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Charge ordering and heavy fermions in the semimetal Yb4As3
ELSEVIER
Physica B 223&224 (1996) 373 375
Charge ordering and heavy fermions in the semimetal Yb4As3 Burkhard Schmidt*, Peter Thalmeier, Peter Fulde Max-Planck-lnstitutfu'r Physik Komplexer Systeme, Dresden, Germany Abstract
We present a model for Yb4As3 which can explain semiquantitatively a number of its unusual properties. Based on the view of the Yb-sites as being aligned on chains, the structural phase transition at Tc ~ 300 K is described by a band Jahn-Teller effect of correlated electrons being driven by a charge ordering of the Yb-ions. The qualitative behavior of the corresponding elastic constant c44 is explained. The heavy-fermion behavior on a scale of T* -~ 50 K is understood within the framework of a slave-boson approximation to the effective low-temperature Hamiltonian.
The rare-earth pnictide Yb4As3 [1-7] is a material with unusual physical properties. At Tc -~ 300 K, a structural phase transition from the anti-Th3P4 structure with space group I7~3d to a trigonal structure labeled R3c occurs, accompanied by a charge ordering of the Yb-ions [8]. The softening of the corresponding elastic constant c44 is observed [9]. Below 100 K, the material shows typical heavy-fermion behavior with a linear specific-heat coefficient ),-~ 200 m J / m o l K 2 and a correspondingly large spin susceptibility Zs. The Sommerfeld-Wilson ratio is found to be of order unity, clearly indicating Fermi-liquid behavior. However, the Hall coefficient at low temperatures indicates a carrier concentration of about 0.001/Yb-atom, suggesting that the heavy-fermion behavior must be of different origin than in Kondolattice systems. In the high-temperature phase the experimental Hall coefficient is (ecRu) -1 = 1 x 1021 cm -3. With a volume f2 = 6.8 x 10 .22 cm 3 per unit cell, this would correspond to 0.25 holes per formula unit Yb4As3 provided we have a one-band system. With four independent bands the application of a generalized model for the Hall coefficient [10] yields one hole per formula unit, as expected from a chemical approach using valence electron counting (assuming trivalent As). In the following, we describe a model which is capable of explaining
* Corresponding author.
a number of the rather unusual properties of the material
[II]. In Yb4As3, all Yb-atoms are aligned on four families of chains oriented along the space diagonals of the cubic unit cell [12]. To reduce the complexity of the resulting 56 4f-bands, we assume that they can be described by four degenerate quasi-one-dimensional bands associated with these chains. Although this model may not be literally true due to the three-dimensionality of the electronic states, it describes the important aspect that a strain coupling to the degenerate band states may easily lead to distortions of the cubic structure accompanied with a repopulation among the involved 4f band states. The strong Coulomb interaction between nearestneighbor Yb-sites shows up in a large deformation potential coupling which removes the associated degeneracy of chain bands (see below). For trivalent As, we have 0.25 4f-holes per Yb-atom. Since the nearest Yb neighbors of an Yb-ion are situated on a triangle normal to the direction of the corresponding chain, the Yb 4f-holes gain Coulomb energy if they order on one of the four chains. At the structural transition, the crystal shrinks in a volume-conserving way in the < 1 1 l)-direction, subdividing the four equivalent chains into one pointing along the main diagonal of the resulting trigonal unit cell, henceforth called short chain, and the three remaining chains, referred to as long chains.
0921-4526/96/$15.00 .25' 1996 Elsevier Science B.V. All rights reserved PII S092 1 - 4 5 2 6 ( 9 6 ) 0 0 1 26-3
B. Schmidt et al. / Physica B 2 2 3 & 2 2 4 (1996) 373 375
374
We describe this transition by an effective Hamiltonian of the form /h
H
=
-
'EE
, u = l (ij)~
~J
•f i'. ~ L , , + ~v Y',
0.5
"
'
0.4
XA,J,,d).o "* "
~=1
-[- NLIIfCO g2 ,
ia (1)
where # labels the summation over the PF = 4 different chains, and ( / j ) denotes a summation over nearestneighbor sites along one chain. The f],~ create f-holes with spin a at site i of chain/~. We choose a bandwidth 4t = 0.2 eV as obtained from LDA calculations [13] and, for simplicity, an effective spin degeneracy of 2S + 1 -= 2. The second term in Eq. (1) describes the volume-conserving coupling of the trigonal strain er (F -= Fs) to the f-bands characterized by a deformation potential A N = d(6ul - (1 - 6 u l ) / ( / A f - 1)). The deformation potential describes the effect of the nearest-neighbor Coulomb repulsion between the Yb-sites. Writing this interaction as an attractive nearest-neighbor Coulomb interaction within the chains with an energy gain ~ V, we have after Hartree Fock decoupling the relations d2/Co = 6V and AEr = -- 2V(n~,= t - ~). We assume that changes of the bandwidths due to the distortion are small and can be neglected. The Hamiltonian (1) does not yet contain the strong on-site Coulomb interactions of the holes at a Yb-site. Therefore, it is reasonable only above T, where the number of holes per Yb-site is ¼ and below T~ as long as one is sufficiently away from half-filling of the short chain. With the above Hamiltonian, one obtains an elastic constant c 4 4 ( T ) whose temperature dependence together with the occupation numbers nu(T) of the short and long chains is shown in Fig. 1. At T,, c44 vanishes, as expected from our mean-field treatment of Eq. (1). The four bands split, shifting holes from the upper threefold degenerate bands corresponding to the long chains into the lower short-chain band (labeled "1"). As the short chain is approaching half-filling, the strong on-site Coulomb interaction in this chain must be taken into account. We incorporate this by an on-site H u b b a r d interaction U ~ 10 eV which can be eliminated by making a canonical transformation onto a t J-like Hamiltonian acting on states with no 4f a4 Yb-sites in the short chain. Introducing an auxiliary boson b~ to parameterize this constraint [14, 15], we write the Hamiltonian in k-space after a mean-field approximation HMr = Z Z Eu,,(k).f~u~fk,', + H .... ,,
(2)
#v ka
E,,.(k) = 6,~e,(k) + rVu,.,
---•
1.0
(3)
with el(k) = (r 2 + Jz/t)e(k) + 2 + Eo, ~,>l(k) = e(k) = - 2t cos(k), and H ..... = N L ( J Z 2 -- 2(1 - rZ)). The mean-field approximation consists of the replacement
0.5
0.0
0.0
0.0
1.0
1.0
2.0
T,q" Fig. 1. Temperature dependence of the elastic constant c44 as obtained from the Hamiltonian (1). Inset: occupation numbers n~(T) per spin direction of the four f-bands.
(b~) ~ r and of a saddle-point approximation to the Hubbard-Stratonovi6field ( ) ~ f i lto f j l a,) - * g . Here 2 is an average Lagrange multiplier enforcing the abovementioned constraint. The remaining parameters are J = 4t2/U, and Eo being the b a n d - C M distance of the lower and upper bands due to the distortion. The matrix Vu,. = V(6,1 + 6,q)(1 - 6 , ~ ) serves for the coupling between the short chain and the other three. For low temperatures, the thermodynamics of this effective Hamiltonian is essentially governed by a renormalization of the quasi-particle mass in the lower band, m*
t -
-
m b -- 6t + gJ
~
100,
(4)
where mb is the bare band mass of the Hamiltonian (1). The fraction 6 by which the filling of the short chain deviates from one hole per site is determined by the coupling matrix Vu,.. This matrix contains effective hopping matrix elements, because the Y b - Y b hopping takes place ia intermediate pnictogen states. These hopping matrix elements are induced by the distortion, and thus vanish as T ~ To. Within our approximation, the variational equations derived from the above Hamiltonian require a critical magnitude [Vc[ -~ (lEo[ - t(2g + 1))/2, lEo[ ~> 2t, for these matrix elements to obtain a solution with a finite self-doping 5 > 0 of the short chain. With our choice of parameters, we have IV[ -~ 0.015 eV. Within our model for Yb4Ass, we are able to understand a n u m b e r of properties of this material. The structural transition at 300 K is effectively driven by the nearest-neighbour Coulomb interaction between the Ybsites, yielding a strong deformation potential with a Grfineisen parameter (2_--A/4t = 25. Due to the transition, the fourfold degenerate effective f-bands split
B. Schmidt et al. /Physica B 223&224 (1996) 373-375
into one corresponding to the short chain, and the three remaining ones. For low temperatures, the large ~,-coefficient of the specific heat results from (spinon-like) excitations within the lower band, which is then almost half-filled, y is obtained formally from the large mass enhancement m*/mb ~-- 100. In the same sense, all other thermodynamic observables like the static magnetic susceptibility are renormalized at low temperatures. Interpreting the resistivity and the Hall coefficient of Yb4As3, we suggest that the same quasi-particles responsible for the low-temperature thermodynamics dominate their temperature dependence for temperatures lower than 100 K. The resistivity of the short chains is presumably higher than the one of the holes in the long chains due to the reduced mobility of the charge carriers. In a crystal with multiple domains the one of the short chains will therefore make the largest contribution to the measured resistivity. (One has to average over the contributions of the individual domains along a certain direction; the resistivity of the long chains is unimportant due to its smallness as compared to that of the short chain.) This might explain why the prefactor A of the T2-term leads to a value for the ratio A/72 like in other heavyfermion systems [16]. As T is further increased towards T¢, the ratio of charge carriers in the short and the long chains approaches one, the mobility in the short chain increases, thus the resistivity is decreasing. For the Hall coefficient, we expect that in the presence of different channels with different conductivities ~u the one with the largest ~r makes the biggest contribution [10], i.e., the holes in the long chains are most effective here. Since the hole density increases with rising T, the Hall coefficient decreases accordingly. The initial increase for small temperatures is possibly an effect beyond the simple expression for R..
375
Acknowledgements We thank K. Kr61as and T. Goto for supplying data prior to publication.
References [1] P. Bonville,A. Ochiai, T. Suzuki and E. Vincent, J. Phys. I France 4 (1994) 595. [2] A. Ochiai, T. Suzuki and T. Kasuya, J. Phys. Soc. Japan 59 (1990) 4129. [3] P.H.P. Reinders, U. Ahlheim, K. Fraas, F. Steglich and T. Suzuki, Physica B 186-188 (1993) 434. [4] T. Suzuki, Physica B 186-188 (1993) 347. [5] T. Suzuki, Jpn. J. Appl. Phys. 8 (1993) 267. [6] A. Ochiai, D.X. Li, Y. Haga, O. Nakamura and T. Suzuki, Physica B 186-188 (1993) 437. [7] T. Kasuya, J. Phys. Soc. Japan 63 (1994)2481;J. Phys. Soc. Japan, in print. [8] K. Kr61as, T. Tomala, A. Ochiai and T. Suzuki, unpublished. [9] T. Goto, unpublished. [10] J.M. Ziman, Principles of the Theory of Solids (Cambridge University Press, London, 1972). [11] P. Fulde, B. Schmidt and P. Thalmeier, Europhys. Lett. 31 (1995) 323. [12] M. O'Keeffe and S. Andersson, Acta Crystalogr. A 33 (1977) 914. [13] K. Takegahara and Y. Kaneta, J. Phys. Soc. Japan 60 (1991) 4009. [14] N. Read and D.M. Newns, J. Phys. C 16 (1983) 3273. [15] P. Lee and N. Nagaosa, Phys. Rev. B 46 (1992) 5621. [16] K. Kadowaki and S.B Woods, Solid State Commun. 58 (1986) 507.
Physica B 223&224 (1996) 373 375
Charge ordering and heavy fermions in the semimetal Yb4As3 Burkhard Schmidt*, Peter Thalmeier, Peter Fulde Max-Planck-lnstitutfu'r Physik Komplexer Systeme, Dresden, Germany Abstract
We present a model for Yb4As3 which can explain semiquantitatively a number of its unusual properties. Based on the view of the Yb-sites as being aligned on chains, the structural phase transition at Tc ~ 300 K is described by a band Jahn-Teller effect of correlated electrons being driven by a charge ordering of the Yb-ions. The qualitative behavior of the corresponding elastic constant c44 is explained. The heavy-fermion behavior on a scale of T* -~ 50 K is understood within the framework of a slave-boson approximation to the effective low-temperature Hamiltonian.
The rare-earth pnictide Yb4As3 [1-7] is a material with unusual physical properties. At Tc -~ 300 K, a structural phase transition from the anti-Th3P4 structure with space group I7~3d to a trigonal structure labeled R3c occurs, accompanied by a charge ordering of the Yb-ions [8]. The softening of the corresponding elastic constant c44 is observed [9]. Below 100 K, the material shows typical heavy-fermion behavior with a linear specific-heat coefficient ),-~ 200 m J / m o l K 2 and a correspondingly large spin susceptibility Zs. The Sommerfeld-Wilson ratio is found to be of order unity, clearly indicating Fermi-liquid behavior. However, the Hall coefficient at low temperatures indicates a carrier concentration of about 0.001/Yb-atom, suggesting that the heavy-fermion behavior must be of different origin than in Kondolattice systems. In the high-temperature phase the experimental Hall coefficient is (ecRu) -1 = 1 x 1021 cm -3. With a volume f2 = 6.8 x 10 .22 cm 3 per unit cell, this would correspond to 0.25 holes per formula unit Yb4As3 provided we have a one-band system. With four independent bands the application of a generalized model for the Hall coefficient [10] yields one hole per formula unit, as expected from a chemical approach using valence electron counting (assuming trivalent As). In the following, we describe a model which is capable of explaining
* Corresponding author.
a number of the rather unusual properties of the material
[II]. In Yb4As3, all Yb-atoms are aligned on four families of chains oriented along the space diagonals of the cubic unit cell [12]. To reduce the complexity of the resulting 56 4f-bands, we assume that they can be described by four degenerate quasi-one-dimensional bands associated with these chains. Although this model may not be literally true due to the three-dimensionality of the electronic states, it describes the important aspect that a strain coupling to the degenerate band states may easily lead to distortions of the cubic structure accompanied with a repopulation among the involved 4f band states. The strong Coulomb interaction between nearestneighbor Yb-sites shows up in a large deformation potential coupling which removes the associated degeneracy of chain bands (see below). For trivalent As, we have 0.25 4f-holes per Yb-atom. Since the nearest Yb neighbors of an Yb-ion are situated on a triangle normal to the direction of the corresponding chain, the Yb 4f-holes gain Coulomb energy if they order on one of the four chains. At the structural transition, the crystal shrinks in a volume-conserving way in the < 1 1 l)-direction, subdividing the four equivalent chains into one pointing along the main diagonal of the resulting trigonal unit cell, henceforth called short chain, and the three remaining chains, referred to as long chains.
0921-4526/96/$15.00 .25' 1996 Elsevier Science B.V. All rights reserved PII S092 1 - 4 5 2 6 ( 9 6 ) 0 0 1 26-3
B. Schmidt et al. / Physica B 2 2 3 & 2 2 4 (1996) 373 375
374
We describe this transition by an effective Hamiltonian of the form /h
H
=
-
'EE
, u = l (ij)~
~J
•f i'. ~ L , , + ~v Y',
0.5
"
'
0.4
XA,J,,d).o "* "
~=1
-[- NLIIfCO g2 ,
ia (1)
where # labels the summation over the PF = 4 different chains, and ( / j ) denotes a summation over nearestneighbor sites along one chain. The f],~ create f-holes with spin a at site i of chain/~. We choose a bandwidth 4t = 0.2 eV as obtained from LDA calculations [13] and, for simplicity, an effective spin degeneracy of 2S + 1 -= 2. The second term in Eq. (1) describes the volume-conserving coupling of the trigonal strain er (F -= Fs) to the f-bands characterized by a deformation potential A N = d(6ul - (1 - 6 u l ) / ( / A f - 1)). The deformation potential describes the effect of the nearest-neighbor Coulomb repulsion between the Yb-sites. Writing this interaction as an attractive nearest-neighbor Coulomb interaction within the chains with an energy gain ~ V, we have after Hartree Fock decoupling the relations d2/Co = 6V and AEr = -- 2V(n~,= t - ~). We assume that changes of the bandwidths due to the distortion are small and can be neglected. The Hamiltonian (1) does not yet contain the strong on-site Coulomb interactions of the holes at a Yb-site. Therefore, it is reasonable only above T, where the number of holes per Yb-site is ¼ and below T~ as long as one is sufficiently away from half-filling of the short chain. With the above Hamiltonian, one obtains an elastic constant c 4 4 ( T ) whose temperature dependence together with the occupation numbers nu(T) of the short and long chains is shown in Fig. 1. At T,, c44 vanishes, as expected from our mean-field treatment of Eq. (1). The four bands split, shifting holes from the upper threefold degenerate bands corresponding to the long chains into the lower short-chain band (labeled "1"). As the short chain is approaching half-filling, the strong on-site Coulomb interaction in this chain must be taken into account. We incorporate this by an on-site H u b b a r d interaction U ~ 10 eV which can be eliminated by making a canonical transformation onto a t J-like Hamiltonian acting on states with no 4f a4 Yb-sites in the short chain. Introducing an auxiliary boson b~ to parameterize this constraint [14, 15], we write the Hamiltonian in k-space after a mean-field approximation HMr = Z Z Eu,,(k).f~u~fk,', + H .... ,,
(2)
#v ka
E,,.(k) = 6,~e,(k) + rVu,.,
---•
1.0
(3)
with el(k) = (r 2 + Jz/t)e(k) + 2 + Eo, ~,>l(k) = e(k) = - 2t cos(k), and H ..... = N L ( J Z 2 -- 2(1 - rZ)). The mean-field approximation consists of the replacement
0.5
0.0
0.0
0.0
1.0
1.0
2.0
T,q" Fig. 1. Temperature dependence of the elastic constant c44 as obtained from the Hamiltonian (1). Inset: occupation numbers n~(T) per spin direction of the four f-bands.
(b~) ~ r and of a saddle-point approximation to the Hubbard-Stratonovi6field ( ) ~ f i lto f j l a,) - * g . Here 2 is an average Lagrange multiplier enforcing the abovementioned constraint. The remaining parameters are J = 4t2/U, and Eo being the b a n d - C M distance of the lower and upper bands due to the distortion. The matrix Vu,. = V(6,1 + 6,q)(1 - 6 , ~ ) serves for the coupling between the short chain and the other three. For low temperatures, the thermodynamics of this effective Hamiltonian is essentially governed by a renormalization of the quasi-particle mass in the lower band, m*
t -
-
m b -- 6t + gJ
~
100,
(4)
where mb is the bare band mass of the Hamiltonian (1). The fraction 6 by which the filling of the short chain deviates from one hole per site is determined by the coupling matrix Vu,.. This matrix contains effective hopping matrix elements, because the Y b - Y b hopping takes place ia intermediate pnictogen states. These hopping matrix elements are induced by the distortion, and thus vanish as T ~ To. Within our approximation, the variational equations derived from the above Hamiltonian require a critical magnitude [Vc[ -~ (lEo[ - t(2g + 1))/2, lEo[ ~> 2t, for these matrix elements to obtain a solution with a finite self-doping 5 > 0 of the short chain. With our choice of parameters, we have IV[ -~ 0.015 eV. Within our model for Yb4Ass, we are able to understand a n u m b e r of properties of this material. The structural transition at 300 K is effectively driven by the nearest-neighbour Coulomb interaction between the Ybsites, yielding a strong deformation potential with a Grfineisen parameter (2_--A/4t = 25. Due to the transition, the fourfold degenerate effective f-bands split
B. Schmidt et al. /Physica B 223&224 (1996) 373-375
into one corresponding to the short chain, and the three remaining ones. For low temperatures, the large ~,-coefficient of the specific heat results from (spinon-like) excitations within the lower band, which is then almost half-filled, y is obtained formally from the large mass enhancement m*/mb ~-- 100. In the same sense, all other thermodynamic observables like the static magnetic susceptibility are renormalized at low temperatures. Interpreting the resistivity and the Hall coefficient of Yb4As3, we suggest that the same quasi-particles responsible for the low-temperature thermodynamics dominate their temperature dependence for temperatures lower than 100 K. The resistivity of the short chains is presumably higher than the one of the holes in the long chains due to the reduced mobility of the charge carriers. In a crystal with multiple domains the one of the short chains will therefore make the largest contribution to the measured resistivity. (One has to average over the contributions of the individual domains along a certain direction; the resistivity of the long chains is unimportant due to its smallness as compared to that of the short chain.) This might explain why the prefactor A of the T2-term leads to a value for the ratio A/72 like in other heavyfermion systems [16]. As T is further increased towards T¢, the ratio of charge carriers in the short and the long chains approaches one, the mobility in the short chain increases, thus the resistivity is decreasing. For the Hall coefficient, we expect that in the presence of different channels with different conductivities ~u the one with the largest ~r makes the biggest contribution [10], i.e., the holes in the long chains are most effective here. Since the hole density increases with rising T, the Hall coefficient decreases accordingly. The initial increase for small temperatures is possibly an effect beyond the simple expression for R..
375
Acknowledgements We thank K. Kr61as and T. Goto for supplying data prior to publication.
References [1] P. Bonville,A. Ochiai, T. Suzuki and E. Vincent, J. Phys. I France 4 (1994) 595. [2] A. Ochiai, T. Suzuki and T. Kasuya, J. Phys. Soc. Japan 59 (1990) 4129. [3] P.H.P. Reinders, U. Ahlheim, K. Fraas, F. Steglich and T. Suzuki, Physica B 186-188 (1993) 434. [4] T. Suzuki, Physica B 186-188 (1993) 347. [5] T. Suzuki, Jpn. J. Appl. Phys. 8 (1993) 267. [6] A. Ochiai, D.X. Li, Y. Haga, O. Nakamura and T. Suzuki, Physica B 186-188 (1993) 437. [7] T. Kasuya, J. Phys. Soc. Japan 63 (1994)2481;J. Phys. Soc. Japan, in print. [8] K. Kr61as, T. Tomala, A. Ochiai and T. Suzuki, unpublished. [9] T. Goto, unpublished. [10] J.M. Ziman, Principles of the Theory of Solids (Cambridge University Press, London, 1972). [11] P. Fulde, B. Schmidt and P. Thalmeier, Europhys. Lett. 31 (1995) 323. [12] M. O'Keeffe and S. Andersson, Acta Crystalogr. A 33 (1977) 914. [13] K. Takegahara and Y. Kaneta, J. Phys. Soc. Japan 60 (1991) 4009. [14] N. Read and D.M. Newns, J. Phys. C 16 (1983) 3273. [15] P. Lee and N. Nagaosa, Phys. Rev. B 46 (1992) 5621. [16] K. Kadowaki and S.B Woods, Solid State Commun. 58 (1986) 507.