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Quantum Monte Carlo study of the Hubbard model doped with nonmagnetic impurities
ARTICLE IN PRESS
Physica B 378–380 (2006) 315–316 www.elsevier.com/locate/physb
Quantum Monte Carlo study of the Hubbard model doped with nonmagnetic impurities Amal Medhi, Saurabh Basu, Charudatt Kadolkar Indian Institute of Technology Guwahati, Guwahati, India
Abstract We calculate the net magnetization for the a dimensional half-filled Hubbard model doped with one and multiple nonmagnetic impurities, using quantum Monte Carlo (QMC) method, to understand the suppression of antiferromagnetic order in strongly correlated electron systems such as La2CuO4 by doping nonmagnetic impurities, e.g. Zn. The QMC results show enhancement of magnetic correlations around an impurity, in agreement with the results obtained in previous studies. The effect of an impurity on the magnetic structure around it and the underlying mechanism is discussed. The situation with additional doping with holes is also examined by carrying out QMC simulation of the Hubbard model at band filling slightly less than half-filling. r 2006 Elsevier B.V. All rights reserved. PACS: 71.10.Hf; 71.27.þa; 75.30.Mb Keywords: Hubbard model; Nonmagnetic impurity
The copper oxide (CuO2) planes in high-T c superconductors have been accepted as having a key role in determining the magnetic and superconducting properties because of the strong electronic correlations [1]. In particular, studies involving doping with impurities and holes revealed interesting effects such as suppression of antiferromagnetic long-range order (AFLRO) by a small amount ð2%Þ of hole doping [2] or by a relatively larger amount ð30%Þ of nonmagnetic impurities (e.g. Zn) [3] in La2CuO4. Thus, it remained of continuing interest for a theoretical investigation of the impurity problem. In this report we employ the quantum Monte Carlo (QMC) method to study the effects of doping in a two-dimensional Hubbard model which satisfactorily explains the physics of the undoped compounds. We report calculations for sublattice magnetization mi ¼ hðni" ni# Þ2 i1=2 , spin–spin correlation function cði; jÞ ¼ hðni" Pni# Þðnj" nj# Þi and spin structure factor SðqÞ ¼ ð1=NÞ ij eiq:ðRi Rj Þ cði; jÞ for a square lattice of size 10 10, doped with nonmagnetic impurities at and away from half-filling. As for the QMC method, we use the Corresponding author. Tel.: +91 361 2582704; fax: +91 361 2690762.
E-mail address: [email protected] (C. Kadolkar). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.333
algorithm of finite temperature grand canonical ensemble simulation of the Hubbard model [4], taking the interaction as U ¼ 4 and inverse temperature b ¼ 6, in units of hopping integral t. The nonmagnetic impurities are incorporated in the P Hubbard Hamiltonian by adding an additional term V ¼ i0 V 0 ni0 , where i0 denote the impurity sites. We have taken the potential V 0 ¼ 20t. Owing to the large negative value of V 0 , the impurity sites are always doubly occupied (occupancy is found to be very close to 2), and hence they remain magnetically inert. At half-filling, with one nonmagnetic impurity the values for the sublattice magnetization mi are found to be 0, 0.882 ð0:003Þ and 0.867 ð0:002Þ at the impurity site, sites neighbouring the impurity and sites far away from the impurity, respectively, thus indicating a slight enhancement of the value near the impurity site. Qualitatively similar behaviour is seen for band filling hni ¼ 0:93. Figs. 1 and 2 show the nearest neighbour (nn) spin correlations around, and along x and diagonal directions with reference to the impurity site, respectively. The nn spin correlations are found to be enhanced in the vicinity of the impurity site which is in agreement with previous studies [5]. The enhancement decreases rapidly as one moves away from the impurity. Qualitatively similar
ARTICLE IN PRESS A. Medhi et al. / Physica B 378–380 (2006) 315–316
316
10
-0.35 10 x 10, U = 4, β = 6
10 x 10, = 1, U = 4, β = 6
= 0.93
8
-0.25 S (π, π)
nn spin correlation
-0.3
= 1.00
-0.2 -0.15
6 4
-0.1 2 -0.05 0
0 0
1
2
3
4
5
0
10
Distance of the 'Squares' from Impurity Fig. 1. cði; jÞ for i, j two nn spins situated on a ‘square’ centring the impurity site, as a function of its distance from the impurity for band filling 1.00 and 0.93. The points in the extreme right of the figures in each case show the corresponding values for the case of no impurity.
0.3 10 x 10, U = 4, β = 6 0.2 nn spin correlation
along diagonal direction
= 1.00 = 0.93
0.1 0 -0.1
along x direction
-0.2 -0.3 0
1
2 Distance from Impurity
3
20
30
40
50
Number of nonmagnetic impurities
4
Fig. 2. cði; jÞ for i, j two nn spin situated on a line along x direction and a line along diagonal direction, passing through the impurity site for band filling 1.00 and 0.93. The values on the x-axis denote the distance of the nn pairs from the impurity site. Again the points in the extreme right of the figures in each case show the corresponding values for the case of no impurity.
behaviour is observed for the hole-doped case. The origin of this effect can be understood on the basis of the fact that configurations with lower energy are favoured more. The four neighbours of the impurity experience restricted hopping because of the impurity. To compensate for this, a spin at any of these sites prefers to have its other three neighbouring sites not to be occupied by the same species of spin, so that it can minimize energy by hopping to those
Fig. 3. Sðp; pÞ as a function of number of impurities.
sites more frequently. This increases the spin correlations near the impurity. On the other hand, the increase in sublattice magnetization at sites near the impurity is because of the reduced probability of double occupancy at those sites. Fig. 3 shows the spin structure factor SðqÞ calculated for wave vector q ¼ ðp; pÞ, as a function of number of randomly positioned impurities. The initial rapid decrease of Sðp; pÞ with increasing number of impurities is due to the suppression of AFLRO. The order seems to get destroyed completely at around 35% of impurity concentration, which is in accordance with experiments [3]. The slow decrease beyond this concentration is due to the impurity-induced dilution of the lattice. Magnetic properties of the 2D Hubbard model at and away from half-filling, doped with nonmagnetic impurities are investigated. The origin of the enhancement of magnetic correlations near impurity is discussed. The amount of nonmagnetic impurity concentration required to destroy the AFLRO is estimated. One of us (A. M.) would like to thank CSIR, India for financial support under the Grant no. F.NO.9/731(022)/ 2003-EMR-.
References [1] [2] [3] [4] [5]
E. Dagotto, Rev. Mod. Phys. 66 (1994) 763. A.H. Takagi, et al., Phys. Rev. B 40 (1989) 2254. B.S.W. Cheong, et al., Phys. Rev. B 44 (1991) 9739. S.R. White, et al., Phys. Rev. B 40 (1989) 506. C.G.B. Martins, et al., Phys. Rev. Lett. 78 (1997) 3563.
Physica B 378–380 (2006) 315–316 www.elsevier.com/locate/physb
Quantum Monte Carlo study of the Hubbard model doped with nonmagnetic impurities Amal Medhi, Saurabh Basu, Charudatt Kadolkar Indian Institute of Technology Guwahati, Guwahati, India
Abstract We calculate the net magnetization for the a dimensional half-filled Hubbard model doped with one and multiple nonmagnetic impurities, using quantum Monte Carlo (QMC) method, to understand the suppression of antiferromagnetic order in strongly correlated electron systems such as La2CuO4 by doping nonmagnetic impurities, e.g. Zn. The QMC results show enhancement of magnetic correlations around an impurity, in agreement with the results obtained in previous studies. The effect of an impurity on the magnetic structure around it and the underlying mechanism is discussed. The situation with additional doping with holes is also examined by carrying out QMC simulation of the Hubbard model at band filling slightly less than half-filling. r 2006 Elsevier B.V. All rights reserved. PACS: 71.10.Hf; 71.27.þa; 75.30.Mb Keywords: Hubbard model; Nonmagnetic impurity
The copper oxide (CuO2) planes in high-T c superconductors have been accepted as having a key role in determining the magnetic and superconducting properties because of the strong electronic correlations [1]. In particular, studies involving doping with impurities and holes revealed interesting effects such as suppression of antiferromagnetic long-range order (AFLRO) by a small amount ð2%Þ of hole doping [2] or by a relatively larger amount ð30%Þ of nonmagnetic impurities (e.g. Zn) [3] in La2CuO4. Thus, it remained of continuing interest for a theoretical investigation of the impurity problem. In this report we employ the quantum Monte Carlo (QMC) method to study the effects of doping in a two-dimensional Hubbard model which satisfactorily explains the physics of the undoped compounds. We report calculations for sublattice magnetization mi ¼ hðni" ni# Þ2 i1=2 , spin–spin correlation function cði; jÞ ¼ hðni" Pni# Þðnj" nj# Þi and spin structure factor SðqÞ ¼ ð1=NÞ ij eiq:ðRi Rj Þ cði; jÞ for a square lattice of size 10 10, doped with nonmagnetic impurities at and away from half-filling. As for the QMC method, we use the Corresponding author. Tel.: +91 361 2582704; fax: +91 361 2690762.
E-mail address: [email protected] (C. Kadolkar). 0921-4526/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2006.01.333
algorithm of finite temperature grand canonical ensemble simulation of the Hubbard model [4], taking the interaction as U ¼ 4 and inverse temperature b ¼ 6, in units of hopping integral t. The nonmagnetic impurities are incorporated in the P Hubbard Hamiltonian by adding an additional term V ¼ i0 V 0 ni0 , where i0 denote the impurity sites. We have taken the potential V 0 ¼ 20t. Owing to the large negative value of V 0 , the impurity sites are always doubly occupied (occupancy is found to be very close to 2), and hence they remain magnetically inert. At half-filling, with one nonmagnetic impurity the values for the sublattice magnetization mi are found to be 0, 0.882 ð0:003Þ and 0.867 ð0:002Þ at the impurity site, sites neighbouring the impurity and sites far away from the impurity, respectively, thus indicating a slight enhancement of the value near the impurity site. Qualitatively similar behaviour is seen for band filling hni ¼ 0:93. Figs. 1 and 2 show the nearest neighbour (nn) spin correlations around, and along x and diagonal directions with reference to the impurity site, respectively. The nn spin correlations are found to be enhanced in the vicinity of the impurity site which is in agreement with previous studies [5]. The enhancement decreases rapidly as one moves away from the impurity. Qualitatively similar
ARTICLE IN PRESS A. Medhi et al. / Physica B 378–380 (2006) 315–316
316
10
-0.35 10 x 10, U = 4, β = 6
10 x 10,
8
-0.25 S (π, π)
nn spin correlation
-0.3
-0.2 -0.15
6 4
-0.1 2 -0.05 0
0 0
1
2
3
4
5
0
10
Distance of the 'Squares' from Impurity Fig. 1. cði; jÞ for i, j two nn spins situated on a ‘square’ centring the impurity site, as a function of its distance from the impurity for band filling 1.00 and 0.93. The points in the extreme right of the figures in each case show the corresponding values for the case of no impurity.
0.3 10 x 10, U = 4, β = 6 0.2 nn spin correlation
along diagonal direction
0.1 0 -0.1
along x direction
-0.2 -0.3 0
1
2 Distance from Impurity
3
20
30
40
50
Number of nonmagnetic impurities
4
Fig. 2. cði; jÞ for i, j two nn spin situated on a line along x direction and a line along diagonal direction, passing through the impurity site for band filling 1.00 and 0.93. The values on the x-axis denote the distance of the nn pairs from the impurity site. Again the points in the extreme right of the figures in each case show the corresponding values for the case of no impurity.
behaviour is observed for the hole-doped case. The origin of this effect can be understood on the basis of the fact that configurations with lower energy are favoured more. The four neighbours of the impurity experience restricted hopping because of the impurity. To compensate for this, a spin at any of these sites prefers to have its other three neighbouring sites not to be occupied by the same species of spin, so that it can minimize energy by hopping to those
Fig. 3. Sðp; pÞ as a function of number of impurities.
sites more frequently. This increases the spin correlations near the impurity. On the other hand, the increase in sublattice magnetization at sites near the impurity is because of the reduced probability of double occupancy at those sites. Fig. 3 shows the spin structure factor SðqÞ calculated for wave vector q ¼ ðp; pÞ, as a function of number of randomly positioned impurities. The initial rapid decrease of Sðp; pÞ with increasing number of impurities is due to the suppression of AFLRO. The order seems to get destroyed completely at around 35% of impurity concentration, which is in accordance with experiments [3]. The slow decrease beyond this concentration is due to the impurity-induced dilution of the lattice. Magnetic properties of the 2D Hubbard model at and away from half-filling, doped with nonmagnetic impurities are investigated. The origin of the enhancement of magnetic correlations near impurity is discussed. The amount of nonmagnetic impurity concentration required to destroy the AFLRO is estimated. One of us (A. M.) would like to thank CSIR, India for financial support under the Grant no. F.NO.9/731(022)/ 2003-EMR-.
References [1] [2] [3] [4] [5]
E. Dagotto, Rev. Mod. Phys. 66 (1994) 763. A.H. Takagi, et al., Phys. Rev. B 40 (1989) 2254. B.S.W. Cheong, et al., Phys. Rev. B 44 (1991) 9739. S.R. White, et al., Phys. Rev. B 40 (1989) 506. C.G.B. Martins, et al., Phys. Rev. Lett. 78 (1997) 3563.