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Vortex lattice distortions in hexagonal unconventional superconductors
14 March 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 186 (1994) 259-264
Vortex lattice distortions in hexagonal unconventional superconductors Yu.S. Barash Department of Theoretical Physics, P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninsky Prospect 53, 117924 Moscow, Russian Federation
A.S. Mel'nikov Department of Solid State Physics, Institutefor Applied Physics, Russian Academy of Sciences, 46 Uljanov Street, 603600 Nizhny Novgorod, Russian Federation
Received 10 December 1993;acceptedfor publication 28 December 1993 Communicatedby V.M. Agranovich
Abstract
A noticeable magnetic field dependence of the vortex lattice distortions in hexagonal unconventional superconductors is obtained below the second order phase transition line H*(T) (H*(T)
A number of experiments show evidence for the complex magnetic field-temperature ( H - T ) phase diagram in the heavy fermion superconductor UPt3 [ 1-5 ]. Several theoretical approaches have been suggested recently to describe this phase diagram and some other unusual properties of UPt3 [6-19 ]. Probably one of the most preferable phenomenological approaches is based on the two-component order parameter which corresponds to one of the two-dimensional representations of the group D6. A splitting of the superconducting transition in UPt3 (which is observed experimentally even in the absence of a magnetic field) is usually explained by taking account of a weak symmetry-breaking field [7,8,1012 ]. For a nonzero magnetic field at least three superconducting phases are shown to exist. There are two phase transition lines in the H - T plane between these phases and there seems to be a tetracritical point
at H~2(T*). One of these phase transitions occurs at low magnetic fields presumably due to the influence of a symmetry-breaking field on the superconducting order parameter. Another transition takes place at high fields H*~0.6Hc2 for Hllc (H*~O.25Hc2 for H ± c ) . In this article we consider only the magnetic phase transition and vortex lattice structure at high fields and so we shall not take into account the influence of a weak symmetry-breaking field mentioned above. The superconducting order parameter for two-dimensional representations of the group D6 has two complex components ~/l and r/2 and the GinzburgLandau functional in a magnetic field can be written as follows,
Yu.S. Barash, A.S. Mel'nikov / PhysicsLettersA 186 (1994)259-264
260
t f G L = J [ _al,li~l.~_l_bl(~li~.)2+b2 [~i~]i
[2
+ KI Pi ~lj PFb + K2P i ~li Pffb + K~PT~ITP:I, + K4P*:ITP:ld d V , P~=Px,
P2=Py,
( 1)
P=-ihV-2eA/c,
a=a(T~-T),bl>O,
b2>-b,,
K~+K2+Kx>IKEI,
KI>IKxl,
/(4>0.
(2)
The sign of the coefficient b2 in ( 1 ) determines the homogeneous superconducting phases which are energetically favourable in the absence of an applied magnetic field. These homogeneous phases exist if the coefficients K~, K2, K3,/(4 meet conditions (2). One can see that for b2>0 these phases (~h, 72) ~ ( 1, + i ) are characterized by the broken time-reversal invariance. For UPt3 there is experimental evidence indicating that precisely these phases exist (see Refs. [2022 ] ). That is why only the case b2> 0 will be considered in this article. We shall also take K2 = K3 since the value (K2-K3)/2K1 is very small if the approximate electron-hole symmetry near the Fermi surface occurs in this compound. The structure of the mixed state in unconventional superconductors described by the Ginzburg-Landau functional (1) has several distinctive features. Recently the solutions of Ginzburg-Landau equations for this model were investigated both in the high field region near He2 and in the low field region near He1. In Ref. [ 9 ] these equations were solved numerically for a magnetic field along the hexagonal axis c and it was shown for the first time that nonaxisymmetric nonsingular vortices may exist near Hc~. For these nonsingular solutions there are no lines within the vortex core where both the order parameter components are zero. In Ref. [ 13 ] the core structure of nonaxisymmetric singular and nonsingular vortices was analyzed analytically for arbitrary orientations of H relative to the crystal axes and specific corrections to the anisotropy of H¢1 due to the complex vortex structure were found. The angular dependence of the upper critical field in this system may also be very unusual at least for a certain range of Ginzburg-Landau coefficients (see Ref. [23] ). It is worth noting that the order parameter eigenfunctions corresponding to the upper critical field for HIIc and HA_ c are quite different. By taking account
of this fact it is natural to expect that there is a great difference between the energetically favourable vortex lattice structures for these field directions. It is in fact confirmed by the theoretical consideration [ 1417 ]. For HIIc the problem of the energetically most favoured structure of the vortex lattice was analyzed in detail taking into account both the lattice period multiplication and the lattice distortions [ 16 ]. A transition from a single-quantum vortex lattice to a two-quantum vortex lattice was shown to exist in a wide range of the Ginzburg-Landau functional coefficients. A coefficient range in which singular vortices are converted into nonsingular ones has also been found. This article is devoted to the theoretical analysis of the magnetic phase transitions and vortex lattice distortions in hexagonal unconventional superconductors for a magnetic f i e l d / / w h i c h is perpendicular to the hexagonal symmetry axis. The case HA_ e was discussed earlier in Refs. [ 15,17-19 ] and the possibility of the second order phase transition was shown. Note that the value H* (T) and the structure of the singular vortex lattices for the regions 1, 5 (see Fig. 1 ) obtained here and in Refs. [ 15,17 ] are the same. At the same time not all possible nonsingular vortex lattices below H * ( T ) were taken into account in Refs. [ 15,17-19 ]. In particular the consideration of field dependent vortex lattice distortions seems to be very important in connection with the recent neutron diffraction measurements in UPt3 [24]. In a conventional superconductor vortex lattice distortions are independent of the applied magnetic field unless the nonlocal electrodynamics is relevant [ 25 ]. However, H/ H~2
1 0.9
1
0.80.7 0.6
2/
3
/ 4
0.5-
I
I
0.06 0.12 0.18 0.24
0,3 0.36 0.42 0.48 0.54
Fig. 1. Phase diagram for C=0.1.
Yu.S. Barash, A.S. Mel"nikov / Physics Letters A 186 (1994) 259-264
the effects of the nonlocal electrodynamics are essential only for temperatures far from Tc and can be neglected near To. On the contrary, the lattice distortion mechanism proposed in this paper does not disappear when T goes to T,. We shall take HIIOy and A llOx. In this case the system of linearized Ginzburg-Landau equations for the functional ( 1 ) has the form
-L2n[(I+2C)(O-i-L'~)n +g~z2Jql=E~/,, z ~2
2
. 02-1
2
2 0 z 0 -Ln[(~xx-i ~-~n)+k~zJth =Et/2,
LH2--
2ell hC
(3)
1(4 k= ~ '
(4'
'
~o , -21t~2E=in
~2 h2Kl a '
2K~
"
izox
2xin(x-xo) ao
~-+-~ [ -
kz-zo
~2
I+2C)k_I)IA[2(
2~zL2n~ 2] :
\ Lu
12> 1) IBI2( I~a°12>
I'
[B[4( 1~o 14) )
+21AI21BI2< I~ao121~012)
For superconductors described by an order parameter which has only one complex component the eigenfunction corresponding to Emi~ may be used as a trial function for the solution in the field range H
I_
(6)
The ql and /~2 sublattices are shifted from one another. From now on we shall assume that the Ginzburg-Landau parameter 2/~ (2 is the London penetration depth, ~ is the coherence length) meets the condition 2/~>> 1. This is undoubtedly true for UPt3. From ( 1 ), (5), (6) we get the following expression for the free energy,
+ (1 + b ) ( I a [ ' ( IVo 14) +
rrhc ~°=]~
ql =A~o =A ~ e x p [ i x p n ( n - 1 ) 4
]"
-
Too -
The upper critical field corresponds to the minimum energy level Emin of this system: He2
/ 2ninx q2 = B~o = B ~ exp~ i ~tpn( n - 1) + - ao ( z - 2 n L 2n/ao)2~
fGL
K2+K3
C=--
261
(5)
+2btQI
IAI21BI2cos(yQ+2yB--2~A),
A= IAI exp(i~,A),
(7)
B = IBI exp(iyB),
Q = IQI exp(i7e) = (¢/~2~a2) , a2
b2
f o = - ~ V, b= b--~, where ( > denote spatial averages and V is the superconductor volume. We shall consider here the case p = 0.5 since only such vortex lattices were observed in Ref. [24]. For conventional uniaxial superconductors the value of the a parameter is determined by the Ginzburg-Landau mass tensor and does not depend on the magnetic field. The changing of the a value with decreasing magnetic field is one of the unusual properties of the mixed state in UPt3. This phenomenon was observed in Ref. [24] at low temperature and one can assume that it occurs near T~ too. We have analyzed expression (7) for f OL numerically as a function of variables a=2nL2/a g, Xo, Zo for different values of parameters b and C. The energetically most favourable lattice structures are found and the results of numerical calculations are presented in the phase diagram (Fig. 1 ) for the particular case C = 0.1. The solid and dashed lines in Fig. 1 correspond to the second and first order phase tran-
262
Yu.S. Barash, A.S. Mel'nikov / Physics Letters A 186 (1994) 259-264
sitions respectively. In region 1 of this phase diagram we have ~/1=0, r/2~0 (we would have t h e 0 , r/2=0 for C < 0 ) . The r/2 lattice in this case is a triangular one expanded along the z-direction by a factor x/~ (p=0.5, a = x / / ~ / 2 ) . In regions 2-5 both the components rh and t/2 are nonzero. For b > 0.44 (region 5) the zeros of these components coincide (Xo= Zo= 0) and so the vortex lattice is a singular one with the parameters p=0.5, e = x / ~ / 2 which do not depend on the applied magnetic field. The case b < 0.44 is more interesting since the free energy minimum corresponds to nonzero values of Xo and Zo and the vortex lattice appears to be nonsingular. For one of the nonsingular phases (region 4) the a parameter does not depend on the magnetic field ( a = 0 . 5 k x / ~ and (Xo, Zo)= (ao/4, - a a o / 2 ) ) . In region 3 the values of Xo, Zo are the same but the tr parameter increases from the value x / / ~ / 2 with the decreasing of the magnetic field below H*. For region 2 we obtained (Xo, Zo) = (ao/2, 0). The tr parameter in this region increases from the value 0.5 k x / ~ just below the second order phase transition line H * ( T ) when H decreases. Note that putting the r/l vortices at the centroid of an r/2 triangle is not energetically favourable for all positive b values. The magnetic field dependence of the a parameter for regions '2 and 3 is shown in Fig. 2. From the transformation properties of ~h and r/2 components (given by expressions (5), ( 6 ) ) with respect to translations [ 17] one can see that the values (Xo, Zo) = (a o/2, 0) and (Xo, Zo) = (ao/ 4 , - t r a o / 2 ) correspond to the order parameter lat4
6
0.5
1
0.4-
2
3
0.3i
i
i
~
q 0.6
i
i
i i.
H/H~2
Fig. 2. Field dependence of the lattice distortion for C=0.1 and for various b values: (1) b.=0; (2) b=0.03; (3) b=0.09; (4) b=0.12; (5) b=0.18; (6) b=0.24.
time the magnetic field itself has a single quantum periodic structure below and above H* for this field direction. It is worth noting that the tr values 0.5 k x / ~ and 0.5x//~ (at p = 0 . 5 ) correspond to two hexagonal vortex lattices which are turned at the angle n / 6 in the plane (x, z) relative to each other and then are expanded along the z-direction by a factor x/~. For a = 0.5 k v / ~ ( a = 0 . 5 x / ~ ) one of the basis vectors of the vortex lattice is parallel to the z-axis (x-axis). These two a values correspond to (a) and (b) (B1 and B3 ) lattice types which were considered in Ref. [25] (see also Ref. [27]). Note that for conventional uniaxial superconductors for H_I_c there exists a continuum of vortex lattices with the same free energy and different orientation relative to the crystal axes [27 ]. From our results one can see that in the case of unconventional hexagonal superconductors this degeneracy may be removed below H * ( T ) . Let us now compare our results with the neutron diffraction experimental data for UPt3 [24]. Since the basis vectors for the shifted sublattices of r/1 and r/2 order parameter components are the same, one can use the conventional expression for the angle a between the x-axis and the first Bragg peak in neutron diffraction experiments: tan a = 1/2a. Therefore the increasing of the parameter tr which occurs in regions 2 and 3 with the magnetic field decreasing results in the decreasing of or. Such a behaviour of ot was observed experimentally for UPt3 (see Ref. [24] ). The a values which may be estimated using these ot measurements are close to a = 0 . 5 x / / ~ if we take the k value from the experimental data [ 4, 5 ] (k ~ 2 ). Since this tr value is energetically favourable for region 3 (not for region 2) of the phase diagram (Fig. 1 ) we can get the range of allowed b values for UPt3. It seems to be important that the value b = 0.17 obtained from experiments on Hcl and He2 measurement in UPt3 [28] also corresponds to region 3. According to the recent experimental results [29,30 ] the b value for UPt3 appears to depend essentially on the concentration of both magnetic and nonmagnetic impurities. Then the gradual change of the vortex lattice distortion considered above occurs only for a certain impurity concentration interval. As for the parameter C, we have carried out our calculations for C = + 0.1, + 0.2, + 0.3. When we increase the absolute value of C the distance between two lower energy levels intices with two flux quanta per unit cell. At the same
YuS. Barash, AS. Mel’nikov /Physics Letters A 186 (1994) 259-264
creases and therefore the second order phase transition occurs at the lower value H*/Hc2. As for the lattice structure below this phase transition line H* ( T), there are no qualitative differences in the phase diagrams for various C values. According to our results the u parameter does not depend on the magnetic field above H*(T) in contrast to its behaviour below the transition line. This feature may be used for the experimental identification of the phase transition. We believe that the experimental data (see Ref. [ 241) cannot yet allow one to make unambiguous conclusions about the a(H) dependence above H*(T), probably due to the insufficient precision of the measurements. The approach used above can be justified only for H*(T) close to Hc2, i.e. for small enough I C] values. Therefore we can analyze the phase diagram in UPt, for Hlc only qualitatively. In addition, for a complete quantitative analysis it is also necessary to take into account the influence of a weak symmetrybreaking field on the phase transitions in UPt,. It is especially important if the parameter C has an extremely small value (see Ref. [ 3 1 ] ). Effects which are analogous to those discussed above should also occur within the framework of the “AB” model proposed in Ref. [ 19 1. It follows from the fact that the Ginzburg-Landau functional of the “AB” model for HI c coincides with expression ( 1) for some particular values of the Ginzburg-Landau coefficients. Further theoretical and experimental work is necessary to find out which of these models is more suitable for UPtS #I. In conclusion, our calculations give evidence for a
xl When this paper was ready for publication we were informed of the results of Ref. [ 32 1. Within the framework of the “AB” model (for a certain set of the Ginzburg-Landau free energy parameters) it was found that the flux lattice evolves continuously in the vicinity of the tetracritical point in the H-T plane. The vortex lattice orientation near H*( T) is the same as the orientation observed in Ref. [ 241 and obtained above for region 3 (Fig. 1). At the same time the behavior of the function a(H) below H*(T) in Ref. [ 321 appears to be different. While we obtain, in accordance with the experimental results, an increase of u with a decrease of the magnetic field, in Ref. [ 321 a decrease of u with a decrease of H is obtained. The important point is that the measurements of Ref. [24] were carried out only at low temperatures. Measurements in the vicinity of T, (and T*) are necessary in order to find out which theory is more preferable for UPt,.
noticeable
263
magnetic
field dependence of vortex latunconventional superconductors for HI c. It is shown that for nonsingular vortex lattices the degeneracy of various lattice types (i.e. (a) and (b) lattices if we use the notations of Ref. [ 251) in anisotropic superconductors is removed. These results give the possible explanation of some unusual phenomena observed in neutron diffraction experiments in UPt, and may be considered as an additional indication of unconventional superconductivity in this compound.
tice distortions
in hexagonal
This work was supported, in part (A&M.), by a Soros Foundation Grant awarded by the American Physical Society. References [ 1] V. Muller et al., Phys. Rev. Lett. 58 (1987) 1224. [2] Y. Quart et al., Solid State Commun. 63 (1987) 599. [3] A. Shenstrom et al., Phys. Rev. Lett. 62 (1989) 332. [4] S. Adenwalla et al., Phys. Rev. L&t. 65 (1990) 2298. [5] G. Bruls et al., Phys. Rev. Lett. 65 (1990) 2294. [6] G.E. Volovik, J. Phys. C 21 (1988) L221. [ 71 R. Joynt, Supercond. Sci. Tech. 1 ( 1988) 2 10. [ 8 ] D.W. Hess, T.A. Tokuyasu and J.A. Sauls, J. Phys. Condens. Matter 1 (1989) 8135. [ 91 T.A. Tokuyasu, D.W. Hess and J.A. Sauls, Phys. Rev. B 41 (1990) 8891. [lo] K. Machida and M. Ozaki, J. Phys. Sot. Japan 58 ( 1989) 2244. [ 111 R. Joynt, V.P. Mineev, G.E. Volovik and M.E. Zhitomirsky, Phys. Rev. B 42 (1990) 2014. [ 121 K Machida and M. Ozaki, Phys. Rev. Lett. 66 ( 1991) 3293; T. Ohmi and K. Machida, Phys. Rev. Lett. 71 ( 1993) 625. [ 131 Yu.S. Barash and A.S. Mel’nikov, Zh. Eksp. Teor. Fiz. 100 (1991) 307 [Sov. Phys. JETP 73 (1991) 1701. [ 141 I. Luk’yanchuk, J. Phys. (Paris) 1 (1991) 1155. [ 15 ] R. Joynt, Europhys. Lett. 16 ( 199 1) 289. [ 161 A.S. Mel’nikov, Zh. Eksp. Teor. Fiz. 101 (1992) 1978 [Sov. Phys. JETP 74 (1992) 10591. [ 171 M.E. Zhitomirsky and LA. Luk’yanchuk, Zh. Eksp. Teor. Fiz. 101 (1992) 1954 [Sov. Phys. JETP 74 (1992) 10461. [ 181 A. Garg, Phys. Rev. Lett. 69 (1992) 676. [ 191 D.-C. Chen and A. Garg, Phys. Rev. Lett. 70 ( 1993) 1689. 1201 Z. Fisk et al., Science 239 (1988) 33. [ 211 C. Broholm et al., Phys. Rev. Lett. 65 ( 1990) 2062. (221 G.M. Lukeet al., Phys. Rev. Lett. 71 (1993) 1466. [23] Yu.S. Barash and A.V. Galaktionov, Phys. Lett. A 164 (1992) 103. [ 241 R.N. Kleiman et al., Phys. Rev. L&t. 69 ( 1992) 3 120. [25] K. Takanaka, in: Anisotropy effects in type II superconductors, ed. H.W. Weber (Plenum, New York, 1977) p. 93.
264
Yu.S. Barash, A.S. Mel'nikov / Physics Letters A 186 (1994) 259-264
[26] G. Lasher, Phys. Rev. A 140 (1965) 523. [27] L.J. Campbell et al., Phys. Rev. B 38 (1988) 2439. [28] Z. Zhao et al., Phys. Rev. B 43 ( 1991 ) 13720. [29 ] T. Vorenkamp et al., Phys. Rev. B 48 ( 1993 ) 6373.
[30] T. Vorenkamp et al., Phys. Rev. B 48 (1993) 6385. [31 ] J.A. Sauls, preprint Nordita 93/71S. [ 32 ] A. Garg and D.-C. Chen, to be published.
ELSEVIER
Physics Letters A 186 (1994) 259-264
Vortex lattice distortions in hexagonal unconventional superconductors Yu.S. Barash Department of Theoretical Physics, P.N. Lebedev Physical Institute, Russian Academy of Sciences, Leninsky Prospect 53, 117924 Moscow, Russian Federation
A.S. Mel'nikov Department of Solid State Physics, Institutefor Applied Physics, Russian Academy of Sciences, 46 Uljanov Street, 603600 Nizhny Novgorod, Russian Federation
Received 10 December 1993;acceptedfor publication 28 December 1993 Communicatedby V.M. Agranovich
Abstract
A noticeable magnetic field dependence of the vortex lattice distortions in hexagonal unconventional superconductors is obtained below the second order phase transition line H*(T) (H*(T)
A number of experiments show evidence for the complex magnetic field-temperature ( H - T ) phase diagram in the heavy fermion superconductor UPt3 [ 1-5 ]. Several theoretical approaches have been suggested recently to describe this phase diagram and some other unusual properties of UPt3 [6-19 ]. Probably one of the most preferable phenomenological approaches is based on the two-component order parameter which corresponds to one of the two-dimensional representations of the group D6. A splitting of the superconducting transition in UPt3 (which is observed experimentally even in the absence of a magnetic field) is usually explained by taking account of a weak symmetry-breaking field [7,8,1012 ]. For a nonzero magnetic field at least three superconducting phases are shown to exist. There are two phase transition lines in the H - T plane between these phases and there seems to be a tetracritical point
at H~2(T*). One of these phase transitions occurs at low magnetic fields presumably due to the influence of a symmetry-breaking field on the superconducting order parameter. Another transition takes place at high fields H*~0.6Hc2 for Hllc (H*~O.25Hc2 for H ± c ) . In this article we consider only the magnetic phase transition and vortex lattice structure at high fields and so we shall not take into account the influence of a weak symmetry-breaking field mentioned above. The superconducting order parameter for two-dimensional representations of the group D6 has two complex components ~/l and r/2 and the GinzburgLandau functional in a magnetic field can be written as follows,
Yu.S. Barash, A.S. Mel'nikov / PhysicsLettersA 186 (1994)259-264
260
t f G L = J [ _al,li~l.~_l_bl(~li~.)2+b2 [~i~]i
[2
+ KI Pi ~lj PFb + K2P i ~li Pffb + K~PT~ITP:I, + K4P*:ITP:ld d V , P~=Px,
P2=Py,
( 1)
P=-ihV-2eA/c,
a=a(T~-T),bl>O,
b2>-b,,
K~+K2+Kx>IKEI,
KI>IKxl,
/(4>0.
(2)
The sign of the coefficient b2 in ( 1 ) determines the homogeneous superconducting phases which are energetically favourable in the absence of an applied magnetic field. These homogeneous phases exist if the coefficients K~, K2, K3,/(4 meet conditions (2). One can see that for b2>0 these phases (~h, 72) ~ ( 1, + i ) are characterized by the broken time-reversal invariance. For UPt3 there is experimental evidence indicating that precisely these phases exist (see Refs. [2022 ] ). That is why only the case b2> 0 will be considered in this article. We shall also take K2 = K3 since the value (K2-K3)/2K1 is very small if the approximate electron-hole symmetry near the Fermi surface occurs in this compound. The structure of the mixed state in unconventional superconductors described by the Ginzburg-Landau functional (1) has several distinctive features. Recently the solutions of Ginzburg-Landau equations for this model were investigated both in the high field region near He2 and in the low field region near He1. In Ref. [ 9 ] these equations were solved numerically for a magnetic field along the hexagonal axis c and it was shown for the first time that nonaxisymmetric nonsingular vortices may exist near Hc~. For these nonsingular solutions there are no lines within the vortex core where both the order parameter components are zero. In Ref. [ 13 ] the core structure of nonaxisymmetric singular and nonsingular vortices was analyzed analytically for arbitrary orientations of H relative to the crystal axes and specific corrections to the anisotropy of H¢1 due to the complex vortex structure were found. The angular dependence of the upper critical field in this system may also be very unusual at least for a certain range of Ginzburg-Landau coefficients (see Ref. [23] ). It is worth noting that the order parameter eigenfunctions corresponding to the upper critical field for HIIc and HA_ c are quite different. By taking account
of this fact it is natural to expect that there is a great difference between the energetically favourable vortex lattice structures for these field directions. It is in fact confirmed by the theoretical consideration [ 1417 ]. For HIIc the problem of the energetically most favoured structure of the vortex lattice was analyzed in detail taking into account both the lattice period multiplication and the lattice distortions [ 16 ]. A transition from a single-quantum vortex lattice to a two-quantum vortex lattice was shown to exist in a wide range of the Ginzburg-Landau functional coefficients. A coefficient range in which singular vortices are converted into nonsingular ones has also been found. This article is devoted to the theoretical analysis of the magnetic phase transitions and vortex lattice distortions in hexagonal unconventional superconductors for a magnetic f i e l d / / w h i c h is perpendicular to the hexagonal symmetry axis. The case HA_ e was discussed earlier in Refs. [ 15,17-19 ] and the possibility of the second order phase transition was shown. Note that the value H* (T) and the structure of the singular vortex lattices for the regions 1, 5 (see Fig. 1 ) obtained here and in Refs. [ 15,17 ] are the same. At the same time not all possible nonsingular vortex lattices below H * ( T ) were taken into account in Refs. [ 15,17-19 ]. In particular the consideration of field dependent vortex lattice distortions seems to be very important in connection with the recent neutron diffraction measurements in UPt3 [24]. In a conventional superconductor vortex lattice distortions are independent of the applied magnetic field unless the nonlocal electrodynamics is relevant [ 25 ]. However, H/ H~2
1 0.9
1
0.80.7 0.6
2/
3
/ 4
0.5-
I
I
0.06 0.12 0.18 0.24
0,3 0.36 0.42 0.48 0.54
Fig. 1. Phase diagram for C=0.1.
Yu.S. Barash, A.S. Mel"nikov / Physics Letters A 186 (1994) 259-264
the effects of the nonlocal electrodynamics are essential only for temperatures far from Tc and can be neglected near To. On the contrary, the lattice distortion mechanism proposed in this paper does not disappear when T goes to T,. We shall take HIIOy and A llOx. In this case the system of linearized Ginzburg-Landau equations for the functional ( 1 ) has the form
-L2n[(I+2C)(O-i-L'~)n +g~z2Jql=E~/,, z ~2
2
. 02-1
2
2 0 z 0 -Ln[(~xx-i ~-~n)+k~zJth =Et/2,
LH2--
2ell hC
(3)
1(4 k= ~ '
(4'
'
~o , -21t~2E=in
~2 h2Kl a '
2K~
"
izox
2xin(x-xo) ao
~-+-~ [ -
kz-zo
~2
I+2C)k_I)IA[2(
2~zL2n~ 2] :
\ Lu
12> 1) IBI2( I~a°12>
I'
[B[4( 1~o 14) )
+21AI21BI2< I~ao121~012)
For superconductors described by an order parameter which has only one complex component the eigenfunction corresponding to Emi~ may be used as a trial function for the solution in the field range H
I_
(6)
The ql and /~2 sublattices are shifted from one another. From now on we shall assume that the Ginzburg-Landau parameter 2/~ (2 is the London penetration depth, ~ is the coherence length) meets the condition 2/~>> 1. This is undoubtedly true for UPt3. From ( 1 ), (5), (6) we get the following expression for the free energy,
+ (1 + b ) ( I a [ ' ( IVo 14) +
rrhc ~°=]~
ql =A~o =A ~ e x p [ i x p n ( n - 1 ) 4
]"
-
Too -
The upper critical field corresponds to the minimum energy level Emin of this system: He2
/ 2ninx q2 = B~o = B ~ exp~ i ~tpn( n - 1) + - ao ( z - 2 n L 2n/ao)2~
fGL
K2+K3
C=--
261
(5)
+2btQI
IAI21BI2cos(yQ+2yB--2~A),
A= IAI exp(i~,A),
(7)
B = IBI exp(iyB),
Q = IQI exp(i7e) = (¢/~2~a2) , a2
b2
f o = - ~ V, b= b--~, where ( > denote spatial averages and V is the superconductor volume. We shall consider here the case p = 0.5 since only such vortex lattices were observed in Ref. [24]. For conventional uniaxial superconductors the value of the a parameter is determined by the Ginzburg-Landau mass tensor and does not depend on the magnetic field. The changing of the a value with decreasing magnetic field is one of the unusual properties of the mixed state in UPt3. This phenomenon was observed in Ref. [24] at low temperature and one can assume that it occurs near T~ too. We have analyzed expression (7) for f OL numerically as a function of variables a=2nL2/a g, Xo, Zo for different values of parameters b and C. The energetically most favourable lattice structures are found and the results of numerical calculations are presented in the phase diagram (Fig. 1 ) for the particular case C = 0.1. The solid and dashed lines in Fig. 1 correspond to the second and first order phase tran-
262
Yu.S. Barash, A.S. Mel'nikov / Physics Letters A 186 (1994) 259-264
sitions respectively. In region 1 of this phase diagram we have ~/1=0, r/2~0 (we would have t h e 0 , r/2=0 for C < 0 ) . The r/2 lattice in this case is a triangular one expanded along the z-direction by a factor x/~ (p=0.5, a = x / / ~ / 2 ) . In regions 2-5 both the components rh and t/2 are nonzero. For b > 0.44 (region 5) the zeros of these components coincide (Xo= Zo= 0) and so the vortex lattice is a singular one with the parameters p=0.5, e = x / ~ / 2 which do not depend on the applied magnetic field. The case b < 0.44 is more interesting since the free energy minimum corresponds to nonzero values of Xo and Zo and the vortex lattice appears to be nonsingular. For one of the nonsingular phases (region 4) the a parameter does not depend on the magnetic field ( a = 0 . 5 k x / ~ and (Xo, Zo)= (ao/4, - a a o / 2 ) ) . In region 3 the values of Xo, Zo are the same but the tr parameter increases from the value x / / ~ / 2 with the decreasing of the magnetic field below H*. For region 2 we obtained (Xo, Zo) = (ao/2, 0). The tr parameter in this region increases from the value 0.5 k x / ~ just below the second order phase transition line H * ( T ) when H decreases. Note that putting the r/l vortices at the centroid of an r/2 triangle is not energetically favourable for all positive b values. The magnetic field dependence of the a parameter for regions '2 and 3 is shown in Fig. 2. From the transformation properties of ~h and r/2 components (given by expressions (5), ( 6 ) ) with respect to translations [ 17] one can see that the values (Xo, Zo) = (a o/2, 0) and (Xo, Zo) = (ao/ 4 , - t r a o / 2 ) correspond to the order parameter lat4
6
0.5
1
0.4-
2
3
0.3i
i
i
~
q 0.6
i
i
i i.
H/H~2
Fig. 2. Field dependence of the lattice distortion for C=0.1 and for various b values: (1) b.=0; (2) b=0.03; (3) b=0.09; (4) b=0.12; (5) b=0.18; (6) b=0.24.
time the magnetic field itself has a single quantum periodic structure below and above H* for this field direction. It is worth noting that the tr values 0.5 k x / ~ and 0.5x//~ (at p = 0 . 5 ) correspond to two hexagonal vortex lattices which are turned at the angle n / 6 in the plane (x, z) relative to each other and then are expanded along the z-direction by a factor x/~. For a = 0.5 k v / ~ ( a = 0 . 5 x / ~ ) one of the basis vectors of the vortex lattice is parallel to the z-axis (x-axis). These two a values correspond to (a) and (b) (B1 and B3 ) lattice types which were considered in Ref. [25] (see also Ref. [27]). Note that for conventional uniaxial superconductors for H_I_c there exists a continuum of vortex lattices with the same free energy and different orientation relative to the crystal axes [27 ]. From our results one can see that in the case of unconventional hexagonal superconductors this degeneracy may be removed below H * ( T ) . Let us now compare our results with the neutron diffraction experimental data for UPt3 [24]. Since the basis vectors for the shifted sublattices of r/1 and r/2 order parameter components are the same, one can use the conventional expression for the angle a between the x-axis and the first Bragg peak in neutron diffraction experiments: tan a = 1/2a. Therefore the increasing of the parameter tr which occurs in regions 2 and 3 with the magnetic field decreasing results in the decreasing of or. Such a behaviour of ot was observed experimentally for UPt3 (see Ref. [24] ). The a values which may be estimated using these ot measurements are close to a = 0 . 5 x / / ~ if we take the k value from the experimental data [ 4, 5 ] (k ~ 2 ). Since this tr value is energetically favourable for region 3 (not for region 2) of the phase diagram (Fig. 1 ) we can get the range of allowed b values for UPt3. It seems to be important that the value b = 0.17 obtained from experiments on Hcl and He2 measurement in UPt3 [28] also corresponds to region 3. According to the recent experimental results [29,30 ] the b value for UPt3 appears to depend essentially on the concentration of both magnetic and nonmagnetic impurities. Then the gradual change of the vortex lattice distortion considered above occurs only for a certain impurity concentration interval. As for the parameter C, we have carried out our calculations for C = + 0.1, + 0.2, + 0.3. When we increase the absolute value of C the distance between two lower energy levels intices with two flux quanta per unit cell. At the same
YuS. Barash, AS. Mel’nikov /Physics Letters A 186 (1994) 259-264
creases and therefore the second order phase transition occurs at the lower value H*/Hc2. As for the lattice structure below this phase transition line H* ( T), there are no qualitative differences in the phase diagrams for various C values. According to our results the u parameter does not depend on the magnetic field above H*(T) in contrast to its behaviour below the transition line. This feature may be used for the experimental identification of the phase transition. We believe that the experimental data (see Ref. [ 241) cannot yet allow one to make unambiguous conclusions about the a(H) dependence above H*(T), probably due to the insufficient precision of the measurements. The approach used above can be justified only for H*(T) close to Hc2, i.e. for small enough I C] values. Therefore we can analyze the phase diagram in UPt, for Hlc only qualitatively. In addition, for a complete quantitative analysis it is also necessary to take into account the influence of a weak symmetrybreaking field on the phase transitions in UPt,. It is especially important if the parameter C has an extremely small value (see Ref. [ 3 1 ] ). Effects which are analogous to those discussed above should also occur within the framework of the “AB” model proposed in Ref. [ 19 1. It follows from the fact that the Ginzburg-Landau functional of the “AB” model for HI c coincides with expression ( 1) for some particular values of the Ginzburg-Landau coefficients. Further theoretical and experimental work is necessary to find out which of these models is more suitable for UPtS #I. In conclusion, our calculations give evidence for a
xl When this paper was ready for publication we were informed of the results of Ref. [ 32 1. Within the framework of the “AB” model (for a certain set of the Ginzburg-Landau free energy parameters) it was found that the flux lattice evolves continuously in the vicinity of the tetracritical point in the H-T plane. The vortex lattice orientation near H*( T) is the same as the orientation observed in Ref. [ 241 and obtained above for region 3 (Fig. 1). At the same time the behavior of the function a(H) below H*(T) in Ref. [ 321 appears to be different. While we obtain, in accordance with the experimental results, an increase of u with a decrease of the magnetic field, in Ref. [ 321 a decrease of u with a decrease of H is obtained. The important point is that the measurements of Ref. [24] were carried out only at low temperatures. Measurements in the vicinity of T, (and T*) are necessary in order to find out which theory is more preferable for UPt,.
noticeable
263
magnetic
field dependence of vortex latunconventional superconductors for HI c. It is shown that for nonsingular vortex lattices the degeneracy of various lattice types (i.e. (a) and (b) lattices if we use the notations of Ref. [ 251) in anisotropic superconductors is removed. These results give the possible explanation of some unusual phenomena observed in neutron diffraction experiments in UPt, and may be considered as an additional indication of unconventional superconductivity in this compound.
tice distortions
in hexagonal
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