Study of Upper Critical Magnetic Field of Superconducting HoMo6Se8

Transcription

1 World Journal of Condened Matter Phyi, 5, 5, 5-7 Publihed Online Augut 5 in SiRe. Study of Upper Critial Magneti Field of Superonduting omo6se8 Tadee Deta, Pooran Singh, Gebregziabher Kahay Department of Phyi, College of Natural Siene, Addi Ababa Univerity, Addi Ababa, Ethiopia Department of Phyi, College of Siene, Bahir Dar Univerity, Bahir Dar, Ethiopia tad4jju@gmail.om, pinghgbpup@yahoo.om, mihige_9@yahoo.om Reeived 5 April 5; aepted 4 July 5; publihed 7 July 5 Copyright 5 by author and Sientifi Reearh Publihing In. Thi work i liened under the Creative Common Attribution International Liene (CC BY). Abtrat Thi work foue on the tudy of mathematial apet of upper ritial magneti field of uperonduting omo 6Se 8. At zero external magneti field, omo 6Se 8 wa found to undergo a tranition from the normal tate to the uperonduting tate at 5.6 K and returned to a normal but magnetially ordered tate between the temperature range of.3 K and.53 K. The main objetive of thi work i to how the temperature dependene of the upper ritial magneti field of uperonduting omo 6Se 8 by uing the Ginzburg-Landau (GL) phenomenologial Equation. We found the diret relationhip between the GL oherene length (ξ GL) and penetration depth ( GL) with temperature. From the GL Equation and the reult obtained for the GL oherene length, the expreion for upper ritial magneti field ( ) i obtained for the uperonduting omo 6Se 8. The reult i plotted a a funtion of temperature. The graph how the linear dependene of upper ritial magneti field ( ) with temperature (T) and our finding i in agreement with experimental obervation. Keyword Ginzburg-Landau Equation, Upper Critial Magneti Field, omo 6Se 8. Introdution Superondutivity i a phenomenon that our at very low temperature. Every uperondutor ha a tranition temperature (T ) below whih it uperondut and above whih it i a normal metal []. In the uperonduting tate, the material ha no eletrial reitane and thu ondut eletriity without loe. On the other hand, in the normal tate, the material doe have reitane and the flow of eletri urrent aompanie with the development of heat and the diipation of energy []. ow to ite thi paper: Deta, T., Singh, P. and Kahay, G. (5) Study of Upper Critial Magneti Field of Superonduting omo 6 Se 8. World Journal of Condened Matter Phyi, 5,

2 T. Deta et al. The era of low-temperature phyi began in 98 when the Duth phyiit eike Kamerlingh Onne firt liquefied helium whih boiled at 4. K at tandard preure. Three year later, in 9, Kamerlingh Onne diovered the phenomenon of uperondutivity while tudying the reitivity of metal at low temperature []. Mot of the fundamental propertie of uperondutor vary from material to material. The uperonduting tate, a any tate of matter, ha it own bai propertie. So, any uperondutor independent of the mehanim of uperondutivity and the material will exhibit thee propertie. The bai propertie of the uperonduting tate are zero reitane, Meiner effet, magneti flux quantization, Joephon effet, the BCS theory, Cooper pair, appearane of an energy gap in elementary exitation energy petrum, Iotope effet and the proximity effet. Every uperonduting tranition i marked by a jump in peifi heat. In the mixed tate, the behavior of type-ii uperondutor ha the ame pattern [3] [4]. omo 6 Se 8 ompound i building blok of the Chevrel-phae rytal truture [5]. The diovery in 984 of the new uperonduting omo 6 Se 8 gave rie to a renewed interet in the interplay of magnetim and uperondutivity. From the experimental reult, a magneti phae tranition (T n.53 K) to a long-period magneti tate ha been oberved via neutron attering in the uperondutor omo 6 Se 8 (T 5.6 K) [6]. The harateriti wave vetor (q ) i trongly temperature dependent and there i no obervation of higher-order atellite. The rare ourrene of ferromagnetim, a found in omo 6 Se 8 and omo 6 Se 8, revealed the trongly ompetitive nature of thee two ooperative phenomena in the form of long-wavelength at low oillatory magneti temperature (< K) and a ferromagneti lok-in tranition that quenhed the uperondutivity. No ign of reentrant behavior (in zero field) wa oberved down to.4 K [6] [7]. Ternary rare-earth uperondutor whih diplay a propenity for ferromagnetim have reeived oniderable experimental a well a theoretial attention reently. In the ae of the new uperondutor, omo 6 Se 8 wa oberved depite the preene of the holmium moment of the rare earth ion in eah unit ell. Thee ompound did not detroy the property of uperondutivity at low temperature. In omo 6 Se 8, the domain oexitene phae urvive till T.3 K (the exhange interation i weaker and the oexitene perit down to T.3 K) [8]. In addition to the uperonduting and ferromagneti domain, a oexitene region i oberved in omo 6 Se 8 in whih the uperonduting tate oexit with a long range modulated magneti order in a narrow region above the reentrant temperature T. The upper ritial magneti field i a very important magneti uperondutivity parameter. Therefore, tarting from it diovery a a uperonduting material, experiment are arried out to analyi the upper ritial magneti field ( ) of omo 6 Se 8. Aording to magnetization meaurement, on both poly and ingle rytalline ample of the ferromagneti uperonduting omo 6 Se 8, the upper ritial magneti field i a turning point from the uperonduting tate to the normal tate. The firt-order phae tranition i the inter hanging point of the uperonduting tate to the normal tate oberved when the external field i applied parallel to the magnetially eay axi (a axi of the hexagonal-rhombohedral rytal lattie truture). The ritial magneti field in type-i uperondutor i the inter hanging point of uperonduting tate into B normal tate. From Maxwell Equation, XE, when the magneti field i frozen, the field i expelled t from the interior of the uperondutor, otherwie uperondutivity will be detroyed by a ritial magneti field ( ), uh that T ( T) ( ) () T T [9]. Equation () yield the expreion of thermodynami ritial magneti field ( ( )). Mathematial Formulation to Find the Upper Critial Magneti Field of omo6se8.. The Bai Ginzburg-Landau Theory Ginzburg-Landau (GL) theory i a mathematial theory ued to deribe uperondutivity. Ginzburg-Landau (GL) theory i ued to explain the differene between Type-I and Type-II uperondutor and enable the alulation of two ritial magneti field and []. Ginzburg-Landau theory wa derived from the BCS miroopi theory by Lev Gorkov, howing that it alo appear in ome limit of miroopi theory and apply- 6

3 T. Deta et al. ing miroopi interpretation of all it parameter. The bai potulate of GL i that if ψ i mall and varie lowly in pae, the free-energy denity ( F ( r )) an be expanded in a erie of the form: β 4 e F Fn + αψ + ψ + i ψ ħ A + () m 8π where α and β are phenomenologial parameter, (β i poitive and the ign of α i temperature dependent), m m i an effetive ma, e q e i the harge of an eletron, A i the magneti vetor potential and B XA i the magneti field [5] []. If ψ, Equation () redue to the free energy of the normal tate; F Fn +. 8π Now, by minimizing the free energy with repet to flutuation in the order parameter and the vetor potential, one arrive at the Ginzburg-Landau Equation given by: αψ β ψ ψ e + + i ħ A m ψ (3) The urrent denity from the amilton energy of partile i given by; e mvd p A m v d e p A m where p iħ v d v d e p A m e i ħ A m where n ψ ( r) ψ ( r) J env d e e J i ψ r r ħ A m ( ) ψ ( ) e e e i i m ψ ψ ψ ψ j ħ A + ħ A (4) e e e m ψ ħ i ψ ψ ħ i ψ j A + A (5) where j i uperurrent denity. Equation (3) determine the order parameter ψ baed on the applied magneti field and Equation (5) yield the uperonduting urrent denity. The Ginzburg-Landau Equation provide omplete information about the uperonduting tate ψ ( r) that give the patial ditribution of the Cooper pair denity taking into aount a A r deribe the loal ditribution of the magneti field poible variation in their onentration, where a ( ) 7

4 T. Deta et al. in the uperondutor. In the abene of external magneti field (at free urfae), there will not be uperonduting urrent(urrent flow) and the Equation for ψ beome: n F F αψ + β ψ ψ (6) Thi Equation ha a trivial olution ψ and it orrepond to normal tate of T > T. Below uperonduting tranition temperature (T ), Equation (6) i expeted to have a non-trivial olution (i.e. ψ ) and the Equation an be rearranged into: ψ α (7) β If the eond part of Equation (3) i poitive, then there i a non zero olution for ψ and thi an be ahieved α α T α T T with n T T. by auming the temperature dependene of α uh that ( ) ( ) α ( T ) β > and ( ) For T > T, the expreion i poitive and the eond part of Equation (3) i negative and only ψ β olve the Ginzburg-Landau Equation. For T < T, the eond part of Equation (3) i poitive and there i a non-trivial olution for ψ. Thu Equation (7) an be expreed a: ψ α ( T T) β ( ) T T α ψ β Equation (8) yield Ginzburg Landau order parameter [5] []. (8).. Calulation of Ginzburg-Landau Coherene Length The Ginzbrug-Landau oherene length (ξ GL ) i a meaure of the ditane in the uperonduting eletron onentration that an not hange dratially in a patially-varying magneti field. The Ginzbrug-Landau oherene length (ξ GL ) i a temperature-dependent a well a a material dependent quantity. In the ae of abene of the magneti vetor potential, Equation (3) redue to: 3 αψ + β ψ + ( iħ ) ψ (9) m Now, let u onider a wave funtion that varie only in the z-diretion with zero applied magneti field. In thi ae, the firt GL Equation i one dimenional. i.e, Auming ψ i real and negleting the term 3 ħ d ψ αψ + β ψ m dx 3 βψ in omparion with α, Equation (), beome: For the plane wave funtion, the olution of Equation () i in the form of, () d ψ αψ ħ () m dx ix ξ ix GL ψ ( x) e exp ξgl Subtituting the value of plane wave funtion into Equation () (in term of ψ ( x) ), we get, ( ) () 8

5 T. Deta et al. ix ħ d ix α exp exp ξgl m dx ξgl ix ħ i ix α exp exp ξgl m ξgl ξgl ħ i ψ αψ m ξgl Thi implie that, ħ α m ξ Solving for ξ GL at uperonduting tate that mean, where α i negative yield, ξ ħ ( ) m α m α T T ħ (3) (4) where α α ( T T ), α α ( T T) and ξ ( ) GL α ħ m T α T T, then we Equation (4) yield the GL oherene length []. Sine α depend on temperature a ( ) an onlude that oherene length i temperature dependent. Now let u onider the ae: Cae (I), For uperonduting tate (T < T ), and Cae (II), For normal tate (T > T ), ξ GL( ) T ξ ; T (5) ξ GL( ) T ξ ; T Cae (III), at T T, the GL theory i not valid Where the length ξ GL() i known a the zero temperature GL oherene length. (6).3. Calulation of Ginzburg-Landau Penetration Depth The urfae urrent flow in a very thin layer of thikne ( GL ) whih i alled the Ginzburg-Landau penetration depth []. The temperature and magneti field dependene of the penetration depth appear quite naturally in Ginzburg-Landau (GL) theory. Like the London model, the GL model i independent of the underlying mehanim for uperondutivity. Ginzburg-Landau theory i tritly valid only in uperonduting phae boundary and i thu not generally appliable at low temperature []. In the Ginzburg-Landau theory, a omplex order parameter(ψ) i a funtion of temperature, magneti field and the patial oordinate [5] []. The total free energy per unit volume of the uperonduting tate in the preene of a magneti field i minimizing thi expreion with repet to the firt GL Equation and with repet to A the urrent denity(gl-ii) Equation; e FGL iħ A ψ + αψ + β ψ ψ (7) m 9

6 T. Deta et al. where m m and e e. Uing Equation (7), we get the expreion for urrent denity, a follow ine ψ Negleting ψ n ψ and α ψ ψ (8) β e ħi e J ψ ψ + ψ ψ ψ m A (9) m e ħi e J ψ ψ + ψ ψ ψ m A () m ψ Equation (), beome; J e m ψ A () Uing Maxwell Equation: Taking the url on both ide of Equation (), we get where B, e X J X A and B XA m ψ From Equation () and Equation (4), we get 4π XB J () 4π X XB ( XJ ) (3) 4π ( B) B ( XJ ) (4) e 4e XJ ψ XA ψ B (5) m m 4π 6πe B ( XJ ) nb (6) m Sine ψ α β n and m, we get 6π en Therefore, 4π B B ( XJ ) (7) m mβ (8) 6πen 6πe α mβ m m (9) 6πe α 6πen 8πen

7 T. Deta et al. where m m and Therefore, n ( T T) α α. β β 8π m β ( ) e α T T (3) For m β (3) 8πe α T L( ) from Equation (3) [9] it follow that, the Ginzburg-Landua penetration depth ( GL(T) ) varie a a funtion of temperature a: T T (3) ( ) ( ) GL T L 4 T T where L() i the London penetration at abolute zero temperature [9]. (33).4. Calulation of the Upper Critial Magneti Field Uing Ginzburg-Landau Theory The upper ritial magneti field (UCMF) i the magneti field whih ompletely uppree uperondutivity in type-ii uperondutor. More properly, the UCMF i a funtion of temperature (and preure) and if not peified abolute zero and tandard preure are implied. Superondting region nuleate pontaneouly within a normal ondutor when the applied magneti field i dereaed below a value denoted by [8]. At the onet of uperondutivity, ψ i mall and we linearize the GL Equation a follow; Sine m m, e e ψ i E ħ A m ψ αψ ψ (34) e, xˆ + yˆ + zˆ x y z and E Ex + Ey + Ez. The upper ritial magneti field ( ) an be alulated by linearizing Equation (34) and ubtituting the value of a: m e iħ xˆ + yˆ + zˆ A ψ αψ (35) x y z The magneti field in a uperonduting region at the onet of uperondutivity i jut the applied field, o A B, x, B x and Equation (35) beome that ( ) m where iħ iħ xˆ + yˆ + zˆ x y z mentum rytal i ħ k. e iħ xˆ + yˆ + zˆ B x ψ αψ (36) x y z P and ħk ħ( kx + ky + kz) P ψ ψ ψ where the eigne value of mo-

8 T. Deta et al. Therefore, m e iħ + ky + kz B x ψ αψ (37) x Sine the expreion of the amiltonian energy given in Equation (37) doe not depend on oordinate (y and z) the orreponding momentum omponent ( k y, k ) are onerved. ħ Eψ αψ ( k + k ) ψ x m y z z e ħ Exψ i Bx ħ + + ψ ( k y + kz ) ψ (39) m x y z m ħ m e x Exψ ψ B ψ + m x m The larget value of the magneti field (B) for whih the olution of Equation (4) of the lowet eigenvalue i given by ħe B ħ k En n+ ħω n+ m m max z α Let u take the mallet eigenvalue n and K z orreponding to the highet field in whih uperondutivity an nuleate in the interior of a bulk ample whih our with the upper ritial magneti field in the oeffiient hange of ign. From Equation (4), we have; where ω i the ylotron frequeny and i given by ine B max, olving for, we get: ħe Bmax ħω α m e B ω m max m α ħ (38) (4) (4) (4) (43) α (44) ħ e ħ T From the relation α α ( T T ) that mean α α ( T T), ξ m ξ ( ) ξ ( ) GL T GL T ξ GL( ) and ξ. T T We obtain the expreion of the temperature dependent upper ritial magneti field ( ), m ħ T ħe m ξgl( ) T ħ T T e ξgl( )

9 T. Deta et al. φ φ T T πξ ( ) πξ GL T GL( ) (45) where h h πħ ħ or πħ h and φ π e e.5. Aniotropi Ma Tenor Model φ h ħ π πe e Now, onider aniotropy in ma, by introduing the effetive ma tenor to the kineti energy term of the GL Equation (3), i.e. where m i an effetive ma tenor whih i given by, αψ β ψ ψ e + + i A ħ m ψ (46) mx m my m z Sine the oherene length ξ GL( ) depend on the effetive ma a ξ, the reulting Equation i m formally idential with the Shrödinger Equation of a partile with harge e e, an iotropi ma tenor m m in a uniform magneti field and the energy level that have the harmoni oillator are given by; Let u onider Newton law of motion under the influene of lorentz fore i.e. α n + ω ( θ) ħ (47) e (48) Fl m v vx where v i the veloity. The upper ritial magneti field an be expreed by uing ylotron frequeny with the lowet free energy; α ħ ω ( ) θ The olution of upper ritial magneti field by applying elliptial orbit travered with ylotron frequeny i given by, ω ( θ ) e B max in θ o θ + mm x z mx The olution of the lowet free energy orrepond to n and by uing Equation (49), we get (49) e B max in θ o θ ħω( θ ) α ħ + mm x z m x 3

10 T. Deta et al. where θ i the angle of the magneti field that make with the z-axi ( T T ) ( T T) α α α. ( ) α T T in θ o θ ħe mm + m x z x From the general expreion of oherene length Equation (45) we have, ħ ξx mxα ( T T) ħ ξz mzα ( T T) (5) (5) (5) and h Uing the expreion for the flux quantization, φ and Equation (45), an be expreed a, e φ in θ o θ π + 4 ξξ x z ξx For field parallel and perpendiular to the ymmetry plane we an write Equation (53) a: φ (53) (54) πξξ πξx z z φ (55) Equation (54) and (55) are the mathematial expreion of the upper ritial magneti field ( ) for field parallel and perpendiular to the ymmetry axi [4]. 3. Reult and Diuion From Equation (5), we obtained the graph that how the relationhip between the GL oherene length and temperature (T) a indiated in Figure. A an be ee from Figure, the GL oherene length inreae with temperature and diverge a T T. ξ GL( ) ha the ame value with the BCS oherene length i.e. ( ξgl( ) ξ ) []. We have already alulated the GL penetration depth in (33), the relationhip between the GL penetration depth and temperature(t) i hown in Figure. From Figure, we oberve that, the inreae of the GL penetration depth with temperature (T) and generally, we oberve that, the penetration depth rie aymptotially a the temperature approahe T. Thu, the penetration of field inreae a the temperature approahe to T. We finally determined the expreion for upper ritial magneti field for uperonduting omo 6 Se 8 uing the GL Equation (45) and by taking experimental data and upper ritial magneti field for parallel and perpendiular, we plot the upper ritial field at parallel and perpendiular to the ymmetry axi a hown in Figure 3 [6] [7]. 4

11 T. Deta et al GL oherene length (m) Temperature (K) Figure. GL oherene length veru temperature (T)..5 7 GL penetration depth (m) Temperature (K) Figure. GL penetration depth veru temperature (T). 5

12 T. Deta et al..5 parallel perpendiular upper ritial magneti field (T) of omo6se Temperature (K) Figure 3. Upper ritial magneti field parallel and perpendiular to the ymmetry axi veru temperature (T). From Figure 3, we an ee that, the upper ritial magneti field dereae a temperature inreae and reahe to zero at the ritial temperature of uperonduting omo 6 Se 8, whih agree with the experimental obervation [6]. And a an be een from $, the upper ritial magneti field ( ) parallel and perpendiular to the ymmetry axi of uperonduting omo 6 Se 8 i inverely proportional to the GL oherene length and fit. 4. Conluion The aim of thi reearh i to determine the upper ritial field of uperonduting omo 6 Se 8 by uing Ginzburg-Landau approah. From the alulation, the effet of oherene length, penetration depth and aniotropy in ma tenor on upper ritial field are onidered in our model. And finally figure are plotted by uing MATLAB ript. From the figure plotted, it an be onluded that the upper ritial magneti field of uperonduting omo 6 Se 8 i inverely related to temperature whih i in agreement with experimental obervation [6]. Referene [] Owen, F.J. and Poole, Jr., C.P. () The New Superondutor. Kluwer Aademi Publiher, New York. [] Mourahkine, A. (4) Room Temperature Superondutivity. Univerity of Cambridge, Cambridge. [3] Patteron, J.D. and Bailey, B.C. () Solid-State Phyi, Introdution to the Theory. [4] Kittel, C. (5) Introdution to Solid State Phyi. John Wiley and Son, In., oboken. [5] Pretemon, S. and Ferrain, P. (7) Bai of Superondutivity. IEEE Tranation on Applied Superondutivity, 3, 4. [6] Lynn, J.W., Gotaa, J.A., Erwin, R.W., Ferrrell, R.A., Bhattaharjee, J.K., Shelton, R.N. and Klavin, P. (984) Temperature Dependent Sinuoidal Magneti Order in the Superondutor omo 6 Se 8. Phyial Review Letter, 5, 33. [7] Gotaa, J.A. and Lynn, J.W. (986) Magneti Field Dependene of the Small Angle Neutron Sattering in omo 6 Se 8. Journal of Magnetim and Magneti Material, 54, [8] Leggett, A.J. (975) A Theoretial Deription of the New Phae of Liquid 3 e. Review of Modern Phyi, 47,

13 T. Deta et al. [9] Maki, K. and Tuneto, T. (964) Pauli Paramagnetim and Superonduting State. Progre of Theoretial Phyi, 3, [] Antoine, J.-P., Govaert, J., Peeter, F., Gerard, J.-M., Gregoire, G., Piraux, L. and Ruelle, P. (5) A Relativiti BCS Theory of Superondutivity Juillet. [] Bulaevkii, L.N., Ginzburg, V.L. and Sobyanin, A.A. (988) The Maroopi Theory of Superondutor with a Short Coherene Length. Journal of Experimental and Theoretial Phyi (JETP), 94,