- Email: [email protected]
Effect of external magnetic field on the occupation probabilities of magneticsuperconductors
Solid State Communications 158 (2013) 13–15
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effect of external magnetic field on the occupation probabilities of magnetic superconductors Brundabana Pradhan n Department of Physics, Government Science College, Malkangiri, Orissa 764048, India
a r t i c l e i n f o
abstract
Article history: Received 2 December 2012 Accepted 16 December 2012 by C. Lacroix Available online 3 January 2013
We present a theoretical model study for the coexistence of superconductivity (SC) and spin density wave (SDW) for high-Tc cuprates in the underdoped region before the onset of the superconductivity in the system. The analytic expressions for the temperature dependence of the SC and SDW order parameters are derived and solved self-consistently. It is observed that in the interplay region both the gap parameters exhibit very strong dependence of their gap values. The effect of the external magnetic field on the occupation probabilities for both itinerant and delocalized electrons is studied. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. High-Tc cuprate superconductors D. Spin density waves
1. Introduction The parent compounds of high-Tc cuprate superconductors are antiferromagnetic (AFM) insulators. Since the discovery of such superconductors, the understanding of the electronic properties would be a key to address the mechanism of it. The electronic structure as well as the orbital character of the doped carriers is a key ingredient for understanding the physics of the cuprates and the mechanism of high-temperature superconductivity. Both theoretical [1–3] and experimental [4–6] studies are going on to know the pairing mechanism and phase diagram of antiferromagnetism and superconductivity (SC) in high-Tc cuprate superconductors.
2. Model and calculations For the calculation of the coexistence of SC and SDW gaps in the underdoped region of the cuprate systems, one has to describe the antiferromagnetic (AFM) ordering of the two-dimensional correlated systems with the Hubbard model on a square lattice in the presence of electron–phonon interaction. In the presence of the external magnetic field, the mean-field Hamiltonian for a longitudinal SDW and SC state is described as H¼
X X ðEk sBÞðcyk, s ck, s Þ þ Ds sðcyk, s ck þ Q s Þ k, s
Dsc
k
n
Tel.: þ91 9437806565. E-mail address: [email protected]
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.12.025
are the effective attractive interaction in BCS limit and the repulsive interaction, respectively. Using Zubarev’s technique of double time single particle Green’s function [7], Green’s functions Aðk, oÞ describing electron–hole pair of up-spin electrons and Bðk, oÞ describing the electron–hole pair of down-spin electrons are calculated as given in [8]. The fourfold band energies 7 oa (a ¼ 124) are observed in the presence of the external magnetic field B as 7 o1 ¼ 8ðBE1k Þ, 7 o2 ¼ 8 ðB þE1k Þ, 7 o3 ¼ 8ðBE2k Þ and 7 o4 ¼ 8 ðB þ E2k Þ, where the quasi-particle bands E1,2k are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by E1,2k ¼ E2k þ ðDs 8 Dsc Þ2 . The SC and SDW order parameters are calculated to be Z g oD Dsc ¼ dE ½F 1 ðTÞF 2 ðTÞ 2 o D k
ð2Þ
and
Ds ¼ g 1
Z
W=2
dEk ½F 1 ðTÞ þ F 2 ðTÞ
ð3Þ
W=2
k, s
X y y ðckm ckk þ ckk ckm Þ
here the conduction electron dispersion is given by Ek ¼ 2t 0 ðcos kx þcos ky Þ in the tight binding approximation, with t0 being the nearest neighbour hopping integral. The external magnetic field B ¼ g J mB H, where gJ and mB are the Lande g-factor for conduction electrons and Bohr magnetron, respectively, and s is þ1 for up-spin and 1 for down-spin electrons. P P Dsc ¼ k V~0 /cykm cykk S and Ds ¼ k, s sV~1 /cyks ck þ Q s S define the SC and SDW gap parameters, respectively, where V~0 and V~1
ð1Þ
respectively with the SC coupling g ¼ Nð0ÞV 0 and the SDW coupling g 1 ¼ Nð0ÞV 1 and Nð0Þ being the density of states (DOS) of conduction electrons and the integration is carried out within the cut-off energy oD . The functions F 1 ðTÞ and F 2 ðTÞ are given by Ds Dsc ff ðo1 Þf ðo2 Þg F 1 ðTÞ ¼ o1 o2
14
B. Pradhan / Solid State Communications 158 (2013) 13–15
and
Ds þ Dsc
o3 o4
0.008
ff ðo3 Þf ðo4 Þg
where f ðoÞ represents the Fermi function corresponding to the band energy o. The effective order parameters D1,2 ¼ ðDs 8 Dsc Þ indicate that they can influence each other either constructively or destructively. Beyond these there may also arise the superconducting gaps D1 and D2 due to the following two types of pairings, defined as X V~0 ½/cyk þ Q m cykk S þ/cyk þ Q k cykm S D1 ¼ ð4Þ
z
b=0 b=0.00092 b=0.00185 b=0.00278
0.006
z, z1
F 2 ðTÞ ¼
z1
0.004
0.002
k
and
D2 ¼
X V~0 ½/cykm cyðk þ Q Þk S þ/cykk cyðk þ Q Þm S
0
ð5Þ
0
0.002
0.004
0.006
k
In performing numerical calculations, the bandwidth of conduction band is taken to be W ¼ 8t 0 , where t0 is the hopping integral and all the quantities entering in Eqs. (2) and (3) are scaled by 2t 0 . Thus, the non-dimensional parameters are the SC gap parameter z ¼ Dsc =2t 0 , the SDW gap parameter z1 ¼ Ds =2t 0 , reduced temperature t ¼ kB T=2t 0 and the magnetic field b ¼ B=2t 0 , the SC coupling constant g ¼ Nð0ÞV 0 and the SDW coupling constant g 1 ¼ Nð0ÞV 1 . The interplay of SC gap z and SDW gap z1 is studied [8]. In the absence of the external magnetic field, it is observed that the reduced SC gap parameter is 2Dsc ð0Þ= kB T c C 3:56 which is comparable to the universal BCS value of 3.52. But in the presence of the external magnetic field the reduced SC gap parameter 2Dsc ð0Þ=kB T c reduces to C 1:456, i.e., the SC gap is suppressed more in the coexistence phase of both the long range orders. Here we present the effect of the external magnetic field on the occupation probabilities of both localized and delocalized electrons involved in the system. Fig. 1 shows the study of spin polarization on the SC and SDW long range orders in the applied external magnetic field parameter b ¼ 0, 0:00092, 0:00185, 0:00278 (i.e., B¼0 T, 4 T, 8 T, 12 T). On the application of the external field, the SC gap is enhanced in the coexistence phase but suppressed in pure superconducting phase for t 4 t s due to the Cooper pairing as expected from the metallic superconductors. However, the SDW gap is suppressed throughout the temperature range on the application of the external magnetic field. Furthermore, two types of SC pairings occur in the SDW states as described in Eqs. (4) and (5). The self-consistent plots for those gaps z1 ¼ D1 =2t 0 and z2 ¼ D2 =2t 0 are shown in Fig. 2. Both the gap values show symmetric properties with z1 positive gap and z2 negative gap which satisfies that D1 þ D2 ¼ 0 and both are suppressed together with the increase in temperature. The application of the external magnetic field suppresses both the gaps throughout the temperature range. The occupation probabilities of the electrons and the effect of the external magnetic field are shown in Figs. 3–5. Fig. 3 shows the effect of the external magnetic field on the occupation probability of itinerant electrons. In the absence of the
2e-07
z
1e-07
z1, z2
3. Results and discussion
0.01
Fig. 1. (Color online) The self-consistent plots of SC gap (z) and SDW gap ðz1 Þ vs. reduced temperature (t) for fixed value of SC coupling g¼ 0.058 and SDW coupling g 1 ¼ 0:028 and for different values of the external magnetic field parameter b ¼ 0, 0:00092, 0:00185, 0:00278.
b=0 b=0.00092 b=0.00185 b=0.00278
1
0
z
-1e-07
2
-2e-07 0
0.002
0.004
0.006
0.008
0.01
t Fig. 2. (Color online) The self-consistent plots of SC gaps ðz1 Þ and ðz2 Þ vs. reduced temperature (t) for different values of the external magnetic field parameter b ¼ 0, 0:00092, 0:00185, 0:00278. 0.504 b=0
n
c
0.502
Occupation Number
respectively and these gaps are also derived from Green’s functions defined earlier [8]. The itinerant electron occupation number ncs in the eg band is P defined as ncs ¼ ð1=NÞ ks /cyks cks S. The delocalized electron occuP pation in the SDW state is defined as nQs ¼ ð1=NÞ ks y /ck þ Q s ck þ Q s S and the net occupation probability of itinerant and delocalized electrons is ns ¼ ncs þnQs : The self-consistent equations are derived from single particle Green’s functions as calculated in our previous paper [8].
0.008
t
0.5
n
0.498
c
n
0.496
0
0.002
0.004
0.006
c
0.008
0.01
t Fig. 3. (Color online) (a) The plots of occupation probabilities of conduction electrons for both up spin and down spin. (b) The occupation probability of conduction electrons in the absence of the external magnetic field and (c) the same in the presence of the external magnetic fields.
external field the conduction electrons show a single band just above 0.5 at zero temperature and falls towards 0.5 with the increase in temperature as in the inset (b). On the application of
B. Pradhan / Solid State Communications 158 (2013) 13–15 0.504 b=0 b=0.00092 b=0.00185 b=0.00278
b=0
n
q
Occupation Number
0.502
0.5
n
0.498
q
n
0.496
0
0.002
0.004
0.006
q
0.008
0.01
t Fig. 4. (Color online) (a) The plots of occupation probabilities of delocalized electrons for both up spin and down spin. (b) The occupation probability of delocalised electrons in the absence of the external magnetic field and (c) the same in the presence of the external magnetic fields.
1.006
b=0 b=0.00092 b=0.00185 b=0.00278
Occupation Number
1.004
15
absence of the external field the occupation probabilities show a single band just below 0.5 at zero temperature and comes close towards 0.5 with the increase in temperature as in the inset (b). Further, the single band splits into two bands corresponding to spin up and spin down as the external field is applied to the system. Again the spin up band is raised up from the mean level and the spin down band is lowered down from the mean level. The whole level also raised with the increase in temperature on the effect of the external field as in the inset (c). When conduction and delocalized electron occupations are added together it shows that the net probability is one as shown in Fig. 5. And the external magnetic field splits the single band into two bands with spin up band raising up and the spin down band lowering down. From the inset (b) it is observed that the levels are splitted even from zero of the temperature which was not seen in the single case. With these observations the Zeeman splitting can be explained.
4. Conclusion In this report we present an extension calculation of our previous work [8] and communicate some results as further study. In the presence of the external magnetic field the interplay of superconductivity and spin density wave gaps is studied. In the coexistence phase, the external magnetic field suppresses the SDW gap throughout the temperature range and the SC gap is enhanced. Beyond the SDW transition temperature the SC gap is suppressed and its transition temperature is decreased. When the external magnetic field is applied, the energy bands exhibit Zeeman splitting.
n
1.002 1 0.998 n
0.996
Acknowledgement
n
0.994 0
0.002
0.004
0.006
0.008
0.01
t Fig. 5. (Color online) (a) The plots of occupation probabilities of both conduction and delocalized electrons for both up spin and down spin. (b) The occupation probability in the absence of the external magnetic field and (c) the same in the presence of the external magnetic fields.
the external magnetic field, the single band of itinerant electrons splits into two bands corresponding to spin up and spin down. The spin up band is raised up from the mean level and the spin down band is lowered down from the mean level. The whole level also lowers with the increase in temperature on the effect of the external field as in the inset (c). Fig. 4 shows the effect of the external magnetic field on the occupation probability of delocalized electrons. Here in the
The author gratefully acknowledges the research facilities offered by the Institute of Physics, Bhubaneswar, India during his short stay.
References [1] D. Senechal, P.-L. Lavertu, M.-A. Marois, A.-M.S. Tremblay, Phys. Rev. Lett. 94 (2005) 156404. [2] M.G. Zacher, W. Hanke, E. Arrigoni, S.-C. Zhang, Phys. Rev. Lett. 85 (2000) 824. [3] B.K. Sahoo, B.N. Panda, Physica C 470 (2010) 547. [4] S. Shimizu, S.-I. Tabata, H. Mukuda, Y. Kitaoka, Phys. Rev. B 83 (2011) 214514. [5] S. Shimizu, H. Mukuda, Y. Kitaoka, H. Kito, Y. Kodama, P.M. Shirage, A. Iyo, J. Phys. Soc. Jpn. 78 (2009) 064075. [6] V.V. Eremenko, V.N. Samovarov, V.L. Vakula, M.Yu. Libin, S.A. Uyutnov, Low Temp. Phys. 26 (2000) 809. [7] D.N. Zubarev, Sov. Phys. Usp. 3 (1960) 320. [8] B. Pradhan, B.K. Raj, G.C. Rout, Physica C 469 (2009) 721.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effect of external magnetic field on the occupation probabilities of magnetic superconductors Brundabana Pradhan n Department of Physics, Government Science College, Malkangiri, Orissa 764048, India
a r t i c l e i n f o
abstract
Article history: Received 2 December 2012 Accepted 16 December 2012 by C. Lacroix Available online 3 January 2013
We present a theoretical model study for the coexistence of superconductivity (SC) and spin density wave (SDW) for high-Tc cuprates in the underdoped region before the onset of the superconductivity in the system. The analytic expressions for the temperature dependence of the SC and SDW order parameters are derived and solved self-consistently. It is observed that in the interplay region both the gap parameters exhibit very strong dependence of their gap values. The effect of the external magnetic field on the occupation probabilities for both itinerant and delocalized electrons is studied. & 2013 Elsevier Ltd. All rights reserved.
Keywords: A. High-Tc cuprate superconductors D. Spin density waves
1. Introduction The parent compounds of high-Tc cuprate superconductors are antiferromagnetic (AFM) insulators. Since the discovery of such superconductors, the understanding of the electronic properties would be a key to address the mechanism of it. The electronic structure as well as the orbital character of the doped carriers is a key ingredient for understanding the physics of the cuprates and the mechanism of high-temperature superconductivity. Both theoretical [1–3] and experimental [4–6] studies are going on to know the pairing mechanism and phase diagram of antiferromagnetism and superconductivity (SC) in high-Tc cuprate superconductors.
2. Model and calculations For the calculation of the coexistence of SC and SDW gaps in the underdoped region of the cuprate systems, one has to describe the antiferromagnetic (AFM) ordering of the two-dimensional correlated systems with the Hubbard model on a square lattice in the presence of electron–phonon interaction. In the presence of the external magnetic field, the mean-field Hamiltonian for a longitudinal SDW and SC state is described as H¼
X X ðEk sBÞðcyk, s ck, s Þ þ Ds sðcyk, s ck þ Q s Þ k, s
Dsc
k
n
Tel.: þ91 9437806565. E-mail address: [email protected]
0038-1098/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.12.025
are the effective attractive interaction in BCS limit and the repulsive interaction, respectively. Using Zubarev’s technique of double time single particle Green’s function [7], Green’s functions Aðk, oÞ describing electron–hole pair of up-spin electrons and Bðk, oÞ describing the electron–hole pair of down-spin electrons are calculated as given in [8]. The fourfold band energies 7 oa (a ¼ 124) are observed in the presence of the external magnetic field B as 7 o1 ¼ 8ðBE1k Þ, 7 o2 ¼ 8 ðB þE1k Þ, 7 o3 ¼ 8ðBE2k Þ and 7 o4 ¼ 8 ðB þ E2k Þ, where the quasi-particle bands E1,2k are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi given by E1,2k ¼ E2k þ ðDs 8 Dsc Þ2 . The SC and SDW order parameters are calculated to be Z g oD Dsc ¼ dE ½F 1 ðTÞF 2 ðTÞ 2 o D k
ð2Þ
and
Ds ¼ g 1
Z
W=2
dEk ½F 1 ðTÞ þ F 2 ðTÞ
ð3Þ
W=2
k, s
X y y ðckm ckk þ ckk ckm Þ
here the conduction electron dispersion is given by Ek ¼ 2t 0 ðcos kx þcos ky Þ in the tight binding approximation, with t0 being the nearest neighbour hopping integral. The external magnetic field B ¼ g J mB H, where gJ and mB are the Lande g-factor for conduction electrons and Bohr magnetron, respectively, and s is þ1 for up-spin and 1 for down-spin electrons. P P Dsc ¼ k V~0 /cykm cykk S and Ds ¼ k, s sV~1 /cyks ck þ Q s S define the SC and SDW gap parameters, respectively, where V~0 and V~1
ð1Þ
respectively with the SC coupling g ¼ Nð0ÞV 0 and the SDW coupling g 1 ¼ Nð0ÞV 1 and Nð0Þ being the density of states (DOS) of conduction electrons and the integration is carried out within the cut-off energy oD . The functions F 1 ðTÞ and F 2 ðTÞ are given by Ds Dsc ff ðo1 Þf ðo2 Þg F 1 ðTÞ ¼ o1 o2
14
B. Pradhan / Solid State Communications 158 (2013) 13–15
and
Ds þ Dsc
o3 o4
0.008
ff ðo3 Þf ðo4 Þg
where f ðoÞ represents the Fermi function corresponding to the band energy o. The effective order parameters D1,2 ¼ ðDs 8 Dsc Þ indicate that they can influence each other either constructively or destructively. Beyond these there may also arise the superconducting gaps D1 and D2 due to the following two types of pairings, defined as X V~0 ½/cyk þ Q m cykk S þ/cyk þ Q k cykm S D1 ¼ ð4Þ
z
b=0 b=0.00092 b=0.00185 b=0.00278
0.006
z, z1
F 2 ðTÞ ¼
z1
0.004
0.002
k
and
D2 ¼
X V~0 ½/cykm cyðk þ Q Þk S þ/cykk cyðk þ Q Þm S
0
ð5Þ
0
0.002
0.004
0.006
k
In performing numerical calculations, the bandwidth of conduction band is taken to be W ¼ 8t 0 , where t0 is the hopping integral and all the quantities entering in Eqs. (2) and (3) are scaled by 2t 0 . Thus, the non-dimensional parameters are the SC gap parameter z ¼ Dsc =2t 0 , the SDW gap parameter z1 ¼ Ds =2t 0 , reduced temperature t ¼ kB T=2t 0 and the magnetic field b ¼ B=2t 0 , the SC coupling constant g ¼ Nð0ÞV 0 and the SDW coupling constant g 1 ¼ Nð0ÞV 1 . The interplay of SC gap z and SDW gap z1 is studied [8]. In the absence of the external magnetic field, it is observed that the reduced SC gap parameter is 2Dsc ð0Þ= kB T c C 3:56 which is comparable to the universal BCS value of 3.52. But in the presence of the external magnetic field the reduced SC gap parameter 2Dsc ð0Þ=kB T c reduces to C 1:456, i.e., the SC gap is suppressed more in the coexistence phase of both the long range orders. Here we present the effect of the external magnetic field on the occupation probabilities of both localized and delocalized electrons involved in the system. Fig. 1 shows the study of spin polarization on the SC and SDW long range orders in the applied external magnetic field parameter b ¼ 0, 0:00092, 0:00185, 0:00278 (i.e., B¼0 T, 4 T, 8 T, 12 T). On the application of the external field, the SC gap is enhanced in the coexistence phase but suppressed in pure superconducting phase for t 4 t s due to the Cooper pairing as expected from the metallic superconductors. However, the SDW gap is suppressed throughout the temperature range on the application of the external magnetic field. Furthermore, two types of SC pairings occur in the SDW states as described in Eqs. (4) and (5). The self-consistent plots for those gaps z1 ¼ D1 =2t 0 and z2 ¼ D2 =2t 0 are shown in Fig. 2. Both the gap values show symmetric properties with z1 positive gap and z2 negative gap which satisfies that D1 þ D2 ¼ 0 and both are suppressed together with the increase in temperature. The application of the external magnetic field suppresses both the gaps throughout the temperature range. The occupation probabilities of the electrons and the effect of the external magnetic field are shown in Figs. 3–5. Fig. 3 shows the effect of the external magnetic field on the occupation probability of itinerant electrons. In the absence of the
2e-07
z
1e-07
z1, z2
3. Results and discussion
0.01
Fig. 1. (Color online) The self-consistent plots of SC gap (z) and SDW gap ðz1 Þ vs. reduced temperature (t) for fixed value of SC coupling g¼ 0.058 and SDW coupling g 1 ¼ 0:028 and for different values of the external magnetic field parameter b ¼ 0, 0:00092, 0:00185, 0:00278.
b=0 b=0.00092 b=0.00185 b=0.00278
1
0
z
-1e-07
2
-2e-07 0
0.002
0.004
0.006
0.008
0.01
t Fig. 2. (Color online) The self-consistent plots of SC gaps ðz1 Þ and ðz2 Þ vs. reduced temperature (t) for different values of the external magnetic field parameter b ¼ 0, 0:00092, 0:00185, 0:00278. 0.504 b=0
n
c
0.502
Occupation Number
respectively and these gaps are also derived from Green’s functions defined earlier [8]. The itinerant electron occupation number ncs in the eg band is P defined as ncs ¼ ð1=NÞ ks /cyks cks S. The delocalized electron occuP pation in the SDW state is defined as nQs ¼ ð1=NÞ ks y /ck þ Q s ck þ Q s S and the net occupation probability of itinerant and delocalized electrons is ns ¼ ncs þnQs : The self-consistent equations are derived from single particle Green’s functions as calculated in our previous paper [8].
0.008
t
0.5
n
0.498
c
n
0.496
0
0.002
0.004
0.006
c
0.008
0.01
t Fig. 3. (Color online) (a) The plots of occupation probabilities of conduction electrons for both up spin and down spin. (b) The occupation probability of conduction electrons in the absence of the external magnetic field and (c) the same in the presence of the external magnetic fields.
external field the conduction electrons show a single band just above 0.5 at zero temperature and falls towards 0.5 with the increase in temperature as in the inset (b). On the application of
B. Pradhan / Solid State Communications 158 (2013) 13–15 0.504 b=0 b=0.00092 b=0.00185 b=0.00278
b=0
n
q
Occupation Number
0.502
0.5
n
0.498
q
n
0.496
0
0.002
0.004
0.006
q
0.008
0.01
t Fig. 4. (Color online) (a) The plots of occupation probabilities of delocalized electrons for both up spin and down spin. (b) The occupation probability of delocalised electrons in the absence of the external magnetic field and (c) the same in the presence of the external magnetic fields.
1.006
b=0 b=0.00092 b=0.00185 b=0.00278
Occupation Number
1.004
15
absence of the external field the occupation probabilities show a single band just below 0.5 at zero temperature and comes close towards 0.5 with the increase in temperature as in the inset (b). Further, the single band splits into two bands corresponding to spin up and spin down as the external field is applied to the system. Again the spin up band is raised up from the mean level and the spin down band is lowered down from the mean level. The whole level also raised with the increase in temperature on the effect of the external field as in the inset (c). When conduction and delocalized electron occupations are added together it shows that the net probability is one as shown in Fig. 5. And the external magnetic field splits the single band into two bands with spin up band raising up and the spin down band lowering down. From the inset (b) it is observed that the levels are splitted even from zero of the temperature which was not seen in the single case. With these observations the Zeeman splitting can be explained.
4. Conclusion In this report we present an extension calculation of our previous work [8] and communicate some results as further study. In the presence of the external magnetic field the interplay of superconductivity and spin density wave gaps is studied. In the coexistence phase, the external magnetic field suppresses the SDW gap throughout the temperature range and the SC gap is enhanced. Beyond the SDW transition temperature the SC gap is suppressed and its transition temperature is decreased. When the external magnetic field is applied, the energy bands exhibit Zeeman splitting.
n
1.002 1 0.998 n
0.996
Acknowledgement
n
0.994 0
0.002
0.004
0.006
0.008
0.01
t Fig. 5. (Color online) (a) The plots of occupation probabilities of both conduction and delocalized electrons for both up spin and down spin. (b) The occupation probability in the absence of the external magnetic field and (c) the same in the presence of the external magnetic fields.
the external magnetic field, the single band of itinerant electrons splits into two bands corresponding to spin up and spin down. The spin up band is raised up from the mean level and the spin down band is lowered down from the mean level. The whole level also lowers with the increase in temperature on the effect of the external field as in the inset (c). Fig. 4 shows the effect of the external magnetic field on the occupation probability of delocalized electrons. Here in the
The author gratefully acknowledges the research facilities offered by the Institute of Physics, Bhubaneswar, India during his short stay.
References [1] D. Senechal, P.-L. Lavertu, M.-A. Marois, A.-M.S. Tremblay, Phys. Rev. Lett. 94 (2005) 156404. [2] M.G. Zacher, W. Hanke, E. Arrigoni, S.-C. Zhang, Phys. Rev. Lett. 85 (2000) 824. [3] B.K. Sahoo, B.N. Panda, Physica C 470 (2010) 547. [4] S. Shimizu, S.-I. Tabata, H. Mukuda, Y. Kitaoka, Phys. Rev. B 83 (2011) 214514. [5] S. Shimizu, H. Mukuda, Y. Kitaoka, H. Kito, Y. Kodama, P.M. Shirage, A. Iyo, J. Phys. Soc. Jpn. 78 (2009) 064075. [6] V.V. Eremenko, V.N. Samovarov, V.L. Vakula, M.Yu. Libin, S.A. Uyutnov, Low Temp. Phys. 26 (2000) 809. [7] D.N. Zubarev, Sov. Phys. Usp. 3 (1960) 320. [8] B. Pradhan, B.K. Raj, G.C. Rout, Physica C 469 (2009) 721.