- Email: [email protected]
Dynamics of a carrier in the antiferromagnetic state with strong Hund coupling
PHYSICA ELSEVIER
Physica C 263 (1996) 142-145
Dynamics of a carrier in the antiferromagnetic state with strong Hund coupling S. Akazawa "'*, J. Inoue b, S. Maekawa b a Kagoshima National College of Technology, Aira-Gun. Kagoshima 899-51, Japan b Department of Applied Physics. Nagoya University, Nagoya 464-01. Japan
Abstract The dynamics of a single carrier added into the 2-dimensional antiferromagnetic state is studied in a model where a strong Hund coupling between localized spins and itinerant electrons coexists and strong electron correlations are also included for itinerant electrons. When the carrier is a hole added into the half-filling band of the itinerant electrons, the lowest energy state occurs at k = ( a t / 2 , -rr/2), while it occurs at k = (0, 0) when the carrier is an electron in the low-density limit. The difference between these lowest energy states originates from the strong electron correlations between the itinerant electrons at half-filling.
Perovskite-type oxides are systems of strong electron correlations and show various interesting phenomena when mobile carriers are introduced. Some Cu-oxides, for example (La-Sr)2CuO 4, show the high-T~ superconductivity [1] by introducing holes into the antiferromagnetic insulating phase of La2CuO 4. Various models, e.g., the Hubbard model, t-J model and spin-fermion model [2,3], have been used to understand the low-energy excitations in the doped Cu-oxides. Another example is Lal_xSr xMnO3, in which the antiferromagnetic insulating phase at x = 0 changes to a metallic ferromagnetic state for 0.2 < x < 0.5, and again to the antiferromagnetic insulating phase for larger x [4]. In these systems, a strong Hund coupling between localized t2g electrons and itinerant eg electrons plays an
* Corresponding author. Fax: +81 995 43 2584; e-mail: [email protected].
essential role in the physical properties [5], and the strong electron correlations between eg electrons themselves are also important [6]. Furthermore, the Mn-oxides have recently attracted considerable attention because of a colossal magnetoresistance near room temperature [7]. However, only a little theoretical work has been done so far [6,8,9]. In order to understand the unusual properties of doped insulators, it is important to clarify how the carriers introduced in the insulating phases move under the strong electron correlations. In this work, we focus our attention on a problem of the motion of carriers in the antiferromagnetic state with strong Hund coupling, using a model where itinerant electrons and localized spins coexist (spin-fermion model). We will calculate the energy momentum relation E(k) of a single mobile carrier using a variational method [3,10] and study the effect of the Hund coupling on E(k). As for the mobile carrier, we consider two cases: one is a hole added at the
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 7 1 2- I
S. Akazawa et al./ Physica C 263 (1996) 142-145
half-filled band of the itinerant electrons with an antiferromagnetic order, and the other is an electron in the low-density limit also with the antiferromagnetic background of the localized spins. The Hamiitonian we use is given by
H=-,
E
E
(i,j)
143
(a)
t t ~ t ~t
.... >
t~
t t ~t
..... >
t ~t
~t
f
i
S,.s
f ~ f xf f (b)
+gESi.si,
(l)
i
where the first, second and third terms denote hopping of itinerant electrons (I > O) between the nearest neighbor (n.n.) sites i and j, antiferromagnetic n.n. exchange interaction ( J > O) between localized spins and the Hund coupling ( K < O) between the localized spin S i and itinerant electron spin s i, respectively. ~ ( ~ i ~ ) is a creation (annihilation) operator of an electron on site i with spin o~ with a constraint that no double occupancy of the same site is permitted. (The constraint is irrelevant for a single 1 electron in the low-density limit.) We assume S = for the localized spins and a 2D square lattice for simplicity. We use a variational method where the Hilbert space is restricted. The variational function is given by a linear combination of many basis vectors which represent the intermediate state in the motion of the carrier starting from the initial state just added in the antiferromagnetic background. Examples of the hopping process are shown in Fig. l(a) for a single electron in the low-density limit and (b) for a single hole at half-filling. We prepare 147 basis vectors for a single electron carrier and 205 for a single hole. The lowest energy branch of E(k) calculated for a single electron in the low-density limit with J / t = 0.7 and K / t = - 10 is shown in Fig. 2(a). Here, the lattice constant is taken to be unity and the origin of E(k) is taken at the energy of the initial state at which the electron is added in the antiferromagnetic state. The shape of E(k) around k = (0, 0) is similar to that of a free electron and the lowest energy appears at k = (0, 0). The results are almost independent of the number of basis vectors and only weakly dependent on K and J. The results can be interpreted in the following way. When I K I < < 4t, the electron can easily hop, irrespective of the potential barrier due to the Hund coupling. When I K I is large, the spins of the itinerant electron and localized
t
~ 0 ~ t .....> t
~ ~ 0 t ..... > f
~ f 0 t x
f ~ t
~ o < ..... f ~ f ~ o < - .... t x
x
~ t
t o
x
Fig. 1. Examples of the hopping process (a) for a single electron in the low-density limit and (b) for a hole (shown by a circle) in the half-filled state. Arrows in the upper and lower columns denote the itinerant electron spins and localized spins, respectively. The crosses denote that the indicated spins are overturned ones.
spin on the same site form the triplet or singlet states, the energies of which are K / 4 and - 3 K / 4 , respectively. Let us assume that an 1' spin electron is initially added on a site with a localized 1' spin. When the 1" spin electron hops to a n.n. site, the spin of the electron and the localized spin are antiparallel in the sense of the classical spin picture. However, by using the state of S z = 0 of the local triplet state, the itinerant electron can hop to further sites without any loss of kinetic energy. Such a process is depicted in Fig. l(a), from which we can see that the number of intermediate states for the hopping of the electron is few. This is the reason for the weak dependence of E(k) on the number of basis vectors. Because the electron leaves no overturned spins behind, J does not affect the motion of the electron, i.e., E(k) is almost independent of J. The dynamics of the electron in the present case with large values of I K I is different from that with positive value of K [3]. In the latter case, with increasing K, the electron spin forms a singlet with the localized spin and the dynamics of the singlet state is nothing but the dynamics of a hole in the half-filled t-J model. The lowest energy branch of E(k) calculated for a hole at half-filling with J / t = 0.7 and K / t = - 10
144
S. Akazawa et al. / Physica C 263 (1996) 142-145 0
0
(a)
- 1 .5
(b)
-1
2
-2
a
-3
-4 (O,O)
- I .6
r4-t .7
-1.7
-1.8
-1.8
-19
-1.9 (IT,0)
(]1/2 ,IT/2 )
(0,O)
~0,0)
(~,0)
(~/2,~/21
(O,O)
k
k
Fig. 2. Calculated results of the lowest-energy dispersion curve with J / t = 0.7 and K / t = - 10 (a) for an electron in the low-density limit and (b) for a hole at half-filling.
is shown in Fig. 2(b). The E ( k ) curve is completely different from that for an electron in the low-density limit; instead, however, it is very similar to that for a hole in the half-filled t - J model [10]. In order to study the effect of K and J on E ( k ) , an energy difference, E(0, 0 ) - E ( ~ / 2 , "rr/2), which corresponds to the band width W of the lowest energy branch, is plotted in Fig. 3 as function of J for several values of K. The following features can be seen in Fig. 3. (1) The carrier becomes more mobile with increasing value of I K I. The band width W / t , however, tends to saturate near 0.5 with further 0.4 K / t =-16
0.2 i 0 0
LLI
J/t Fig. 3. Calculated results of E(0, 0 ) - E ( ' r r / 2 , 7 r / 2 ) for a hole at half-filling as a function of J / t with several values of K / t .
increase of K around J / t ~ 1, although the result is not shown explicitly in Fig. 3. (2) For smaller values of J / t , E(0, 0 ) - E(-rr/2, r r / 2 ) is negative. (3) With increasing J / t , W / t increases till J / t ~ 1.0, while it decreases with further increase of J / t . The features above can be easily interpreted by considering the hopping process of the hole in the system. An example of the process is shown in Fig. l(b). As the hole hops to a n.n. site, there remains an antiparallel alignment of the itinerant electron spin and localized spin on the trace of the hole. The itinerant electron spin is overtumed by the transverse component of K S i • s i to recover the same spin state with the initial antiferromagnetic state. By this process, the localized spin is overturned, but it can be remedied by the transverse component of JSi. S ~, and the background spin state can be the same as the initial antiferromagnetic state. Repeating these processes, the hole can hop further and further. Thus, the larger values of [K [ and J are favorable for the motion of the hole. Actually, without K, the hole can hardly move. However, the longitudinal component of KSi. s i and JSi. S i are the barriers for the motion of the hole, and thereby W / t saturates or decreases with further increase of [K I/t and J / t . The effect of the barrier is stronger for the J term than for the K term because the number of spins contributing to the barrier is more in the J term than
S. Akazawa et al./ Physica C 263 (1996) 142-145
in the K term. There is another process for the hole to hop, i.e., one and a half rotations around a plaquett. Because this process makes k = (0, 0) the ground state, E(0, 0) - E(Ir/2, ~r/2) becomes negative for small values of J / t , as in the t - J model. The results obtained in the present model are almost the same as those in the t - J model except for the role of K. The Hund coupling makes the band width W smaller than that in the t-J model, that is, W ~ 0.5J at most in the present model, while W ~ 2 J in the t-J model. In conclusion, the dynamics of a mobile carrier in the antiferromagnetic state with Hund coupling is strongly dependent on whether the carrier is an electron in the low-density limit or a hole in the half-filled state. In the former case, E ( k ) is more or less free-electron-like, even for a large value of [ K I. This is because the S z = 0 state of the local triplet state can contribute to the hopping of the electron. In the latter case, however, the shape of E ( k ) is almost the same as that for a hole in the half-filled t - J model. Strong electron correlations between itinerant electrons, the Hund coupling and the exchange interaction between localized spins play important roles in the dynamics of the hole in this case.
145
Acknowledgements The numerical calculations were performed at the Computer Center of Nagoya University. One of us (SA) expresses great thanks for support from the Computer Center of Nagoya University. This work has been supported by a Grant-in-Aid for Scientific Research on Priority Areas from the Ministry of Education, Science and Culture of Japan.
References [1] J.G. Bednorz and K.A. Miiller, Z. Phys. B 64(1986) 189. [2] See, e.g., Physics of High-Temperature Superconductors, eds. S. Maekawa and M. Sato, Springer Solid Sta~ Phys. 106 (1991). [3] J. Inoue, S. Akazawa and S. Maekawa, Physica B 186-188 (1993) 956. [4] E.O. Wollen and W.C. Koehler, Phys. Rev. I00 (1955) 545. [5] P.-G. de Gennes, Phys. Rev. 118 (1960) 141. [6] J. Inoue and S. Maekawa, Phys. Rev. Lett. 74 (1995) 3407. [7] Y. Tokura, A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. Kido and N. Furukawa, J. Phys. Soc. Jpn. 63 (1994) 3931. [8] N. Furukawa, J. Phys. Soc. Jpn. 63 (1994) 3214. [9] A.J. Millis, P.B. Littlewood and B.I. Shraiman, Phys. Rev. Lett. 74 (1995) 5144. [10] J. lnoue and S. Maekawa, J. Phys. Soc. Jpn. 59 (1990) 2110.
Physica C 263 (1996) 142-145
Dynamics of a carrier in the antiferromagnetic state with strong Hund coupling S. Akazawa "'*, J. Inoue b, S. Maekawa b a Kagoshima National College of Technology, Aira-Gun. Kagoshima 899-51, Japan b Department of Applied Physics. Nagoya University, Nagoya 464-01. Japan
Abstract The dynamics of a single carrier added into the 2-dimensional antiferromagnetic state is studied in a model where a strong Hund coupling between localized spins and itinerant electrons coexists and strong electron correlations are also included for itinerant electrons. When the carrier is a hole added into the half-filling band of the itinerant electrons, the lowest energy state occurs at k = ( a t / 2 , -rr/2), while it occurs at k = (0, 0) when the carrier is an electron in the low-density limit. The difference between these lowest energy states originates from the strong electron correlations between the itinerant electrons at half-filling.
Perovskite-type oxides are systems of strong electron correlations and show various interesting phenomena when mobile carriers are introduced. Some Cu-oxides, for example (La-Sr)2CuO 4, show the high-T~ superconductivity [1] by introducing holes into the antiferromagnetic insulating phase of La2CuO 4. Various models, e.g., the Hubbard model, t-J model and spin-fermion model [2,3], have been used to understand the low-energy excitations in the doped Cu-oxides. Another example is Lal_xSr xMnO3, in which the antiferromagnetic insulating phase at x = 0 changes to a metallic ferromagnetic state for 0.2 < x < 0.5, and again to the antiferromagnetic insulating phase for larger x [4]. In these systems, a strong Hund coupling between localized t2g electrons and itinerant eg electrons plays an
* Corresponding author. Fax: +81 995 43 2584; e-mail: [email protected].
essential role in the physical properties [5], and the strong electron correlations between eg electrons themselves are also important [6]. Furthermore, the Mn-oxides have recently attracted considerable attention because of a colossal magnetoresistance near room temperature [7]. However, only a little theoretical work has been done so far [6,8,9]. In order to understand the unusual properties of doped insulators, it is important to clarify how the carriers introduced in the insulating phases move under the strong electron correlations. In this work, we focus our attention on a problem of the motion of carriers in the antiferromagnetic state with strong Hund coupling, using a model where itinerant electrons and localized spins coexist (spin-fermion model). We will calculate the energy momentum relation E(k) of a single mobile carrier using a variational method [3,10] and study the effect of the Hund coupling on E(k). As for the mobile carrier, we consider two cases: one is a hole added at the
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All fights reserved SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 7 1 2- I
S. Akazawa et al./ Physica C 263 (1996) 142-145
half-filled band of the itinerant electrons with an antiferromagnetic order, and the other is an electron in the low-density limit also with the antiferromagnetic background of the localized spins. The Hamiitonian we use is given by
H=-,
E
E
(i,j)
143
(a)
t t ~ t ~t
.... >
t~
t t ~t
..... >
t ~t
~t
f
i
S,.s
f ~ f xf f (b)
+gESi.si,
(l)
i
where the first, second and third terms denote hopping of itinerant electrons (I > O) between the nearest neighbor (n.n.) sites i and j, antiferromagnetic n.n. exchange interaction ( J > O) between localized spins and the Hund coupling ( K < O) between the localized spin S i and itinerant electron spin s i, respectively. ~ ( ~ i ~ ) is a creation (annihilation) operator of an electron on site i with spin o~ with a constraint that no double occupancy of the same site is permitted. (The constraint is irrelevant for a single 1 electron in the low-density limit.) We assume S = for the localized spins and a 2D square lattice for simplicity. We use a variational method where the Hilbert space is restricted. The variational function is given by a linear combination of many basis vectors which represent the intermediate state in the motion of the carrier starting from the initial state just added in the antiferromagnetic background. Examples of the hopping process are shown in Fig. l(a) for a single electron in the low-density limit and (b) for a single hole at half-filling. We prepare 147 basis vectors for a single electron carrier and 205 for a single hole. The lowest energy branch of E(k) calculated for a single electron in the low-density limit with J / t = 0.7 and K / t = - 10 is shown in Fig. 2(a). Here, the lattice constant is taken to be unity and the origin of E(k) is taken at the energy of the initial state at which the electron is added in the antiferromagnetic state. The shape of E(k) around k = (0, 0) is similar to that of a free electron and the lowest energy appears at k = (0, 0). The results are almost independent of the number of basis vectors and only weakly dependent on K and J. The results can be interpreted in the following way. When I K I < < 4t, the electron can easily hop, irrespective of the potential barrier due to the Hund coupling. When I K I is large, the spins of the itinerant electron and localized
t
~ 0 ~ t .....> t
~ ~ 0 t ..... > f
~ f 0 t x
f ~ t
~ o < ..... f ~ f ~ o < - .... t x
x
~ t
t o
x
Fig. 1. Examples of the hopping process (a) for a single electron in the low-density limit and (b) for a hole (shown by a circle) in the half-filled state. Arrows in the upper and lower columns denote the itinerant electron spins and localized spins, respectively. The crosses denote that the indicated spins are overturned ones.
spin on the same site form the triplet or singlet states, the energies of which are K / 4 and - 3 K / 4 , respectively. Let us assume that an 1' spin electron is initially added on a site with a localized 1' spin. When the 1" spin electron hops to a n.n. site, the spin of the electron and the localized spin are antiparallel in the sense of the classical spin picture. However, by using the state of S z = 0 of the local triplet state, the itinerant electron can hop to further sites without any loss of kinetic energy. Such a process is depicted in Fig. l(a), from which we can see that the number of intermediate states for the hopping of the electron is few. This is the reason for the weak dependence of E(k) on the number of basis vectors. Because the electron leaves no overturned spins behind, J does not affect the motion of the electron, i.e., E(k) is almost independent of J. The dynamics of the electron in the present case with large values of I K I is different from that with positive value of K [3]. In the latter case, with increasing K, the electron spin forms a singlet with the localized spin and the dynamics of the singlet state is nothing but the dynamics of a hole in the half-filled t-J model. The lowest energy branch of E(k) calculated for a hole at half-filling with J / t = 0.7 and K / t = - 10
144
S. Akazawa et al. / Physica C 263 (1996) 142-145 0
0
(a)
- 1 .5
(b)
-1
2
-2
a
-3
-4 (O,O)
- I .6
r4-t .7
-1.7
-1.8
-1.8
-19
-1.9 (IT,0)
(]1/2 ,IT/2 )
(0,O)
~0,0)
(~,0)
(~/2,~/21
(O,O)
k
k
Fig. 2. Calculated results of the lowest-energy dispersion curve with J / t = 0.7 and K / t = - 10 (a) for an electron in the low-density limit and (b) for a hole at half-filling.
is shown in Fig. 2(b). The E ( k ) curve is completely different from that for an electron in the low-density limit; instead, however, it is very similar to that for a hole in the half-filled t - J model [10]. In order to study the effect of K and J on E ( k ) , an energy difference, E(0, 0 ) - E ( ~ / 2 , "rr/2), which corresponds to the band width W of the lowest energy branch, is plotted in Fig. 3 as function of J for several values of K. The following features can be seen in Fig. 3. (1) The carrier becomes more mobile with increasing value of I K I. The band width W / t , however, tends to saturate near 0.5 with further 0.4 K / t =-16
0.2 i 0 0
LLI
J/t Fig. 3. Calculated results of E(0, 0 ) - E ( ' r r / 2 , 7 r / 2 ) for a hole at half-filling as a function of J / t with several values of K / t .
increase of K around J / t ~ 1, although the result is not shown explicitly in Fig. 3. (2) For smaller values of J / t , E(0, 0 ) - E(-rr/2, r r / 2 ) is negative. (3) With increasing J / t , W / t increases till J / t ~ 1.0, while it decreases with further increase of J / t . The features above can be easily interpreted by considering the hopping process of the hole in the system. An example of the process is shown in Fig. l(b). As the hole hops to a n.n. site, there remains an antiparallel alignment of the itinerant electron spin and localized spin on the trace of the hole. The itinerant electron spin is overtumed by the transverse component of K S i • s i to recover the same spin state with the initial antiferromagnetic state. By this process, the localized spin is overturned, but it can be remedied by the transverse component of JSi. S ~, and the background spin state can be the same as the initial antiferromagnetic state. Repeating these processes, the hole can hop further and further. Thus, the larger values of [K [ and J are favorable for the motion of the hole. Actually, without K, the hole can hardly move. However, the longitudinal component of KSi. s i and JSi. S i are the barriers for the motion of the hole, and thereby W / t saturates or decreases with further increase of [K I/t and J / t . The effect of the barrier is stronger for the J term than for the K term because the number of spins contributing to the barrier is more in the J term than
S. Akazawa et al./ Physica C 263 (1996) 142-145
in the K term. There is another process for the hole to hop, i.e., one and a half rotations around a plaquett. Because this process makes k = (0, 0) the ground state, E(0, 0) - E(Ir/2, ~r/2) becomes negative for small values of J / t , as in the t - J model. The results obtained in the present model are almost the same as those in the t - J model except for the role of K. The Hund coupling makes the band width W smaller than that in the t-J model, that is, W ~ 0.5J at most in the present model, while W ~ 2 J in the t-J model. In conclusion, the dynamics of a mobile carrier in the antiferromagnetic state with Hund coupling is strongly dependent on whether the carrier is an electron in the low-density limit or a hole in the half-filled state. In the former case, E ( k ) is more or less free-electron-like, even for a large value of [ K I. This is because the S z = 0 state of the local triplet state can contribute to the hopping of the electron. In the latter case, however, the shape of E ( k ) is almost the same as that for a hole in the half-filled t - J model. Strong electron correlations between itinerant electrons, the Hund coupling and the exchange interaction between localized spins play important roles in the dynamics of the hole in this case.
145
Acknowledgements The numerical calculations were performed at the Computer Center of Nagoya University. One of us (SA) expresses great thanks for support from the Computer Center of Nagoya University. This work has been supported by a Grant-in-Aid for Scientific Research on Priority Areas from the Ministry of Education, Science and Culture of Japan.
References [1] J.G. Bednorz and K.A. Miiller, Z. Phys. B 64(1986) 189. [2] See, e.g., Physics of High-Temperature Superconductors, eds. S. Maekawa and M. Sato, Springer Solid Sta~ Phys. 106 (1991). [3] J. Inoue, S. Akazawa and S. Maekawa, Physica B 186-188 (1993) 956. [4] E.O. Wollen and W.C. Koehler, Phys. Rev. I00 (1955) 545. [5] P.-G. de Gennes, Phys. Rev. 118 (1960) 141. [6] J. Inoue and S. Maekawa, Phys. Rev. Lett. 74 (1995) 3407. [7] Y. Tokura, A. Urushibara, Y. Moritomo, T. Arima, A. Asamitsu, G. Kido and N. Furukawa, J. Phys. Soc. Jpn. 63 (1994) 3931. [8] N. Furukawa, J. Phys. Soc. Jpn. 63 (1994) 3214. [9] A.J. Millis, P.B. Littlewood and B.I. Shraiman, Phys. Rev. Lett. 74 (1995) 5144. [10] J. lnoue and S. Maekawa, J. Phys. Soc. Jpn. 59 (1990) 2110.