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Range of validity of the Landau theory of phase transitions for very weak itinerant ferromagnets
Volume 39A, number 2
PHYSICS LETTERS
24 April 1972
RANGE OF VALIDITY OF THE LANDAU THEORY OF PHASE TRANSITIONS FOR VERY WEAK ITINERANT FERROMAGNETS C. HARGITAI*, S. SHTRIKMAN** and E.P. WOHLFARTH Department of Mathematics, Imperial College, London, S. W. Z, U.IC
Received 1 March 1972 The range of temperatures about the Curie temperature where the Landau theory of phase transitions fails is estimated explicitly for very weak itinerant ferromagnets. The results are extended to finite fields and are then related to magnetic isotherms in the form of Arrott plots.
The Landau theory of phase transitions [1] has been applied with success to the very weak itinerant limit of Stoner theory [2], using two or sometimes three [3] terms in the expansion of the free energy as a function of the magnetization M. However, the Landau theory in the form used in [2] is essentially a mean field theory and thus neglects all fluctuations in M [e.g. 4]. Hence it seems necessary to asses the importance of these fluctuations. Within the Landau theory, Ginzburg [5] assessed the range of its validity by calculating the temperature range AT about the Curie temperature Te within which the fluctuations of M become larger than M itself. The result (for T < T c is [4,5]
eG explicitly for a given single particle band structure. Here we obtain the value of eG for a parabolic band for which [2, 6]
e G = A T / T c = (1/32rt 2) ( k B / A C k 3 ) 2 ,
eG = 72/7r 6 ~ 7%.
(1)
where AC is the specific heat discontinuity at Tc within the Landau theory and k characterizes the correhtion length ~ which occurs in the Omstein-Zernike relation for the susceptibility, x(q) = X(0) (1 + q2~2)-1
(2)
The relation between ~ and k is = XleG 1-1/2 (T~
(3)
At present e G has only been estimated very crudely for iron [4]. However, for a very weak itinerant ferromagnet, enough is known [2, 6] in order to calculate *On leave from Central Research Institute for Physics, Budapest, Hungary. **On leave from Department of Electronics, Weizmann Institute, Rehovot, Israel.
~ - 2 = 4 8 3'kF~'o, 22
(4)
where k F is the Fermi vector for the non-ferromagnetic case, ~'o is the relative magnetization at 0°K and 3' is related to the density of states curve [2], having a value 2/27 in the present example. Further [2]
AC= M2o/Xorc,
(5)
where M o is the saturation magnetization at T= 0, ×o the high field susceptibility and Tc the Curie temperature. Hence, (6)
This result is remarkable in that the temperature range of validity of the Landau theory, scaled to To, is here independent of all characteristic quantities such as the electron density and the relative magnetization at 0°K. Results of magnetic measurements on very weak itinerant ferromagnets are conveniently displayed on Arrott plots, giving the dependence o f M 2 on H/M, where H is the applied field [2]. Hence the range of validity of the Landau theory needs to be examined for H :/: 0. This was recently done by Menyhard [7] who, by comparing thermal averages of terms in the free energy which are of fourth and second order in the fluctuations of M, estimated e M = A TIT c for T ~ Te. It was found that the Landau theory is valid as long as the susceptibility X is less than a constant value Xc which depends [7] on the Landau parameters. 87
Volume 39A, number 2
PHYSICS LETTERS
For the very weak itinerant model and using only 2 terms in the free energy expansion in terms of the uniform magnetization M [2], H =-A M + BM 3, where
(7)
A = - (1 / 2Xo) (1 - T 2 / Tc2), B = 1 / (2XoMo2). Hence with x = H/M, y = BM 2, the reduced Arrott plots are given by y =-A
+x.
(8)
The Menyhard result of a constant susceptibility then defines a critical line in the x, y plane given by y ___l ( x c - 1 _ x),
(9)
within which the Landau theory breaks down. It follows from eqs. ( 7 - 9 ) that Xc = Xo / (2e M )"
(10)
The magnetitude of e M may be calculated and conveniently compared with eG, defined above, for T < T c ; it is found that em = 729/2~r 8, i.e. e M /eG = 81/16rr2~0"5,
(11)
which is satiafyingly close to unity considering the differing definitions of e. In any case, the actual numbers obtained for the various criteria are not too important since the breakdown of the Landau theory occurs with reference not only to the criteria themselves, but also to the quantity being measured (magnetization, specific heat, etc.) The result (6) implies that the Landau theory should break down within about 7% of Tc. For ZrZn 2 O [2], Tc ~- 20°K so that AT ~ 1.5 K. The measured Arrott plots [8] are indeed linear where measured, which is well outside the corresponding triangle defreed by eq.(8). It would thus be of interest to measure magnetic properties closer to Tc to see whether a breakdown of the Landau theory might be manifested there by the appearance of the usual [4] critical phenomena. Such measurements should be carried out in fields less than the largest critical value [7] obtained from eqs. (7), (9) and (10) and given by* H c = 2 (32--eM)3/2 (Mo/Xo).
(12)
*Note that eq. (12) gives the maximum field on the line given by eq. (9), the minimum field being zero when x = 0 or y = 0.
88
24 April 1972
For ZrZn 2 [2, 8], this amounts to only about 600 Oe where, however, the value of the parameter X used corresponds to a parabolic band. On the other hand, ×o--1 ~ M o 2 [2] so t h a t H c "-Mo3. Hence critical phenomena should be sought in materials with rather larger magnetizations. Mathon [9] had earlier assessed the breakdown of Landau theory in the regime of concentrations c close to the critical value c o for which Tc = 0. It is now clear that, since here [10] Tc (c) ~ I c - c o I 1/2, the relevant range of concentrations is immediately related to the corresponding range of temperatures about Tc(C) given by e G or e M. Since the high field susceptibility Xo diverges as I c - c o I - 1 near c o [9] so does the critical susceptibility Xo defined above [7] and given by eq. (10). Hence the region of breakdown of the Landau theory tends to zero as c ~ c o [9]. Finally, it may be noted that for different band structures the correlation length parameter X may be calculated from the expansion of the susceptibility X (q) [ 11 ]. The value of X- 2 in eq. (4) may depend quite sensitively on the band structure and since e"~h - 6 , the range of applicability of the Landau theory may depend even more so on this structure. We are grateful to D.M. Edwards for very valuable advice regarding the correlation length in very weak itinerant ferromagnets.
References [ 1 ] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon 1958). [2] D.M. Edwards and E.P. Wohlfarth, Proc. Roy. Soc. 303 (1968) 127. E.P. Wohlfarth, J. Appl. Phys. 39 (1968) 1061. [3] M.J. Besnus, Y. Gottehrer and G. Munschy, Phys. Stat. Solidi (b) 49 (1972) 597. [4] L.P. Kadanoff et al., Rev. Mod. Phys. 39 (1967) 395. [5] V.L. Ginzburg, Soviet Phys. Solid State 2 (1960) 1824. [6] J.A. Hertz, Int. J. Magnetism 1 (1971) 253. [7] N. Menyhard, Solid State Comm. 8 (1970) 1337. [8] S. Ogawa and N. Sakamoto, J. Phys. Soc. Japan 22 (1967) 1214. [9] J. Mathon, Proc. Roy. Soc. 306 (1968) 355. [10] J. Mathon and E.P. Wohlfarth, Phys. Stat. Sol. 30 (1968) K 131. [11] D. Lipton, J. Phys. F 1(1971)469.
PHYSICS LETTERS
24 April 1972
RANGE OF VALIDITY OF THE LANDAU THEORY OF PHASE TRANSITIONS FOR VERY WEAK ITINERANT FERROMAGNETS C. HARGITAI*, S. SHTRIKMAN** and E.P. WOHLFARTH Department of Mathematics, Imperial College, London, S. W. Z, U.IC
Received 1 March 1972 The range of temperatures about the Curie temperature where the Landau theory of phase transitions fails is estimated explicitly for very weak itinerant ferromagnets. The results are extended to finite fields and are then related to magnetic isotherms in the form of Arrott plots.
The Landau theory of phase transitions [1] has been applied with success to the very weak itinerant limit of Stoner theory [2], using two or sometimes three [3] terms in the expansion of the free energy as a function of the magnetization M. However, the Landau theory in the form used in [2] is essentially a mean field theory and thus neglects all fluctuations in M [e.g. 4]. Hence it seems necessary to asses the importance of these fluctuations. Within the Landau theory, Ginzburg [5] assessed the range of its validity by calculating the temperature range AT about the Curie temperature Te within which the fluctuations of M become larger than M itself. The result (for T < T c is [4,5]
eG explicitly for a given single particle band structure. Here we obtain the value of eG for a parabolic band for which [2, 6]
e G = A T / T c = (1/32rt 2) ( k B / A C k 3 ) 2 ,
eG = 72/7r 6 ~ 7%.
(1)
where AC is the specific heat discontinuity at Tc within the Landau theory and k characterizes the correhtion length ~ which occurs in the Omstein-Zernike relation for the susceptibility, x(q) = X(0) (1 + q2~2)-1
(2)
The relation between ~ and k is = XleG 1-1/2 (T~
(3)
At present e G has only been estimated very crudely for iron [4]. However, for a very weak itinerant ferromagnet, enough is known [2, 6] in order to calculate *On leave from Central Research Institute for Physics, Budapest, Hungary. **On leave from Department of Electronics, Weizmann Institute, Rehovot, Israel.
~ - 2 = 4 8 3'kF~'o, 22
(4)
where k F is the Fermi vector for the non-ferromagnetic case, ~'o is the relative magnetization at 0°K and 3' is related to the density of states curve [2], having a value 2/27 in the present example. Further [2]
AC= M2o/Xorc,
(5)
where M o is the saturation magnetization at T= 0, ×o the high field susceptibility and Tc the Curie temperature. Hence, (6)
This result is remarkable in that the temperature range of validity of the Landau theory, scaled to To, is here independent of all characteristic quantities such as the electron density and the relative magnetization at 0°K. Results of magnetic measurements on very weak itinerant ferromagnets are conveniently displayed on Arrott plots, giving the dependence o f M 2 on H/M, where H is the applied field [2]. Hence the range of validity of the Landau theory needs to be examined for H :/: 0. This was recently done by Menyhard [7] who, by comparing thermal averages of terms in the free energy which are of fourth and second order in the fluctuations of M, estimated e M = A TIT c for T ~ Te. It was found that the Landau theory is valid as long as the susceptibility X is less than a constant value Xc which depends [7] on the Landau parameters. 87
Volume 39A, number 2
PHYSICS LETTERS
For the very weak itinerant model and using only 2 terms in the free energy expansion in terms of the uniform magnetization M [2], H =-A M + BM 3, where
(7)
A = - (1 / 2Xo) (1 - T 2 / Tc2), B = 1 / (2XoMo2). Hence with x = H/M, y = BM 2, the reduced Arrott plots are given by y =-A
+x.
(8)
The Menyhard result of a constant susceptibility then defines a critical line in the x, y plane given by y ___l ( x c - 1 _ x),
(9)
within which the Landau theory breaks down. It follows from eqs. ( 7 - 9 ) that Xc = Xo / (2e M )"
(10)
The magnetitude of e M may be calculated and conveniently compared with eG, defined above, for T < T c ; it is found that em = 729/2~r 8, i.e. e M /eG = 81/16rr2~0"5,
(11)
which is satiafyingly close to unity considering the differing definitions of e. In any case, the actual numbers obtained for the various criteria are not too important since the breakdown of the Landau theory occurs with reference not only to the criteria themselves, but also to the quantity being measured (magnetization, specific heat, etc.) The result (6) implies that the Landau theory should break down within about 7% of Tc. For ZrZn 2 O [2], Tc ~- 20°K so that AT ~ 1.5 K. The measured Arrott plots [8] are indeed linear where measured, which is well outside the corresponding triangle defreed by eq.(8). It would thus be of interest to measure magnetic properties closer to Tc to see whether a breakdown of the Landau theory might be manifested there by the appearance of the usual [4] critical phenomena. Such measurements should be carried out in fields less than the largest critical value [7] obtained from eqs. (7), (9) and (10) and given by* H c = 2 (32--eM)3/2 (Mo/Xo).
(12)
*Note that eq. (12) gives the maximum field on the line given by eq. (9), the minimum field being zero when x = 0 or y = 0.
88
24 April 1972
For ZrZn 2 [2, 8], this amounts to only about 600 Oe where, however, the value of the parameter X used corresponds to a parabolic band. On the other hand, ×o--1 ~ M o 2 [2] so t h a t H c "-Mo3. Hence critical phenomena should be sought in materials with rather larger magnetizations. Mathon [9] had earlier assessed the breakdown of Landau theory in the regime of concentrations c close to the critical value c o for which Tc = 0. It is now clear that, since here [10] Tc (c) ~ I c - c o I 1/2, the relevant range of concentrations is immediately related to the corresponding range of temperatures about Tc(C) given by e G or e M. Since the high field susceptibility Xo diverges as I c - c o I - 1 near c o [9] so does the critical susceptibility Xo defined above [7] and given by eq. (10). Hence the region of breakdown of the Landau theory tends to zero as c ~ c o [9]. Finally, it may be noted that for different band structures the correlation length parameter X may be calculated from the expansion of the susceptibility X (q) [ 11 ]. The value of X- 2 in eq. (4) may depend quite sensitively on the band structure and since e"~h - 6 , the range of applicability of the Landau theory may depend even more so on this structure. We are grateful to D.M. Edwards for very valuable advice regarding the correlation length in very weak itinerant ferromagnets.
References [ 1 ] L.D. Landau and E.M. Lifshitz, Statistical physics (Pergamon 1958). [2] D.M. Edwards and E.P. Wohlfarth, Proc. Roy. Soc. 303 (1968) 127. E.P. Wohlfarth, J. Appl. Phys. 39 (1968) 1061. [3] M.J. Besnus, Y. Gottehrer and G. Munschy, Phys. Stat. Solidi (b) 49 (1972) 597. [4] L.P. Kadanoff et al., Rev. Mod. Phys. 39 (1967) 395. [5] V.L. Ginzburg, Soviet Phys. Solid State 2 (1960) 1824. [6] J.A. Hertz, Int. J. Magnetism 1 (1971) 253. [7] N. Menyhard, Solid State Comm. 8 (1970) 1337. [8] S. Ogawa and N. Sakamoto, J. Phys. Soc. Japan 22 (1967) 1214. [9] J. Mathon, Proc. Roy. Soc. 306 (1968) 355. [10] J. Mathon and E.P. Wohlfarth, Phys. Stat. Sol. 30 (1968) K 131. [11] D. Lipton, J. Phys. F 1(1971)469.