lattice structures in which the type of combined chiral orbital and charge order discussed here can occur. In TiSe2 both the absence of a mirror plane in the Ti layer and the spatial orientation of the Ti-3d orbitals are crucial in establishing the chiral pattern. Many other charge density wave materials, even within the dichalcogenide family, lack these prerequisite characteristics. One noteworthy example of a material that does meet all requirements is 1T–TiTe2. If it can be induced to undergo a charge density wave transition under doping, its charge ordered state is expected to be chiral.
Acknowledgments This work was supported by the US DOE, Office of Science, under Contract No. DE-AC02-06CH11357. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
J. Ishioka, et al., Phys. Rev. Lett. 105 (2010) 176401. J. van Wezel, P. Littlewood, Physics 3 (2010) 87. J. van Wezel, Europhys. Lett. 96 (2011) 67011. F. diSalvo, J. Waszczak, Phys. Rev. B 17 (1978) 3801. A. Zunger, A.J. Freeman, Phys. Rev. B 17 (1978) 1839. C. Monney, et al., Europhys. Lett. 92 (2010) 47003. K. Rossnagel, et al., Phys. Rev. B 65 (2002) 235101. M.H. Whangbo, et al., J. Am. Chem. Soc. 114 (1992) 9587. H.P. Hughes, J. Phys. C 10 (1977) L319. K. Motizuki, et al., Solid State Commun. 40 (1981) 995. N. Suzuki, et al., J. Phys. Soc. Jpn. 54 (1985) 4668. G. Li, W. Hu, et al., Phys. Rev. Lett. 99 (2007) 027404. J. Wilson, Solid State Commun. 22 (1997) 551. H. Cercellier, et al., Phys. Rev. Lett. 99 (2007) 146403.
1782
[15] [16] [17] [18] [19]
J. van Wezel / Physica B 407 (2012) 1779–1782
C. Monney, et al., Phys. Rev. B 79 (2009) 45116. J. van Wezel, et al., Phys. Rev. B 81 (2010) 165109. J. van Wezel, et al., Europhys. Lett. 89 (2010) 47004. J. van Wezel, et al., Phys. Status Solidi 247 (2010) 592. R.E. Peierls, Quantum Theory of Solids, Oxford University Press, Oxford, 1955.
¨ [20] G. Gruner, Density Waves in Solids, Addison-Wesley, Reading, 1994. [21] W.L. McMillan, Phys. Rev. B 12 (1975) 1187. [22] If the argument of the inverse cosine is less than 1, the lowest energy solution instead becomes the non-chiral triple-q mode given by j1 ¼ j2 ¼ j3 ¼ p=2.