Int. J. hem. Sci.: 8(3,, 473-5 PHOOIDUED SUPERODUIIY I UPRAES AUJ UWAL, S. L. KAKAI *a an M. L. KALRA Deartment of Physics, University ollege of Science, M. L. Sukhaia University, UDAIPUR 33 (RAJ. IDIA a Institute of echnology an Management,. H. o. 79, hilwara y-pass, hittor Roa, HILWARA 3 (RAJ. IDIA ASRA he hotoinuce suerconucting hase transition temerature in high- curates with two ban Hamiltonian has been investigate. It is shown to ossess two suerconucting gas. he suerconuctivity is cause by the interban reulsive electron-lattice an coulomb interaction. he hoto- excitation rouces a change of the hole oing ( an as a result, the shift of. he stuy of free energy, critical fiel an electronic secific heat base on this moel is also resente. Key wors: Green s function, an holes, Ga arameter, Secific heat, Free energy, ritical fiel, arrier concentration. P A S 74. z, 74.5 Fy, 74.7 h, 74.7 k IRODUIO It is of funamental imortance to exlain the oing eenence of the suerconucting roerties of high- curates. Recent exerimental investigations on high- curates having comositions in the semiconuctor regime, have shown light inuce changes, which are inicative of hoto-inuce suerconuctivity -3. In some cases, there is an increase of the critical temerature with raiation osage 4. he most interesting result is the growth of the absolute value of the iamagnetic moment almost linearly with raiation osage, which saturates beyon a certain stage 4. Yu et al. have shown that the onset of transient hoto-inuce suerconuctivity, at high excitation levels is a real henomenon. * Author for corresonence; E-mail: anuj_nuwal@yahoo.com; kalraml@yahoo.com; slkakani@yahoo.com
474 A. uwal et al.: Photoinuce Suerconuctivity. he Moel Hamiltonian he moel Hamiltonian has the form 5 - H H o H o H...( Where ( H...( ( H...(3 an H...(4 where an are momentum labels in the an bans, resectively with energies an ; μ is the common chemical otential. Each ban has its roer airing interaction an, while the air interchange between the two bans is assure by term. We have assume, an we efine the following quantities : μ μ (5 ow H in equation ( rea as H...(6
Int. J. hem. Sci.: 8(3, 475 We stuy the Hamiltonian ( with the Green s function technique an following the equation of motion metho. Green s functions In orer to stuy the hysical roerties, we efine the following normal an anomalous Green s functions 6-5 : (a G (, τ τ ( τ ( τ τ (b G (, τ τ ( τ ( τ τ (c f, τ τ ( τ ( τ τ ( f, τ τ ( τ ( τ τ (e f (, τ τ ( τ ( τ τ (f f (, τ τ ( τ ( τ...(7 τ hese Green s functions satisfy the following equations: ( ω, δ...(8, ( ω, δ...(9, ( ω,,...( ( ω,,...( ( ω,,...( ( ω,,...(3 ( ω δ,,...(4 ( ω δ,,...(5
476 A. uwal et al.: Photoinuce Suerconuctivity. o solve the above equations, we have assume ( an ( an hen ( (...(6 Finally, one obtains the Green s functions by solving coule equations (8 to (5 as : Green s function for -holes << >>, <<, >> Green s function for -holes << >>, <<, >> he orrelation Functions ( ω E P ( ω ( ω E P ( ω E ( ω ( ω E...(7...(8...(9...( We obtain the correlation functions for the Green s functions given by equations (7 an (8 as -4 -
Int. J. hem. Sci.: 8(3, 477 < > ( α α [ f α f ( α ] (...( Where α, α...( an f (α an f (α are Fermi functions. < > α P f ( α α ( α α [ f ( α f ( ]...(3 Similarly, correlation functions for Green s functions (9 an ( for holes are obtaine. One can efine the two suerconucting orer arameters relate to the correlation functions corresoning to Green s functions <<, - >> an <<, - >> for an holes, resectively. In a similar manner, free energy an critical magnetic fiel can also be efine relate to both an holes. Suerconucting Orer Parameters Ga arameter is the suerconucting orer arameter, which can be etermine self consistently from the ga equation. We obtaine the suerconucting orer arameter for an holes as below: ( For holes hω α α tanh tanh (...(4 ( α α ( For holes: In a similar manner, we can obtain the exression for suerconucting orer arameter for holes.
478 A. uwal et al.: Photoinuce Suerconuctivity. hω α α tanh tanh (...(5 ( α α We observe that exressions (4 an (5 reuces to stanar S 6. α an α...(6 Where α an α for holes is given by ( an for holes is given by (6. Using equations (4 an (5, one can stuy the behavior of suerconucting orer arameter with temerature for both; an holes. Deenences of Suerconucting Gas on emerature ( an arrier oncentration on ( One can stuy the eenence of on the hole concentration n h an chemical otential (μ for the system Ya u 3 O 7-x from this moel. he effective chemical otential (μ corresons to the average carrier concentration (n h. he maximum of (n h corresons to chemical otential (μ lying in the common region of both bans roughly in the mile between E o an E c. From two ban moel, one can stuy the eenence of the suerconucting transition temerature on the hole concentration etermine by chemical otential (μ by stuying: (a Deenence of chemical otential (μ on critical temerature. (b ariation of hole concentration (n h with. Deenence of hemical Potential (μ on ritical emerature We calculate the suerconucting transition temerature for E o < μ < E, In Matrix form, suerconucting orer arameter can be written as 5 -
Int. J. hem. Sci.: 8(3, 479 ( j j i i j G...(7 j Where i an j reresent an holes. here are two suerconucting gas for an holes in our interban moel. he exressions for the eenence of suerconucting gas at on the hole concentration can be erive. One can write the equations for suerconucting gas for an holes as follows - ( tanh...(8 k ( tanh...(9 k Where an are airing interaction of an bans, resectively, while the air interchange between the two bans is assure by the term. he quantity has been suose to be oerative an constant in the energy interval for higher ban an lower ban, keeing in min the integration ranges, the ga orer arameter satisfy the system. Since ~ <<, so an can be neglecte. On further simlification of equations (8 an (9, one obtains ( ( tanh k tanh k...(3 At, (, so equation (3 takes the form
48 A. uwal et al.: Photoinuce Suerconuctivity. tanh tanh ( ( k k We have assume that tanh tanh ( ( k k...(3 Deenence of Hole oncentration (n h on ritical emerature ( Konsin an coworkers 3,7- stuie the eenence of carrier concentration on. For stuying the oing eenence on the chemical otential, we have [3, ]- E E f E E f E E E ( ( ρ ρ...(3 Here f (E {ex [(E-μ/k ] } -, is the number of holes er cell an E is the with of the broa ban. Using the conition of electroneutrality, we obtain the equation for the chemical otential as 3 - k E k E k E E k k E k E μ μ μ μ ρ ρ ex ex log ex ex log...(33 Where is the total number of holes er cell. he relation connecting n h an reas as n h, where E ρ.
Int. J. hem. Sci.: 8(3, 48 Deenence of hemical Potential Shift (μ on Hole oncentration (n h We have calculate the hole concentration eenence of the chemical otential shift μ for our system Ya u 3 O 7-x with the same fitting arameters use in calculation of hole concentration. Accoring to Konsin an Sorkin, the seuoga E g inuces the shift of the chemical otential through the eenence of the ensity of states ρ on E g. As a result, the electron chemical otential shift with oing is obtaine as - he chemical otential equals - Where E μ ( n E ( n h μ h...(34 μ n μ ( n E ( n E (...(35 ( h h g h g μ ( nh nh ρ E ρe ( ρ ρ...(36 g ( nh h hcr.3 ( n n e at.9 n. 58...(37 he eenence of E g (n h is erive from electronic secific heat. We obtaine critical concentration for our system Ya u 3 O 7-x is n hcr.58 an E g ( ~.3 e. he calculate oing eenences of the chemical otential shifts μ(n h for our system Ya u 3 O 7-x is resente in Figure 5 an theoretical reicte values are shown in able 6. In uner-oe an otimally-oe regions, the chemical otential shift μ(n h is strongly suresse by the seuoga E g (n h u to n hcr.58. he arameters are moel eenent for the calculation of μ(n h an (n h, but the goo agreement of the theoretical an exerimental curves of (n h an μ(n h for Ya u 3 O 7-x is obtaine. h
48 A. uwal et al.: Photoinuce Suerconuctivity. As follows from our calculations at low temeratures, the shift μ(n h in Ya u 3 O 7-x is unaffecte by the seuoga. he influence of the seuoga E g ( on the suerconucting transition temerature is weak because the contributions of E g ( to the chemical otential μ an to the gas, are aroximately comensate for in the equation of. Physical Proerties of Suerconuctors Electronic Secific heat ( es For holes he electronic secific heat er atom of a suerconuctor is etermine. -5, We obtain es ( hω α ex( α { ex( α } ( α α ex( α α ex( α { ex( α } { ex( α }...(38 For holes Where α an α are given by equation (. as- Similarly, one can write the exression for electronic secific heat es for holes, es ( hω α ex( α { ex( α } ( α α ex( α α ex( α { ex( α } { ex( α }...(39 Where α an α are given by equation (6. he jums of secific heat at is obtaine as - es en 9.4 R k...(4
Int. J. hem. Sci.: 8(3, 483 Where R is given by R γ [ ρ γ ρ γ ] κ log Γ < μ < [( μ Γ ( Γ μ 4γ ( k π ] 4 ( μ Γ ( Γ μ 4γ ( k π κ log [ ] 3 Γ,4 γ κ log [( μ Γ3 ( Γ4 μ 4γ ( k π ] 4 ( μ Γ ( Γ μ 4γ ( k π κ log 3 4 [ ]...(4 With ρ ρ κ ρρ 4 he secific heat in the normal hase is obtaine as en π Rk...(4 3 Where R is given by - [ ρ ρ ] R Γ3 < μ < Γ, 4 One can easily show that the relative jum of the secific heat at equals en.43 R R...(43 R he ratio characterizes the ossible eviation of the relative jum of the secific R heat in the resent moel from the S one. Free Energy It is well establishe that the thermal roerties of a metal changes sharly as the temerature is allowe to ecrease through the transition temerature to a suerconucting
484 A. uwal et al.: Photoinuce Suerconuctivity. state. he entroy ecreases remarkably on cooling the suerconuctors below the critical temerature. he free energy can easily be efine for the suerconucting transition as it is relate by the entroy, hence it also exhibit a similar behaviour 3. Obviously, the entroy as well as the free energy ifference in the normal state is always greater than the entroy in the suerconucting state. ( ( F F S...(44 Where F s is free energy in suerconucting state an F is free energy in normal state an is the volume er unit cell an is airing interaction. One can write exressions for the free energy ifference for -holes an -holes searately. One obtains the final exression for free energy ifference as e e F F S ω ω h h log ( log ( 4...(45 With the hel of the above relation, we can calculate the free energy ifference for - holes. Similarly, one obtains the free energy ifference for holes as e e F F S ω ω h h log ( log ( 4...(46 Where α an α
Int. J. hem. Sci.: 8(3, 485 ritical Fiel (H he critical fiel H ( calle the thermoynamic fiel for the transition between the normal to suerconucting state an is relate to the free energy ifference is given as, F S F H c 8π...(47 Hence, the low temerature critical fiel is obtaine as - 8 F S F H c π...(48 One can obtain the relation for critical fiel for both tyes of holes. umerical alculations alues of various arameters aearing in equation are given in able. Using these values, we have mae stuy of various arameters for the system Y a u 3 O 7-x. able : alues of various arameters for Y a u 3 O 7-x Parameter *Suerconucting transition temerature ( c 88 K alue * Phonon energy ( h ω.6 J * Phonon energy ( h ω.3 J *Density of states at the Fermi surface ( 4.95 9 J atom * Pairing interaction for holes.3 9 J atom * Pairing interaction for holes.38 9 J atom *umber of electrons er unit cell ( n ~ 5 * oltzmann constant ( k.38 3 J
486 A. uwal et al.: Photoinuce Suerconuctivity. Suerconucting Orer Parameter ( o stuy the behavior of suerconucting orer arameter for the system Ya u 3 O 7-x, one fins following two ifferent situations: (a he suerconucting orer arameter ue to -holes only, (b he suerconucting orer arameter ue to -holes only. he S Orer Parameter for -Holes One obtains the exression for as - ( hω tanh k...(49 Using the following changes in variables x x - J, hω y, hω y an after simlification, one obtains -.63 y e 6 6 y.396 x y.396 x y 396 e. x...(5 Solving equations (5 numerically, one can stuy the variation of S orer arameter with temerature in the absence of -holes. he values obtaine from equations (5 are eicte in able an the behavior is shown in Figure for holes. able : Deenence of suerconucting orer arameter ( (-holes with temerature S. o emerature (K x J.37.35 3 3.33 4 4.3 ont
Int. J. hem. Sci.: 8(3, 487 S. o emerature (K x J 5 5. 6 6. 7 7.7 8 8. 9 88..5 Suerconucting orer arameter ( Joule.5.5 3 4 5 6 7 8 9 emerature (K Fig. : ehavior of suerconucting orer arameter ( with temerature for - holes he S Orer Parameter for -Holes Similarly, one obtains the following exression for suerconucting orer arameter for holes -.53 y 94..4 94..4 y. 769 x y. 769. 769 e x y e x (5 Solving equation (5, one can stuy the variation of S orer arameter with temerature in the absence of -holes. he values obtaine from equation (5 are eicte in able 3 an the behaviour is shown in Figure for holes.
488 A. uwal et al.: Photoinuce Suerconuctivity. able 3: Deenence of suerconucting orer arameter ( (-holes with temerature S. o. emerature (K x J.9.8 3 3.5 4 4. 5 5.94 6 6.77 7 7.5 8 8.6 9 88..5 Suerconucting orer arameter ( Joule.5.5 3 4 5 6 7 8 9 emerature (K Fig. : ehavior of suerconucting orer arameter ( with temerature for holes. One can stuy the variation of S orer arameter of both; -holes an -holes with temerature. he values are eicte in able 4 an the behaviour is shown in Figure 3 for both an holes.
Int. J. hem. Sci.: 8(3, 489 able 4: Deenence of suerconucting orer arameter ( (for & holes with temerature S. o. emerature (K ( x J 4.46 4.43 3 3 4.38 4 4 4.3 5 5 4.4 6 6 3.77 7 7 3. 8 8.6 9 88. Suerconucting orer arameter ( 4.5 4 3.5 3.5.5.5 3 4 5 6 7 8 9 emerature (K Fig. 3: ehavior of suerconucting orer arameter ( with temerature for an holes Electronic Secifc Heat On the basis of equation (4 an (4, one can show that the relative jum of the secific heat at equals
49 A. uwal et al.: Photoinuce Suerconuctivity. en.43 R R he ratio R R characterizes the ossible eviation of the relative jum of the secific heat in the resent moel from the S one. he function R reveals only a weak eenence on n h, it is unerstanable that the curve vs n h is mainly etermine by the function (n h staning outsie the R in the formula for (4. he resemblance between the eenence of an on n h on (c.f. Figure 4 an Figure 6 reflects also this circumstance. able 5: ariation of hole concentration (n h with critical temerature ( an chemical otential (μ S. o hemical otential (μ ritical tem. ( K P ρ E Hole concentration n h - o. 88.3.5.85.9. 88.5..85.395 3.3 88.6.45.85.6 4.4 88.5.6.85.85 5.5 88..85.85. 6.6 87.69 3.3.85.5 7.7 87. 3.35.85.4 8.8 86. 3.44.85.65 9.9 85. 3.645.85.83. 83.7 3.84.85.6. 8. 3.873.85.58
Int. J. hem. Sci.: 8(3, 49 ritical temerature ( 9 8 7 6 5 4 3 * * * * * * heoretical * Exerimental.5.5.5 Hole concentration (n h Fig. 4: ariation of hole concentration (n h with critical temerature ( able 6: Deenence of chemical otential shift μ on hole concentration n h S. o. Hole concentration (n h μ o (n h E g (n h E g (O μ (n h μ μ (n h E.9.83.8.3 -...395.9.5.3 -.96.4 3.6.399.9.3 -.9 -.9 4.85.57.79.3 -.86 -.86 5..64.66.3 -.38 -.8 6.5.7.533.3 -.475 -.375 7.4.83.43.3 -.57 -.47 8.65.938.74.3 -.665 -.565 9.83.46.44.3 -.76 -.66.6.49..3 -.85 -.75.58.66..3 -.866 -.766
49 A. uwal et al.: Photoinuce Suerconuctivity. hemical otential ( μ.. -. -. -.3 -.4 -.5 -.6 -.7 -.8.5.5.5 Hole concentration (n h Fig. 5: Deenence of chemical otential shift (μ on hole concentration (n h o carry out numerical estimates, we utilize the same sets of fitting arameters as use in calculation written in critical fiel. he theoretical eenence of on hole concentration (n h for Ya u 3 O 7-x is eicte in able 7 an behaviour is shown in Fig. 6. able 7: Deenence of with hole concentration n h S. o. ritical temerature (K Hole concentration n h - o 9.4 Rk en π 6 3 R k 6 88.3.9 6.8 4. 88.5.395 6. 4. 3 88.6.6 6. 4.5 4 88.5.85 6. 4. 5 88.. 6.8 4.97 6 87.69.5 6.4 4.73 7 87..4 6.9 4.4 8 86..65 6.3 3.999 9 85..83 5.96 3.953 83.7.6 5.86 3.888 8..58 --- 3.88
Int. J. hem. Sci.: 8(3, 493 6. heoretical (e/mol-k 6.5 6. 6.5 6 5.95 5.9 5.85.5.5.5 Hole concentration (n h Fig. 6: Deenence of with hole concentration n h Free Energy he free energy ifference for holes is given in equation (45, making the use of hω y, hω y an following values from able, one obtains for holes F F 4 S 9 y.56 (.56 x..6 log 3.76 4.56 y 3 (. e.6 log.76 y y 57.97 y (..6 log 57.97 e y y (5 Similarly for holes, We obtain form equation (46, FS F 9 (.35 x. 4.69 y 3 (. e.6 log.76 y y 47.4 y (..3 log 47.4 e y log 4.69 y.6 3.76 y (53
494 A. uwal et al.: Photoinuce Suerconuctivity. he values of free energy ifference at various temeratures, obtaine from equation (5 are shown in able 8 an Figure 7 for holes an obtaine from equation (53 shown in able 9 an Figure 8 for holes. able 8: ariation of free energy ifference F - F S for -holes with temerature emerature F S- F S. o. x -3 Joule/mole (K. 9.98 3 3 9.8 4 4 9.58 5 5 8.87 6 6 7.55 7 7 5.8 8 8 3.5 9 88. Free energy ifference (F s-f n/ 9 8 7 6 5 4 3 3 4 5 6 7 8 9 emerature (K Fig. 7: ariation of free energy ifference with temerature for holes.
Int. J. hem. Sci.: 8(3, 495 able 9: ariation of free energy ifference F - F S for -holes with temerature S. o. emerature (K F - F S x -3 Joule/mole 6.75 6.638 3 3 6.455 4 4 6.93 5 5 5.95 6 6 5.84 7 7 3.968 8 8.44 9 88.946 Free energy ifference -3 (F s-f n/ x Joule/mol 7 6 5 4 3 3 4 5 6 7 8 9 emerature (K Fig. 8: ariation of free energy ifference with temerature for holes ritical Fiel (H Using the relation (48 an substituting the value of free energy ifference from Eq. (5 an (53, one can obtain the relation for critical fiel for both tyes of holes.
496 A. uwal et al.: Photoinuce Suerconuctivity. For holes, it is ( log ( log ( 4 8 e e H c π hω hω (54 an for holes, ( log ( log ( 4 8 e e H c π hω hω (55 he relation (54 an (55 are reucible to stanar S relation. he values of critical fiel at various temeratures, obtaine from equation (54 are shown in able an Figure 9 for holes an obtaine from equation (55 are shown in able an Figure for holes. able : ariation of critical fiel (H for -holes S. o. emerature (K ritical fiel (H x - esla 5.64 5.6 3 3 4.96 4 4 4.95 5 5 4.7 6 6 4.354 7 7 3.87 8 8.965 9 88.736
Int. J. hem. Sci.: 8(3, 497 - ritical fiel (H x esla 5.5 5 4.5 4 3.5 3.5.5 3 4 5 6 7 8 9 emerature (K Fig. 9: ariation of critical fiel with temerature for holes able : ariation of critical fiel (H for -holes S. o. emerature (K ritical Fiel (H x - esla 4.8 4.83 3 3 4.6 4 4 3.947 5 5 3.85 6 6 3.573 7 7 3.57 8 8.476 9 88.54
498 A. uwal et al.: Photoinuce Suerconuctivity. - ritical fiel (H x esla 4.5 4 3.5 3.5.5 3 4 5 6 7 8 9 emerature (K Fig. : ariation of critical fiel with temerature for holes Deenence of hemical Potential (μ on ritical emerature ( : For Ya u 3 O 7-x suerconuctors, we use the best fitting arameters value for numerical estimation: ( Density of states for holes.65 e -, ( Density of states for holes.5 e -, he air interchange between the two bans.8 e, E ut off energy of lower ban. e, E o energy of higher ban. e, E o energy of lower ban. e, hω.6 x - Joule. e, hω.3 x - Joule.8 e. he small ifference between E c an E o is taken keeing in min the uncertainty in efining the effective bottom of the ban E c. he arameters are moel eenent for the calculation of (n h. hese quantitative characteristic seem to be reasonable at least they are of the orer roose 3, for high curates.
Int. J. hem. Sci.: 8(3, 499 Making use of hω y, hω y an hω y, hω y, with above arameters an limits of integration on equation (3, for eenence of critical temerature ( on chemical otential eenence (μ, one obtains.336. μ μ y y. μ. tanh.865. μ y y.8 tanh.865 (56 Solving numerically equation (56, we get able 5 for critical temerature ( with resect to ifferent oing arameters (μ in interval E o < μ < E. Deenence of arrier oncentration (n h with ritical emerature y substituting numerical values of chemical otential μ an critical temerature in equation (33 with fitting arameters, we get eenence of carrier concentration n h on critical temerature. he calculate eenence of carrier concentration (n h on critical temerature for system Ya u 3 O 7-x is shown in able 5 an eicte in Fig. 4. DISUSSIO AD OLUSIOS We have mae a stuy of hoto-inuce high curate suerconuctivity by canonical two-ban S Hamiltonian containing a Fermi surface of an holes. Following Green s function techniques an equation of motion metho, we have shown that the system ossess two suerconucting gas. It is also evient from the stuy that the system amits a recursor hase of ooer air rolets that unergoes a hase locking transition at a critical temerature below the mean fiel solution 5. In the two-comonent moel, the suerconuctivity is cause by the interban reulsive electron-lattice an oulomb interaction. he hotoexcitation rouces a change of the hole oing ( an as a result, there is shift of the high temerature curate suerconuctors. We have mae self-consistent stuies of the oing eenences of suerconucting transition temerature ( an chemical otential (μ shift in Ya u 3 O 7-x. Stuy reveals that the hase sace for air-transfer scattering between the overlaing bans is governe by the osition of chemical otential, i.e. the eenence of the electron chemical otential shift of n h regulates the hase shift for air transfer scattering between the overlaing ban an narrow bans an leas to the observe oing eenences of. Exeriments clearly
5 A. uwal et al.: Photoinuce Suerconuctivity. len suort to the roose oing eenence of μ(n h an (n h in the comoun Ya u 3 O 7-x. he fitting of the theory with exerimental ata shows one of the articiating hole bans to be narrow, as execte from theory. Accoring to our moel, there are two suerconucting gas on the system Ya u 3 O 7-x an temerature eenences of these gas are of S tye. We have also obtaine the free energy exression an critical magnetic fiel, which shows that in the interban moel, the suerconucting state is stable at >. he moel fairly exlains the observe features in Ya u 3 O 7-x. REFEREES. G. Yu,. H. Lee, A. J. Heeger,. Herron an E. M. Mcarron,, Phys. Rev. Lett., 67, 58 (99.. P. Konsin an. Sorkin, J. Physics: onference Series, 97, 8 (8. 3. P. Konsin,. Kistoffel an. Sorking, J. Phys. onens. Matter,, 6533 (998. 4.. M. Kreines an. I. Kuinov, Mo. Phys. Lett., 6, 89 (99. 5.. K. hakraverty, Phy. Rev.., 48, 6 (993. 6. R. Lal, Ajay Hota an S. K. Joshi, Phy. Rev., 57, 66 (998. 7. A. Prata, Ajay an A. S. riathi, J. Suercon., 9, 595 (997. 8. S. Khanka an P. Singh, Phys. Status Solii, 44, 699 (6. 9... Das, Phys. Rev., 46, 645 (99.. S. L. Kakani an U.. Uahyaya, J. Low em. Phys., 5, (983.. S. L. Kakani an U.. Uahyaya, Phys. Status Solii, 5, 86 (984.. S. L. Kakani an U.. Uahyaya, Phys. Status Solii, 35, 35 (986. 3. S. L. Kakani an U.. Uahyaya, Phys. Status Solii, A 99, 5 (987. 4. S. L. Kakani an U.. Uahyaya, J. Low em. Phys., 7, 5 (988. 5. A. Fetter an J. D. Walecka, Quantum heory of Many Particle Physics, McGraw- Hill, ew York (97. 6. K. P. Sinha, Physica,, 8 (993. 7. P. Konsin,. Kristoffel an. Or, Phys. Lett., A 43, 83 (99.
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