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Induced ferromagnetism in ErRh4B4
5011d 5tate C0mmun1cat10n5, V01. 46, N0.2, pp. 133-137, 1983. Pr1nted 1n 6reat 8r1ta1n.
0038-1098/83/140133--05503.00/0 Per9am0n Pre55 Ltd.
1NDUCED FERR0MA6NE715M 1N ErRh484 L.N. 8u1aev5k11, A.1.8u2d1n, D.1. Kh0m5k11 and 5.V. Panjuk0v Phy51c5 P.N. Le6edev 1n5t1tute, M05c0w, U.5.5.R. (Rece1ved 6 Decem6er 1982 6y KM. A9ran0v1ch) 70 exp1a1n the exper1menta1 data 0n the c0ex15tence 0f 5uperc0nduct1v1ty and ma9net1c 0rder1n9 1n ErRh484 we 5upp05e that the temperature 0f the tran51t10n t0 the ferr0ma9net1c 5tate 1n the a65ence 0f 5uperc0nduct1v1ty var1e5 0ver the 5amp1e5 due t0 the 5tre55e5 and th15 tran51t10n 15 15t 0rder 0ne 0r very c105e t0 1t. We exp1a1n the 0r191n 0f ferr0ma9net1c peak5 (F), 5ate111te5 peak5 and 6r0ad 11ne5near F-peak5 1n the framew0rk 0f the the0ry, wh1ch pred1ct5 the d0ma1n-11ke ma9net1c 5tructure 1n the c0ex15tence 5tate. 7he h19her 5ate111te5 peak5 1nten51t1e51n 5can5 are ca1cu1ated a55um1n9 the 1rre9u1ar1ty 0f d0ma1n 5tructure. 1.1N7R0DUC710N 51NHA et a1. [1,2] have 5tud1ed the ma9net1c 5tructure 1n a 51n91ecry5ta1 0f ma9net1c 5uperc0nduct0r ErRh484 w1th cr1t1ca1 temperature 7e1 ~ 9 K. 0n c0011n9 6e10w 1.2 K the 9r0wth 0f ferr0ma9net1c (F) peak5 and the1r 5ate111te5 (5) wa5 065erved, the p051t10n 0f 5ate111te5, 1.e. wavevect0r 0f 1nh0m09ene0u5 ma9net1c 5tructure Q 6e1n9 pract1ca11y 1ndependent 0n temperature. At 0.7 K the 5ate111te5a5 we11 a5 5uperc0nduct1v1ty d15appear a6rupt1y, the 5uperc0nduct1v1ty 6e1n9 detected 6y re515t1v1tymea5urement5.8e10w 0.7 K the c0herent ma9net1c 5tructure 15 character12ed 6y the F-peak5 0n1y. At any temperature the net 1nten51ty 18 0f 5ate111te5 0f a 91ven F-peak wa5 much 5ma11er than the F-peak 1nten51ty 10 (15 ~< 0.1 10), and the t0ta1 1nten51ty 0f a11 the peak5 (5 and F) a60ve 0.85 K wa5 much 5ma11er than 1t w0u1d 6e 1f the 2nd 0rder pha5e tran51t10n take5 p1ace at 1.2 K. 8e51de5 the dependence 10(7) d0e5 n0t c0rre5p0nd t0 the 2nd 0rder ma9net1c pha5e tran51t10n 6ecau5e d210/d7 2 > 0 at 7 > 0.7 K and 10 r15e5 very 5teep1y (fr0m 0.13 10(0) up t0 0.7 10(0) a5 temperature decrea5e5 fr0m 0.85 t0 0.7 K. M0re0ver 6e10w 1.2 K d0wn t0 the 10we5t temperature the 6r0ad 6au551an 11ne (8-11ne) wa5 065erved near F-peak5.1t 5eem5 t0 c0me fr0m the 5ma11percenta9e 0f 5ma11 ferr0ma9net1c re910n5 0f 512e ~ 100 A 1nc0herent w1th the ma1n ferr0ma9net1c 5tructure. 7he auth0r5 [1, 2] c0nc1ude that 6e10w 1.2K the 5amp1e c0n515t5 0f ferr0ma9net1c n0rma1 (FN) re910n5 w1th 5maU 1nc0herent ferr0ma9net1c re910n5, wh1ch 91veF-peak and 8-11ne c0rre5p0nd1n91y, a5 we11 a5 5uperc0nduct1n9 re910n5 w1th 1nh0m09ene0u5 ma9net1c 5tructure at 7 > 0.7 K, the 1atter 6e1n9 re5p0n5161e f0r 5-peak5. 7h15 p1cture 15 1n a c0ntrad1ct10n w1th the pred1ct10n5 0f the the0r1e5 0f 1nh0m09ene0u5 ma9net1c 133
5tructure 1n 5uperc0nduct0r5 w1th 2nd 0rder ferr0ma9net1c tran51t10n at 0 1n the a65ence 0f 5uperc0nduct1v1ty [ 3 - 5 ] . 7he the0ry wh1ch take5 1nt0 acc0unt the exchan9e (EX) and e1ectr0ma9net1c (EM) 1nteract10n5 0f 10ca112ed m0ment5 (LM) and e1ectr0n5 a5 we11a5 ma9net1c an150tr0py 0f ea5y ax15 type [3(a)] pred1ct5 f0r 0 •¢ 7et the ma9net1c tran51t10n at 7 M ~ 0 1nt0 the 5uperc0nduct1n9 5tate w1th 51nu501da11ym0du1ated ma9net1c 5tructure 0f LM, the wavevect0r 0f m0du1at10n Q 6e1n9 QM ~ ( a2 ~000/0ex) 1/3. 7h15 15 Ander50n-5uh1 re5u1t [6], a 15 ma9net1c 5t1ffne55 0f 0rder 0f at0m1c 1en9th, 60 15 5uperc0nduct1n9 c0rre1at10n 1en9th, 00 15 the ferr0ma9net1c 9r0und 5tate ener9y per 0ne LM (0f 0rder 0f 7M) and 0ex 15the c0ntr16ut10n fr0m 10n9 wave1en9th EX 1nteract10n t0 0 0 . 0 n c0011n9 6e10w 7m the 51nu501da1 5tructure tran5f0rm5 very rap1d1y 1nt0 the tran5ver5e 0ne d1men510n5a1 d0ma1n ma9net1c 5tructure (D5 pha5e), the th1ckne55 0f d0ma1n5 6e1n9 d = -~ "~ rr (~0p/0exn) 1/2 -~ rr (00a~0/0ex) 1/2,
(1)
where p 15 the d0ma1n wa11 5urface ener9y and n 15 LM c0ncentrat10n. 7he expre5510n (a) 15 va11d 1f
0ex >> 0m d2/~r2 ~,~,,
(2)
where ~,L 15 the L0nd0n penetrat10n depth and 0m 15 the 10n9 wave1en9th EM c0ntr16ut10n t0 00 (0,n = 27rn/~2 ,/a 15 ma9net1c m0ment 0f LM). 7he EM 1nteract10n 151ne55ent1a11n the 1nterp1ay 0f 5uperc0nduct1v1ty and ma9net1c 0rder1n9 a510n9 a5 (2) 15 fu1f111ed.0n c0011n9 the 15t 0rder tran51t10n D5 -~ F N 0ccur5 when the n0ma112ed m0ment 5(7) (5(0) = 1) reache5 the va1ue5e2 wh1ch 15 determ1ned 6y the expre5510n (1n pure ca5e)
134
1NDUCED FERR0MA6NE715M 1N ErRh484 h05e2
5e2 1n - -
A0
~
3UF
0ex = h~V(0),
22rh0d •
A 0 = 1.76 7c1
(3)
where N(0) 15 the e1ectr0n den51ty 0f 5tate5 per 0ne LM. 0n warm1n9 FN 5tate 5urv1ve5up t0 the 0verheat1n9 temperature 7e(~) wh1ch c01nc1de5 pract1ca11y w1th 7M. 1n th15 p1cture 1n D5-pha5e 0n1y 5-peak5 and 1n F N 5tate 0n1y F-peak5 5h0u1d 6e 065erved. 8e51de5 the net 1nten51ty 15 ha5 t0 6e pract1ca11y the 5ame a5 the 1nten51ty 0f F-peak5 wh1ch w0u1d 6e 065erved at the 5ame temperature 1n the a65ence 0f 5uperc0nduct1v1ty (e.9.1n 0verheated FAr 5tate) and 60th 5h0u1d f0110w 54uared 8r1110u1n curve. 7he 6ehav10ur 0f H0M0658 5amp1e5 [7] 15 1n acc0rdance w1th th15 p1cture, 6ut that 0f ErRh,84 15 n0t [1, 2, 8]. 7he the0ry 6a5ed 0n EM 1nteract10n [4, 5] 15 va11d 1f the 1ne4ua11ty 0pp051te t0 e4uat10n (2) 15 fu1f111ed. 1n the framew0rk 0f 5uch a the0ry w1th 7M "¢ 7c1, 1n the pre5ence 0f an150tr0py, the 51nu501da1 ma9net1c 5tructure w1th wavevect0r Qm ~ (0 0a 2 60 X2L10m)115 appear5 at 7M, and 0n c0011n9 1t tran5f0rm5 rap1d1y 1nt0 D5 pha5e w1th d0ma1n th1ckne55 d ~ rr (00a~0X2L/0ra) 114 (We5upp05e QM~0 >> 1 and ~01d >> 1, the D5-pha5e 15 ana1090u5 t0 the 11near p01ar12ed 501ut10n 1ntr0duced 1n [4]). 0n c0011n9 the D5 pha5e may 90 1nt0 the 5p0ntane0u5 vertex 5tate 0r FN-5tate, the 1atter 6e1n9 p055161e at temperature5, where 4nnp5(7) >/~e2(7).
(4)
Here H~,2 15 the 0r61ta1 upper cr1t1ca1 ma9net1c f1e1d wh1ch may 6e 06ta1ned 6y extrap01at10n 0f He2 (7) near 7e1 t0 the 10wer temperature5 [9]. 7h15 p1cture a/50 d0e5 n0t exp1a1n the 51mu1tane0u5 ex15tence 0f 60th F- and 5-peak5 1n ErRh484 and temperature dependence 0f the1r 1nten51t1e5.M0re0ver the EM the0ry [3, 4] cann0t exp1a1n the tran51t10n t0 FN 5tate 6ecau5e e4uat10n (3) 15 n0t fu1f111ed1n ErRh484 (42r/an ~- 6.5 K0e and/-/~e2( 7 >> 7~t) ~ 11-14 K0e acc0rd1n9 t0 data [10, 161 ). We remark that 1n ErRh484 the c0nd1t10n (2) 15 certa1n1y fu1f111ed6ecau5e 0ex ~ 0 . 5 - 1 . 3 K , 0m ~ 1.8K, d = 50A. and ;~L = 750A fr0m the data [16] 0n H. 7he va1ue 0ex 15 e5t1mated 6y A6r1k050v-60rk0v re1at10n d7¢/(x)/dx rr20ex/2 where 7et (x) 15 the cr1t1ca1temperature 0f c0mp0und5 ErxY1-xRh484, the data [11] f0r 7e1 (x) 9et 0ex ~ 0.5 K. 7he va1ue 0ex may 6e 06ta1ned a150 fr0m the data [8] 0n the n0rma112ed 1nten51t1e5 0f F-peak 10 at 7 = 2 and 5 K 1n the pre5ence 0f ma9net1c f1e1dH = He2(7). At the5e temperature5 He2 15 determ1ned 6y the parama9net1c effect 0n1y due t0 the 1ne4ua11ty He2 ,~H~e2 (5ee data [16] ) 1.e. 6y the exchan9e f1e1dh 0f LM act1n9 0n e1ectr0n5. 50 we 9et h = h05 ~. 1.3 7e1 [9] and 0ex = h~N(0). F0r 5= ~ ~ 0.22 and N(0) -~-6.02 eV -1/Er (5ee [ 10] ) we 06ta1n 0ex ~ 1.3 K.
V01.46, N0.2
2.7HE MA1N A55UMP710N5 70 exp1a1n the exper1menta1 data 0n ErRh484 we a55ume that (1) 7he cr1t1ca1temperature 0 0f the tran51t10n N ~ FN 1n the a65ence 0f 5uperc0nduct1v1ty var1e5 0ver the 5amp1e5 5tud1ed due t0 the 1nh0m09ene0u5 5tre55e5. We 5ee 6e10w that th15 ma1n a55umpt10n a110w5 t0 exp1a1n the 51mu1tane0u5 ex15tence 0fF- and 5-peak5. (2) 7he 1ncrea5e 0f F-peak 1nten51ty 10(7) 1n the 1nterva1 0.7 K < 7 < 0.85 K 15 very 5teep 1n the 51n91e cry5ta1 [1, 2] and 1t d0e5 n0t c0rre5p0nd t0 the 54uared 8r1110u1n curve. 7he 5uperc0nduct1v1ty 1t5e1fd0e5 n0t effect pract1ca11y the va1ue 0f 5p0ntane0u5 m0ment5 [3], and 15(7) >>10(7). 50 10(7) character12e5 the ferr0ma9net1c 0rder1n9 wh1ch w0u1d 6e 065erved 1n the a65ence 0f 5uperc0nduct1v1ty. 7he d15tr16ut10n 0f 0 can 0n1y d1m1n15hthe 510pe 0f the 10(7) curve. 7hu5 t0 exp1a1n the dependence 10(7) we a55ume that the tran51t10n N ~ FAr at 0 5h0u1d 6e the 15t 0rder 0ne (0f 1nduced ferr0ma9net15m type) 0r at 1ea5t c105e t0 1t. 5uch a 5u99e5t10n wa5 made 6y 8ehr0021 et a1. [12] fr0m ma9net1c mea5urement5 0n p01ycry5ta111ne5amp1e5. A 5tr0n9 ev1dence 1n fav0ur 0f 1t 15 pr0v1ded 6y ma9net1c mea5urement5 d0ne 6y M00k et a1. [2] and Cra6tree eta1. [16]. Fr0m the data 0f [2] 0ne can 5ee that there ex15t a hy5tere515 1n the dependence 0f ma9net12at10n 0n ma9net1c f1e1d 1n the n0rma1 5tate at h19h en0u9h temperature5 (at 1 and 2K a60ve therm0dynam1c cr1t1ca1 f1e1d), wherea5 there 15n0 hy5tere515 1n ferr0ma9net1c pha5e at 0.43 K [ 16]. 7hu5 the 0r191n 0f th15 hy5tere515 15 n0t the u5ua1 d0ma1n effect5 6ut ev1dent1y the meta5ta6111ty 1n the 0verheat1n9 re910n 1n 15t 0rder tran51t10n at 7M. At 5t111h19her temperature5 (0ut51de the 0verheat1n9 re910n) hy5tere515 15a65ent. 8e51de5 fr0m the data 0f [2] 0ne can 5ee that at 10w H the dependence M(1-1) 15 c0ncave wh1ch 15 a150 typ1ca1 0f 15t 0rder ma9net1c tran51t10n. 5pec1f1c heat mea5urement5 0f W001fet a1. [13] 5h0w that the 10w temperature 6ehav10ur 0f ErRh484 15 determ1ned 6y tw0 10we5t d0u61et5 w1th the ener9y 5p11tt1n9 ~ 1 K. 7he 1nduced ma9net15m may 6e cau5ed 1n th15 ca5e 6y the c0ntr16ut10n 0f the h19he5t d0u61et t0 the m0ment a10n9 6-ax15 0•)- 1n the p1cture 0f 1nduced ma9net15m 0ne can a150 exp1a1n the d1fference 0f 5aturat10n m0ment at 10w temperature5 and at h19h f1e1d [16]. 1t 15 kn0wn that the 5p0ntane0u5 m0ment 1n ErRh484 11e51n the 6a5a1 p1ane, w1th tw0 a pr10r1 e4u1va1ent ax15 1n 1t, a and 6.51nha et a1. [1] have n0t1ced, h0wever, that actua11y the ma9net12at10n wa5 0r1ented a10n9 0ne 0f the5e ax15, 6, pre5uma61y 6ecau5e 0f the 5119ht 5tre55e5 pre5ent 1n the 5amp1e. 5uch a 5tr0n9 dependence 0f ma9net1c character15t1c5 0n ax1a1 5tra1n5 15 typ1ca1 0f 1nduced ma9net15m and
Vol. 46, No. 2
INDUCED FERROMAGNETISM
it also confirms our assumptions. We see below that together with first one this allows us to explain the small intensity of satellites in comparison with F-peaks. At first we consider how the second assumption modifies the results of the theory and then we take into account the distribution of 0 to obtain the final picture. 3. NUCLEATION PROCESS OF DS PHASE FORMATION As long as the transition is 2nd order one magnetic domain structure establishes itself starting from the sinusoidally modulated structure with infinitely small amplitude at TM. As the magnetic transition approaches the 1st order one, growth of magnetization S on cooling below TM in DS phase become more steep. So the interval near TM, where s is small and Q depends strongly on temperature, narrows. Besides, T,, approaches TM. Let us now consider how the magnetic ordering appears in the presence of superconductivity in the case of 1st order magnetic transition N -+ FN (see [3b] for more detailed discussion). The DS phase forms when the jump of magnetization s at 0 is less than sc2 which is determined by equations (1) and (3). At first, the ferromagnetic region appears in the form of small rode-like nucleus due to the easy axis anisotropy [ 141. Its subsequent growth depends crucially on the interplay with superconductivity: the nucleus will grow in the form of thin plate unbounded in two directions but with the thickness d not exceeding go. Indeed this form is energetically most favourable: it permits to gain as much as possible magnetic energy without too much less of superconducting condensation energy (as far as d 4 [A fhe superconductivity will not be destroyed by such a magnetic domain). The resulting value of d can be estimated as follows: we minimize the sum of magnetic energy of the plate and the change of superconducting energy due to the magnetic ordering in
+ p + fe,,m2(T)
d2 -
135
TM at which the plate may appear. For &F(T) one gets: &Y(r) = (e,,s2p/~,t2p2 = eos2(~/t;o)l/2, i.e. = - e (a/,$o)1’2 and d,,,, = (eoaEo/e,,)1’2. TM-e The magnetization in the plate is along easy axis, the normal to the surface of plate should be perpendicular to the easy axis (i.e. in xz plane) owing to the EM energy and its final direction in xz plane should minimize the domain wall energy p. Now when the growth of the first place was stopped by superconductivity, the second nuclei can appear. It is evident that the exchange and magnetic field of the first plate decrease the activation energy for the second nuclei if it is located near the first one and its moment is opposite to the moment of the first plate. The second nuclei then also grows transforming into the second plate adjoining to the first one. Repetition of this process gives the one-dimensional transverse domain-like magnetic structure in the superconducting state, the creation of the first critical ferromagnetic nuclei being the bottle neck. The characteristics of this domain superconducting (OS) phase itself are the same as in the case of the 2nd order transition, the only difference is the independence of the vector Q = n/d on the temperature near TM. Experimentally moment is along b-axis and Q = (0.042 a*, O,O.S55c*) [ 11. The values 2d = 100 A is obtained by use of equation (1) with to = 200 A from the data [ 161 on Hc2 near T,., and a reasonable values of eoleex x 1 anda% 1 a. The domain structure should be characterized by the satellites peaks (2~ + l)Q (K integer), in monocrystal the higher peak intensities being 1, = (2~ + l)-2Zo in the case of ideal domain structure. Thus the 3Q-peak intensity should be 11% of the Q-peak. Experimentally 3Q-peak was not found, i.e. its intensity was less than 2% of the Q-peak [ 11. This discrepance may be explained by irregularity of domain structure which is expected due to internal stresses and the nucleation process of DS phase formation in the case of 1st order transition. Taking the Gaussian for the domain thickness distribution d we obtain the dependence Zsk on q = (2~ + 1)Q + 9 (a*, 0, c*) near (10 1) Bragg peak in the form
4‘0 Iskb?)
Here 6 Yis the gram of magnetic energy (per unit volume), o is an area of the plate. The EM contribution is neglected here due to the condition (2) which may be checked after calculation. Minimizing .&l w.r.t. d one gets the thickness dM of ferromagnetic plate in superconductor, i.e. dM = ~oS.F(T)/e,S2(T)n. The condition Fpl(dM, T) < 0 determines the temperature
IN ErRh4B4
A0
-A;:
, hk = (2K
+
1)2ho.
+Tq
Thus the 3Q-peak is 81 times weaker at r) = 0 and 9 times broader than Q-peak. In measurements of Sinha et al. [ 1, 21 the resolution was about the same as A0 and so the search of 3Q-peak was unsuccessful. 4. PHASE TRANSITIONS Thus below the temperature TM = e the DS phase forms regardless of the type of transition at TM. On
136
1NDUCED FERR0MA6NE715M 1N ErRh484
V01. 46, N0.2
further c0011n9 the 15t 0rder tran51t10n D5 ~ FN 0ccur5 at 7e2 when 5(7, 0) = 5c2 and acc0rd1n9 t0 the 5ec0nd a55umpt10n the 1nterva1 0f D5 pha5e ex15tence may 6e very narr0w even 1f 5e2 15 0f 0rder 0f un1ty (we 9et 5e2 ~ 0.8 fr0m the data 0f [1, 2], the the0ret1ca1 e5t1mat10n 6e1n9 ~ 1 fr0m e4uat10n (3) at 0ex ~ 1.3 K and vF ~ 5.5 7c1~0/h= 1.4 x 107 cm5ec-1). We remark that 50me part 0f d0ma1n5 may rema1n 1n FN pha5e 6e10w 7e2 due t0 the p1nn1n9 0f d0ma1n wa1150n 1mperfect10n5, the5e ••wr0n9•• d0ma1n5 1n ferr0ma9net1c pha5e may 91ve 6r0ad 11ne (8-11ne) near F-peak5. N0w t0 exp1a1n the temperature dependence 0f the F- and 5-peak 1nten51t1e5we mu5t 1nv0ke the f1r5t a55umpt10n. Acc0rd1n9 t0 1t the tran51t10n5 5 -+ D5 and D5 ~ FN take p1ace at d1fferent temperature5 1n the d1fferent part5 0f the 5amp1e. At 50me temperature 7 the f0110w1n9three pha5e5 may 6e pre5ent 1n the 5amp1e:
51mp1e p1cture 5h0u1d 6e 065erved: 0n c0011n9 0n1y 5ate111te5 (2~r + 1)Q appear 5teep1y 6e10w 7M and at 7e2 they are rep1aced 5udden1y 6y F-peak5 (p055161y w1th 8-11ne). 7he 1nten51ty 0f Q-peak a60ve 7e2 5h0u1d 6e 0n1y 5119ht1y1e55than 1nten51ty 0fF-peak 6e10w 7c2. 7he f0110w1n9exper1ment5 may e1uc1date f1na11y the p1cture pre5ented a60ve. 7he type 0f the tran51t10n N ~ FN at 0 1n the a65ence 0f 5uperc0nduct1v1ty may 6e 065erved 1n 5amp1e5 under m1cr0wave 0r 1a5er 1rrad1at10n 5uff1c1ent t0 de5tr0y the C00per pa1r1n9. 1n 5uch a 5amp1e 0n1y F peak5 5h0u1d 6e pre5ent 1n neutr0n 5catter1n9 6e10w 0 w1th hy5tere515 1n near 0. 7he mea5urement5 0f ma9net12at10n and re515t1v1ty under un1ax1a1 pre55ure may check the a55umpt10n 0f the 5tr0n9 dependence 0f 0 0n 5tre55.7he d1rect 5tudy 0f the exact nature 0f 10w-1y1n9cry5ta1 f1e1d 1eve151n ErRh484 (e.9. v1a E5R) w0u1d 6e a150 very 1nf0rmat1ve: 1t w0u1d perm1t 0ne t0 ca1cu1ate (1) n0nma9net1c 5uperc0nduct1n9 pha5e 1n re910n5 the0ret1ca11y a11 the character15t1c5 0f ma9net1c w1th 7M < 7, (2) 5uperc0nduct1n9 D5 pha5e, wh1ch 91ve5 5ate111te5, tran51t10n. 7he d1rect c0nf1rmat10n 0f d0ma1n-11ke ma9net1c 5tructure 1n the c0ex15tence 5tate may c0me 1n the re910n5 where 0 < 5(7, 0) < 5c2, fr0m the m0re accurate 5tudy 0f 5ate111te5peak5 1n (3) n0rma1 ferr0ma9net1c pha5e w1th re51dua1 neutr0n 5catter1n9 0r fr0m the 1nterna1 f1e1d5 d0ma1n5, wh1ch 91ve5F-peak5 and 8-11ne5, 1n re910n5 1nve5t19at10n5 (e.9. v1a NMR). where 5(7, 0) > 5e2. 1n th15 p1cture the 5ma11ne550f the re1at1ve 5ate111te5 1nten51ty 18/10 re5u1t5 fr0m the rap1d 1ncrea5e 0f 5(7, 0) fr0m 0 t0 5e2 1n the narr0w 1nterva1 (7M, 7e2) and the fact 0f 0-d15tr16ut10n 1n the 1nterva1 wh1ch 15 much 6r0ader than (7• -- 7c2). We 9et 1n the crude appr0x1mat10n
8 -18(7 ) ~, -f1 52D5(7M d/°(7) ~ 5~8Nf(7), d7
-
-
7e2)f(7), (7)
where 5205and 5~N are the mean va1ue5 0f 52 1n D5 and FN pha5e c0rre5p0nd1n91y and f•(0) 15 the d15tr16ut10n funct10n 0f 0.50 15(7) and the effect1ve heat capac1ty c(7) are pr0p0rt10na1 t0 f(7) ~ d10/d7. Exper1menta1 data [1, 13] are 1n acc0rdance w1th th15 re1at10n5. 7he tran51t10n t0 the n0rma1 5tate 1n re515t1v1ty 15 perc01at10n 0ne, 1t 0ccur5 when FN pha5e 15 75% 0f the wh01e v01ume 0f the 5amp1e. 7he ma9net1c f1e1d 1ncrea5e5 the va1ue 0f 7e2 (5ee [15] ). 7h15 15 a rea50n 0f 1ncrea51n9 0f10 and decrea51n9 0f18 1n the ma9net1c f1e1d 06ta1ned 6y 51nha et a1. [ 1, 2]. 50 0ur a55umpt10n5 a110w u5 t0 exp1a1n the ma1n feature5 0f ErRh484 6ehav10ur. We 5ee that many 0f them are cau5ed 6y the 1mperfect10n5 0f the 5amp1e5 and n0t 6y the 1nterp1ay 0f 5uperc0nduct1n9 and ma9net1c 0rder1n9 1t5e1f. 1n perfect 51n91ecry5ta15 m0re
REFERENCE5 1.
5.K. 51nha, 6.W. Cra6tree, D.6. H1nk5 • H. M00k, Phy5. Rev. Lett. 48, 950 (1982). 2. H.A. M00k, 0.A. Pr1n91e, 5. Kawara2ak1, 5.K. 51nha, 6.W. Cra6tree, D.6. H1nk5, M.8. Map1e 2. F15k, D.5. J0hn5t0n • L.D. W001f, Pr0c. 4th C0nf. 0n 5uperc0nduct1v1ty 1n d- and f-6and Meta15, p. 201 Kar15ruhe (1982). 3. (a) L.N. 8u1aev5k11, A.1.8u2d1n, 5.V. Panjuk0v • M.L. Ku11c. 5011d 5tate C0mmun. (6) 5u6m1tted t0 Phy5. Rev. 8. 4. H.5.6reen51de, E.1. 810unt • C.M. Varma, Phy5. Rev. Lett. 46, 49 (1981). 5. M. 7ach1k1,Phy51ca 109 • 1108, 1699 (1982). 6. P.W. Ander50n • H. 5uh1, Phy5. Rev. 116,898 (1959). 7. J.W. Lynn, 6.5h1rane, W. 7h0m11n50n, R.N. 5he1t0n • D.E. M0nct0n,Phy5. Rev. 824, 3817 (1981). 8. D.E. M0nct0n, D.8. McWhan, P.H. 5hm1dt, 6. 5h1rane, W. 7h0m11n50n, M.8. Map1e, H.8. McKay • L.D. W001f, 2. F15k; D.C. J0hn5t0n, Phy5. Rev. Lett. 45, 2060 (1981). 9. D. 5a1nt-Jame5, 6.5arma • E.J. 7h0ma5, 7ype 2 5uperc0nduct0r5, Ch. 6. Per9am0n Pre55 (1969). 10. H.R. 0tt, W.A. Fert19, D.C. J0hn5t0n, M.8. Map1e • 8.7. Matt1a5, J. L0w. 7emp. Phy5. 33, 159 (1978). 11. H.8. MacKay, L.D. W001f, M.8. Map1e • D.C. J0hn5t0n, J. L0w. 7emp. Phy5. 41,639 (1980). 12. F. 8ehr0021, 6. W. Cra6tree, C.A. Cam6e11, M. Levy, D.K. 5n1der, D.5. J0hn50n • 8.7. Matt1a5, 5011d5tate C0mmun. 39, 1041 (1981).
V01. 46, N0.2 13. 14.
1NDUCED FERR0MA6NE715M 1N ErRh484
L.D. W001f, D.C. J0hn5t0n, H.8. McKay, R.W. McCa11um• M.8. Map1e.J. L0w. 7emp. Phy5. 35,651 (1979). A.1.Pr1v0r0t5k11, U5p. F12. Nauk (50v. Phy5. U5p.} 108, 43 (1972).
137
15.
L.N. 8u1aev5k11, A.1.8u2d1n • 5.V. Panjuk0v, 5011d5tate C0mmun. 43,135 (1982);JE7P, 83,
16.
768 (1982). 6.W. Cra6tree, F. 8ehr0021, 5.A. Camp6e11• D.6. H1nk5.Phy5. Rev. Lett. 49, 1342 (1982).
0038-1098/83/140133--05503.00/0 Per9am0n Pre55 Ltd.
1NDUCED FERR0MA6NE715M 1N ErRh484 L.N. 8u1aev5k11, A.1.8u2d1n, D.1. Kh0m5k11 and 5.V. Panjuk0v Phy51c5 P.N. Le6edev 1n5t1tute, M05c0w, U.5.5.R. (Rece1ved 6 Decem6er 1982 6y KM. A9ran0v1ch) 70 exp1a1n the exper1menta1 data 0n the c0ex15tence 0f 5uperc0nduct1v1ty and ma9net1c 0rder1n9 1n ErRh484 we 5upp05e that the temperature 0f the tran51t10n t0 the ferr0ma9net1c 5tate 1n the a65ence 0f 5uperc0nduct1v1ty var1e5 0ver the 5amp1e5 due t0 the 5tre55e5 and th15 tran51t10n 15 15t 0rder 0ne 0r very c105e t0 1t. We exp1a1n the 0r191n 0f ferr0ma9net1c peak5 (F), 5ate111te5 peak5 and 6r0ad 11ne5near F-peak5 1n the framew0rk 0f the the0ry, wh1ch pred1ct5 the d0ma1n-11ke ma9net1c 5tructure 1n the c0ex15tence 5tate. 7he h19her 5ate111te5 peak5 1nten51t1e51n 5can5 are ca1cu1ated a55um1n9 the 1rre9u1ar1ty 0f d0ma1n 5tructure. 1.1N7R0DUC710N 51NHA et a1. [1,2] have 5tud1ed the ma9net1c 5tructure 1n a 51n91ecry5ta1 0f ma9net1c 5uperc0nduct0r ErRh484 w1th cr1t1ca1 temperature 7e1 ~ 9 K. 0n c0011n9 6e10w 1.2 K the 9r0wth 0f ferr0ma9net1c (F) peak5 and the1r 5ate111te5 (5) wa5 065erved, the p051t10n 0f 5ate111te5, 1.e. wavevect0r 0f 1nh0m09ene0u5 ma9net1c 5tructure Q 6e1n9 pract1ca11y 1ndependent 0n temperature. At 0.7 K the 5ate111te5a5 we11 a5 5uperc0nduct1v1ty d15appear a6rupt1y, the 5uperc0nduct1v1ty 6e1n9 detected 6y re515t1v1tymea5urement5.8e10w 0.7 K the c0herent ma9net1c 5tructure 15 character12ed 6y the F-peak5 0n1y. At any temperature the net 1nten51ty 18 0f 5ate111te5 0f a 91ven F-peak wa5 much 5ma11er than the F-peak 1nten51ty 10 (15 ~< 0.1 10), and the t0ta1 1nten51ty 0f a11 the peak5 (5 and F) a60ve 0.85 K wa5 much 5ma11er than 1t w0u1d 6e 1f the 2nd 0rder pha5e tran51t10n take5 p1ace at 1.2 K. 8e51de5 the dependence 10(7) d0e5 n0t c0rre5p0nd t0 the 2nd 0rder ma9net1c pha5e tran51t10n 6ecau5e d210/d7 2 > 0 at 7 > 0.7 K and 10 r15e5 very 5teep1y (fr0m 0.13 10(0) up t0 0.7 10(0) a5 temperature decrea5e5 fr0m 0.85 t0 0.7 K. M0re0ver 6e10w 1.2 K d0wn t0 the 10we5t temperature the 6r0ad 6au551an 11ne (8-11ne) wa5 065erved near F-peak5.1t 5eem5 t0 c0me fr0m the 5ma11percenta9e 0f 5ma11 ferr0ma9net1c re910n5 0f 512e ~ 100 A 1nc0herent w1th the ma1n ferr0ma9net1c 5tructure. 7he auth0r5 [1, 2] c0nc1ude that 6e10w 1.2K the 5amp1e c0n515t5 0f ferr0ma9net1c n0rma1 (FN) re910n5 w1th 5maU 1nc0herent ferr0ma9net1c re910n5, wh1ch 91veF-peak and 8-11ne c0rre5p0nd1n91y, a5 we11 a5 5uperc0nduct1n9 re910n5 w1th 1nh0m09ene0u5 ma9net1c 5tructure at 7 > 0.7 K, the 1atter 6e1n9 re5p0n5161e f0r 5-peak5. 7h15 p1cture 15 1n a c0ntrad1ct10n w1th the pred1ct10n5 0f the the0r1e5 0f 1nh0m09ene0u5 ma9net1c 133
5tructure 1n 5uperc0nduct0r5 w1th 2nd 0rder ferr0ma9net1c tran51t10n at 0 1n the a65ence 0f 5uperc0nduct1v1ty [ 3 - 5 ] . 7he the0ry wh1ch take5 1nt0 acc0unt the exchan9e (EX) and e1ectr0ma9net1c (EM) 1nteract10n5 0f 10ca112ed m0ment5 (LM) and e1ectr0n5 a5 we11a5 ma9net1c an150tr0py 0f ea5y ax15 type [3(a)] pred1ct5 f0r 0 •¢ 7et the ma9net1c tran51t10n at 7 M ~ 0 1nt0 the 5uperc0nduct1n9 5tate w1th 51nu501da11ym0du1ated ma9net1c 5tructure 0f LM, the wavevect0r 0f m0du1at10n Q 6e1n9 QM ~ ( a2 ~000/0ex) 1/3. 7h15 15 Ander50n-5uh1 re5u1t [6], a 15 ma9net1c 5t1ffne55 0f 0rder 0f at0m1c 1en9th, 60 15 5uperc0nduct1n9 c0rre1at10n 1en9th, 00 15 the ferr0ma9net1c 9r0und 5tate ener9y per 0ne LM (0f 0rder 0f 7M) and 0ex 15the c0ntr16ut10n fr0m 10n9 wave1en9th EX 1nteract10n t0 0 0 . 0 n c0011n9 6e10w 7m the 51nu501da1 5tructure tran5f0rm5 very rap1d1y 1nt0 the tran5ver5e 0ne d1men510n5a1 d0ma1n ma9net1c 5tructure (D5 pha5e), the th1ckne55 0f d0ma1n5 6e1n9 d = -~ "~ rr (~0p/0exn) 1/2 -~ rr (00a~0/0ex) 1/2,
(1)
where p 15 the d0ma1n wa11 5urface ener9y and n 15 LM c0ncentrat10n. 7he expre5510n (a) 15 va11d 1f
0ex >> 0m d2/~r2 ~,~,,
(2)
where ~,L 15 the L0nd0n penetrat10n depth and 0m 15 the 10n9 wave1en9th EM c0ntr16ut10n t0 00 (0,n = 27rn/~2 ,/a 15 ma9net1c m0ment 0f LM). 7he EM 1nteract10n 151ne55ent1a11n the 1nterp1ay 0f 5uperc0nduct1v1ty and ma9net1c 0rder1n9 a510n9 a5 (2) 15 fu1f111ed.0n c0011n9 the 15t 0rder tran51t10n D5 -~ F N 0ccur5 when the n0ma112ed m0ment 5(7) (5(0) = 1) reache5 the va1ue5e2 wh1ch 15 determ1ned 6y the expre5510n (1n pure ca5e)
134
1NDUCED FERR0MA6NE715M 1N ErRh484 h05e2
5e2 1n - -
A0
~
3UF
0ex = h~V(0),
22rh0d •
A 0 = 1.76 7c1
(3)
where N(0) 15 the e1ectr0n den51ty 0f 5tate5 per 0ne LM. 0n warm1n9 FN 5tate 5urv1ve5up t0 the 0verheat1n9 temperature 7e(~) wh1ch c01nc1de5 pract1ca11y w1th 7M. 1n th15 p1cture 1n D5-pha5e 0n1y 5-peak5 and 1n F N 5tate 0n1y F-peak5 5h0u1d 6e 065erved. 8e51de5 the net 1nten51ty 15 ha5 t0 6e pract1ca11y the 5ame a5 the 1nten51ty 0f F-peak5 wh1ch w0u1d 6e 065erved at the 5ame temperature 1n the a65ence 0f 5uperc0nduct1v1ty (e.9.1n 0verheated FAr 5tate) and 60th 5h0u1d f0110w 54uared 8r1110u1n curve. 7he 6ehav10ur 0f H0M0658 5amp1e5 [7] 15 1n acc0rdance w1th th15 p1cture, 6ut that 0f ErRh,84 15 n0t [1, 2, 8]. 7he the0ry 6a5ed 0n EM 1nteract10n [4, 5] 15 va11d 1f the 1ne4ua11ty 0pp051te t0 e4uat10n (2) 15 fu1f111ed. 1n the framew0rk 0f 5uch a the0ry w1th 7M "¢ 7c1, 1n the pre5ence 0f an150tr0py, the 51nu501da1 ma9net1c 5tructure w1th wavevect0r Qm ~ (0 0a 2 60 X2L10m)115 appear5 at 7M, and 0n c0011n9 1t tran5f0rm5 rap1d1y 1nt0 D5 pha5e w1th d0ma1n th1ckne55 d ~ rr (00a~0X2L/0ra) 114 (We5upp05e QM~0 >> 1 and ~01d >> 1, the D5-pha5e 15 ana1090u5 t0 the 11near p01ar12ed 501ut10n 1ntr0duced 1n [4]). 0n c0011n9 the D5 pha5e may 90 1nt0 the 5p0ntane0u5 vertex 5tate 0r FN-5tate, the 1atter 6e1n9 p055161e at temperature5, where 4nnp5(7) >/~e2(7).
(4)
Here H~,2 15 the 0r61ta1 upper cr1t1ca1 ma9net1c f1e1d wh1ch may 6e 06ta1ned 6y extrap01at10n 0f He2 (7) near 7e1 t0 the 10wer temperature5 [9]. 7h15 p1cture a/50 d0e5 n0t exp1a1n the 51mu1tane0u5 ex15tence 0f 60th F- and 5-peak5 1n ErRh484 and temperature dependence 0f the1r 1nten51t1e5.M0re0ver the EM the0ry [3, 4] cann0t exp1a1n the tran51t10n t0 FN 5tate 6ecau5e e4uat10n (3) 15 n0t fu1f111ed1n ErRh484 (42r/an ~- 6.5 K0e and/-/~e2( 7 >> 7~t) ~ 11-14 K0e acc0rd1n9 t0 data [10, 161 ). We remark that 1n ErRh484 the c0nd1t10n (2) 15 certa1n1y fu1f111ed6ecau5e 0ex ~ 0 . 5 - 1 . 3 K , 0m ~ 1.8K, d = 50A. and ;~L = 750A fr0m the data [16] 0n H. 7he va1ue 0ex 15 e5t1mated 6y A6r1k050v-60rk0v re1at10n d7¢/(x)/dx rr20ex/2 where 7et (x) 15 the cr1t1ca1temperature 0f c0mp0und5 ErxY1-xRh484, the data [11] f0r 7e1 (x) 9et 0ex ~ 0.5 K. 7he va1ue 0ex may 6e 06ta1ned a150 fr0m the data [8] 0n the n0rma112ed 1nten51t1e5 0f F-peak 10 at 7 = 2 and 5 K 1n the pre5ence 0f ma9net1c f1e1dH = He2(7). At the5e temperature5 He2 15 determ1ned 6y the parama9net1c effect 0n1y due t0 the 1ne4ua11ty He2 ,~H~e2 (5ee data [16] ) 1.e. 6y the exchan9e f1e1dh 0f LM act1n9 0n e1ectr0n5. 50 we 9et h = h05 ~. 1.3 7e1 [9] and 0ex = h~N(0). F0r 5= ~ ~ 0.22 and N(0) -~-6.02 eV -1/Er (5ee [ 10] ) we 06ta1n 0ex ~ 1.3 K.
V01.46, N0.2
2.7HE MA1N A55UMP710N5 70 exp1a1n the exper1menta1 data 0n ErRh484 we a55ume that (1) 7he cr1t1ca1temperature 0 0f the tran51t10n N ~ FN 1n the a65ence 0f 5uperc0nduct1v1ty var1e5 0ver the 5amp1e5 5tud1ed due t0 the 1nh0m09ene0u5 5tre55e5. We 5ee 6e10w that th15 ma1n a55umpt10n a110w5 t0 exp1a1n the 51mu1tane0u5 ex15tence 0fF- and 5-peak5. (2) 7he 1ncrea5e 0f F-peak 1nten51ty 10(7) 1n the 1nterva1 0.7 K < 7 < 0.85 K 15 very 5teep 1n the 51n91e cry5ta1 [1, 2] and 1t d0e5 n0t c0rre5p0nd t0 the 54uared 8r1110u1n curve. 7he 5uperc0nduct1v1ty 1t5e1fd0e5 n0t effect pract1ca11y the va1ue 0f 5p0ntane0u5 m0ment5 [3], and 15(7) >>10(7). 50 10(7) character12e5 the ferr0ma9net1c 0rder1n9 wh1ch w0u1d 6e 065erved 1n the a65ence 0f 5uperc0nduct1v1ty. 7he d15tr16ut10n 0f 0 can 0n1y d1m1n15hthe 510pe 0f the 10(7) curve. 7hu5 t0 exp1a1n the dependence 10(7) we a55ume that the tran51t10n N ~ FAr at 0 5h0u1d 6e the 15t 0rder 0ne (0f 1nduced ferr0ma9net15m type) 0r at 1ea5t c105e t0 1t. 5uch a 5u99e5t10n wa5 made 6y 8ehr0021 et a1. [12] fr0m ma9net1c mea5urement5 0n p01ycry5ta111ne5amp1e5. A 5tr0n9 ev1dence 1n fav0ur 0f 1t 15 pr0v1ded 6y ma9net1c mea5urement5 d0ne 6y M00k et a1. [2] and Cra6tree eta1. [16]. Fr0m the data 0f [2] 0ne can 5ee that there ex15t a hy5tere515 1n the dependence 0f ma9net12at10n 0n ma9net1c f1e1d 1n the n0rma1 5tate at h19h en0u9h temperature5 (at 1 and 2K a60ve therm0dynam1c cr1t1ca1 f1e1d), wherea5 there 15n0 hy5tere515 1n ferr0ma9net1c pha5e at 0.43 K [ 16]. 7hu5 the 0r191n 0f th15 hy5tere515 15 n0t the u5ua1 d0ma1n effect5 6ut ev1dent1y the meta5ta6111ty 1n the 0verheat1n9 re910n 1n 15t 0rder tran51t10n at 7M. At 5t111h19her temperature5 (0ut51de the 0verheat1n9 re910n) hy5tere515 15a65ent. 8e51de5 fr0m the data 0f [2] 0ne can 5ee that at 10w H the dependence M(1-1) 15 c0ncave wh1ch 15 a150 typ1ca1 0f 15t 0rder ma9net1c tran51t10n. 5pec1f1c heat mea5urement5 0f W001fet a1. [13] 5h0w that the 10w temperature 6ehav10ur 0f ErRh484 15 determ1ned 6y tw0 10we5t d0u61et5 w1th the ener9y 5p11tt1n9 ~ 1 K. 7he 1nduced ma9net15m may 6e cau5ed 1n th15 ca5e 6y the c0ntr16ut10n 0f the h19he5t d0u61et t0 the m0ment a10n9 6-ax15 0•)- 1n the p1cture 0f 1nduced ma9net15m 0ne can a150 exp1a1n the d1fference 0f 5aturat10n m0ment at 10w temperature5 and at h19h f1e1d [16]. 1t 15 kn0wn that the 5p0ntane0u5 m0ment 1n ErRh484 11e51n the 6a5a1 p1ane, w1th tw0 a pr10r1 e4u1va1ent ax15 1n 1t, a and 6.51nha et a1. [1] have n0t1ced, h0wever, that actua11y the ma9net12at10n wa5 0r1ented a10n9 0ne 0f the5e ax15, 6, pre5uma61y 6ecau5e 0f the 5119ht 5tre55e5 pre5ent 1n the 5amp1e. 5uch a 5tr0n9 dependence 0f ma9net1c character15t1c5 0n ax1a1 5tra1n5 15 typ1ca1 0f 1nduced ma9net15m and
Vol. 46, No. 2
INDUCED FERROMAGNETISM
it also confirms our assumptions. We see below that together with first one this allows us to explain the small intensity of satellites in comparison with F-peaks. At first we consider how the second assumption modifies the results of the theory and then we take into account the distribution of 0 to obtain the final picture. 3. NUCLEATION PROCESS OF DS PHASE FORMATION As long as the transition is 2nd order one magnetic domain structure establishes itself starting from the sinusoidally modulated structure with infinitely small amplitude at TM. As the magnetic transition approaches the 1st order one, growth of magnetization S on cooling below TM in DS phase become more steep. So the interval near TM, where s is small and Q depends strongly on temperature, narrows. Besides, T,, approaches TM. Let us now consider how the magnetic ordering appears in the presence of superconductivity in the case of 1st order magnetic transition N -+ FN (see [3b] for more detailed discussion). The DS phase forms when the jump of magnetization s at 0 is less than sc2 which is determined by equations (1) and (3). At first, the ferromagnetic region appears in the form of small rode-like nucleus due to the easy axis anisotropy [ 141. Its subsequent growth depends crucially on the interplay with superconductivity: the nucleus will grow in the form of thin plate unbounded in two directions but with the thickness d not exceeding go. Indeed this form is energetically most favourable: it permits to gain as much as possible magnetic energy without too much less of superconducting condensation energy (as far as d 4 [A fhe superconductivity will not be destroyed by such a magnetic domain). The resulting value of d can be estimated as follows: we minimize the sum of magnetic energy of the plate and the change of superconducting energy due to the magnetic ordering in
+ p + fe,,m2(T)
d2 -
135
TM at which the plate may appear. For &F(T) one gets: &Y(r) = (e,,s2p/~,t2p2 = eos2(~/t;o)l/2, i.e. = - e (a/,$o)1’2 and d,,,, = (eoaEo/e,,)1’2. TM-e The magnetization in the plate is along easy axis, the normal to the surface of plate should be perpendicular to the easy axis (i.e. in xz plane) owing to the EM energy and its final direction in xz plane should minimize the domain wall energy p. Now when the growth of the first place was stopped by superconductivity, the second nuclei can appear. It is evident that the exchange and magnetic field of the first plate decrease the activation energy for the second nuclei if it is located near the first one and its moment is opposite to the moment of the first plate. The second nuclei then also grows transforming into the second plate adjoining to the first one. Repetition of this process gives the one-dimensional transverse domain-like magnetic structure in the superconducting state, the creation of the first critical ferromagnetic nuclei being the bottle neck. The characteristics of this domain superconducting (OS) phase itself are the same as in the case of the 2nd order transition, the only difference is the independence of the vector Q = n/d on the temperature near TM. Experimentally moment is along b-axis and Q = (0.042 a*, O,O.S55c*) [ 11. The values 2d = 100 A is obtained by use of equation (1) with to = 200 A from the data [ 161 on Hc2 near T,., and a reasonable values of eoleex x 1 anda% 1 a. The domain structure should be characterized by the satellites peaks (2~ + l)Q (K integer), in monocrystal the higher peak intensities being 1, = (2~ + l)-2Zo in the case of ideal domain structure. Thus the 3Q-peak intensity should be 11% of the Q-peak. Experimentally 3Q-peak was not found, i.e. its intensity was less than 2% of the Q-peak [ 11. This discrepance may be explained by irregularity of domain structure which is expected due to internal stresses and the nucleation process of DS phase formation in the case of 1st order transition. Taking the Gaussian for the domain thickness distribution d we obtain the dependence Zsk on q = (2~ + 1)Q + 9 (a*, 0, c*) near (10 1) Bragg peak in the form
4‘0 Iskb?)
Here 6 Yis the gram of magnetic energy (per unit volume), o is an area of the plate. The EM contribution is neglected here due to the condition (2) which may be checked after calculation. Minimizing .&l w.r.t. d one gets the thickness dM of ferromagnetic plate in superconductor, i.e. dM = ~oS.F(T)/e,S2(T)n. The condition Fpl(dM, T) < 0 determines the temperature
IN ErRh4B4
A0
-A;:
, hk = (2K
+
1)2ho.
+Tq
Thus the 3Q-peak is 81 times weaker at r) = 0 and 9 times broader than Q-peak. In measurements of Sinha et al. [ 1, 21 the resolution was about the same as A0 and so the search of 3Q-peak was unsuccessful. 4. PHASE TRANSITIONS Thus below the temperature TM = e the DS phase forms regardless of the type of transition at TM. On
136
1NDUCED FERR0MA6NE715M 1N ErRh484
V01. 46, N0.2
further c0011n9 the 15t 0rder tran51t10n D5 ~ FN 0ccur5 at 7e2 when 5(7, 0) = 5c2 and acc0rd1n9 t0 the 5ec0nd a55umpt10n the 1nterva1 0f D5 pha5e ex15tence may 6e very narr0w even 1f 5e2 15 0f 0rder 0f un1ty (we 9et 5e2 ~ 0.8 fr0m the data 0f [1, 2], the the0ret1ca1 e5t1mat10n 6e1n9 ~ 1 fr0m e4uat10n (3) at 0ex ~ 1.3 K and vF ~ 5.5 7c1~0/h= 1.4 x 107 cm5ec-1). We remark that 50me part 0f d0ma1n5 may rema1n 1n FN pha5e 6e10w 7e2 due t0 the p1nn1n9 0f d0ma1n wa1150n 1mperfect10n5, the5e ••wr0n9•• d0ma1n5 1n ferr0ma9net1c pha5e may 91ve 6r0ad 11ne (8-11ne) near F-peak5. N0w t0 exp1a1n the temperature dependence 0f the F- and 5-peak 1nten51t1e5we mu5t 1nv0ke the f1r5t a55umpt10n. Acc0rd1n9 t0 1t the tran51t10n5 5 -+ D5 and D5 ~ FN take p1ace at d1fferent temperature5 1n the d1fferent part5 0f the 5amp1e. At 50me temperature 7 the f0110w1n9three pha5e5 may 6e pre5ent 1n the 5amp1e:
51mp1e p1cture 5h0u1d 6e 065erved: 0n c0011n9 0n1y 5ate111te5 (2~r + 1)Q appear 5teep1y 6e10w 7M and at 7e2 they are rep1aced 5udden1y 6y F-peak5 (p055161y w1th 8-11ne). 7he 1nten51ty 0f Q-peak a60ve 7e2 5h0u1d 6e 0n1y 5119ht1y1e55than 1nten51ty 0fF-peak 6e10w 7c2. 7he f0110w1n9exper1ment5 may e1uc1date f1na11y the p1cture pre5ented a60ve. 7he type 0f the tran51t10n N ~ FN at 0 1n the a65ence 0f 5uperc0nduct1v1ty may 6e 065erved 1n 5amp1e5 under m1cr0wave 0r 1a5er 1rrad1at10n 5uff1c1ent t0 de5tr0y the C00per pa1r1n9. 1n 5uch a 5amp1e 0n1y F peak5 5h0u1d 6e pre5ent 1n neutr0n 5catter1n9 6e10w 0 w1th hy5tere515 1n near 0. 7he mea5urement5 0f ma9net12at10n and re515t1v1ty under un1ax1a1 pre55ure may check the a55umpt10n 0f the 5tr0n9 dependence 0f 0 0n 5tre55.7he d1rect 5tudy 0f the exact nature 0f 10w-1y1n9cry5ta1 f1e1d 1eve151n ErRh484 (e.9. v1a E5R) w0u1d 6e a150 very 1nf0rmat1ve: 1t w0u1d perm1t 0ne t0 ca1cu1ate (1) n0nma9net1c 5uperc0nduct1n9 pha5e 1n re910n5 the0ret1ca11y a11 the character15t1c5 0f ma9net1c w1th 7M < 7, (2) 5uperc0nduct1n9 D5 pha5e, wh1ch 91ve5 5ate111te5, tran51t10n. 7he d1rect c0nf1rmat10n 0f d0ma1n-11ke ma9net1c 5tructure 1n the c0ex15tence 5tate may c0me 1n the re910n5 where 0 < 5(7, 0) < 5c2, fr0m the m0re accurate 5tudy 0f 5ate111te5peak5 1n (3) n0rma1 ferr0ma9net1c pha5e w1th re51dua1 neutr0n 5catter1n9 0r fr0m the 1nterna1 f1e1d5 d0ma1n5, wh1ch 91ve5F-peak5 and 8-11ne5, 1n re910n5 1nve5t19at10n5 (e.9. v1a NMR). where 5(7, 0) > 5e2. 1n th15 p1cture the 5ma11ne550f the re1at1ve 5ate111te5 1nten51ty 18/10 re5u1t5 fr0m the rap1d 1ncrea5e 0f 5(7, 0) fr0m 0 t0 5e2 1n the narr0w 1nterva1 (7M, 7e2) and the fact 0f 0-d15tr16ut10n 1n the 1nterva1 wh1ch 15 much 6r0ader than (7• -- 7c2). We 9et 1n the crude appr0x1mat10n
8 -18(7 ) ~, -f1 52D5(7M d/°(7) ~ 5~8Nf(7), d7
-
-
7e2)f(7), (7)
where 5205and 5~N are the mean va1ue5 0f 52 1n D5 and FN pha5e c0rre5p0nd1n91y and f•(0) 15 the d15tr16ut10n funct10n 0f 0.50 15(7) and the effect1ve heat capac1ty c(7) are pr0p0rt10na1 t0 f(7) ~ d10/d7. Exper1menta1 data [1, 13] are 1n acc0rdance w1th th15 re1at10n5. 7he tran51t10n t0 the n0rma1 5tate 1n re515t1v1ty 15 perc01at10n 0ne, 1t 0ccur5 when FN pha5e 15 75% 0f the wh01e v01ume 0f the 5amp1e. 7he ma9net1c f1e1d 1ncrea5e5 the va1ue 0f 7e2 (5ee [15] ). 7h15 15 a rea50n 0f 1ncrea51n9 0f10 and decrea51n9 0f18 1n the ma9net1c f1e1d 06ta1ned 6y 51nha et a1. [ 1, 2]. 50 0ur a55umpt10n5 a110w u5 t0 exp1a1n the ma1n feature5 0f ErRh484 6ehav10ur. We 5ee that many 0f them are cau5ed 6y the 1mperfect10n5 0f the 5amp1e5 and n0t 6y the 1nterp1ay 0f 5uperc0nduct1n9 and ma9net1c 0rder1n9 1t5e1f. 1n perfect 51n91ecry5ta15 m0re
REFERENCE5 1.
5.K. 51nha, 6.W. Cra6tree, D.6. H1nk5 • H. M00k, Phy5. Rev. Lett. 48, 950 (1982). 2. H.A. M00k, 0.A. Pr1n91e, 5. Kawara2ak1, 5.K. 51nha, 6.W. Cra6tree, D.6. H1nk5, M.8. Map1e 2. F15k, D.5. J0hn5t0n • L.D. W001f, Pr0c. 4th C0nf. 0n 5uperc0nduct1v1ty 1n d- and f-6and Meta15, p. 201 Kar15ruhe (1982). 3. (a) L.N. 8u1aev5k11, A.1.8u2d1n, 5.V. Panjuk0v • M.L. Ku11c. 5011d 5tate C0mmun. (6) 5u6m1tted t0 Phy5. Rev. 8. 4. H.5.6reen51de, E.1. 810unt • C.M. Varma, Phy5. Rev. Lett. 46, 49 (1981). 5. M. 7ach1k1,Phy51ca 109 • 1108, 1699 (1982). 6. P.W. Ander50n • H. 5uh1, Phy5. Rev. 116,898 (1959). 7. J.W. Lynn, 6.5h1rane, W. 7h0m11n50n, R.N. 5he1t0n • D.E. M0nct0n,Phy5. Rev. 824, 3817 (1981). 8. D.E. M0nct0n, D.8. McWhan, P.H. 5hm1dt, 6. 5h1rane, W. 7h0m11n50n, M.8. Map1e, H.8. McKay • L.D. W001f, 2. F15k; D.C. J0hn5t0n, Phy5. Rev. Lett. 45, 2060 (1981). 9. D. 5a1nt-Jame5, 6.5arma • E.J. 7h0ma5, 7ype 2 5uperc0nduct0r5, Ch. 6. Per9am0n Pre55 (1969). 10. H.R. 0tt, W.A. Fert19, D.C. J0hn5t0n, M.8. Map1e • 8.7. Matt1a5, J. L0w. 7emp. Phy5. 33, 159 (1978). 11. H.8. MacKay, L.D. W001f, M.8. Map1e • D.C. J0hn5t0n, J. L0w. 7emp. Phy5. 41,639 (1980). 12. F. 8ehr0021, 6. W. Cra6tree, C.A. Cam6e11, M. Levy, D.K. 5n1der, D.5. J0hn50n • 8.7. Matt1a5, 5011d5tate C0mmun. 39, 1041 (1981).
V01. 46, N0.2 13. 14.
1NDUCED FERR0MA6NE715M 1N ErRh484
L.D. W001f, D.C. J0hn5t0n, H.8. McKay, R.W. McCa11um• M.8. Map1e.J. L0w. 7emp. Phy5. 35,651 (1979). A.1.Pr1v0r0t5k11, U5p. F12. Nauk (50v. Phy5. U5p.} 108, 43 (1972).
137
15.
L.N. 8u1aev5k11, A.1.8u2d1n • 5.V. Panjuk0v, 5011d5tate C0mmun. 43,135 (1982);JE7P, 83,
16.
768 (1982). 6.W. Cra6tree, F. 8ehr0021, 5.A. Camp6e11• D.6. H1nk5.Phy5. Rev. Lett. 49, 1342 (1982).