The magnetic phase diagrams of all dilute Cr alloys can be explained by a simple theoretical model with coupled spin- and charge-density waves and a finite electron reservoir. If the charge-density wave and electron reservoir are sufficiently large, the paramagnetic to commensurate spin-density wave transition becomes strongly first order, as found in Cr{sub 1- x}Fe{sub x} and Cr{sub 1-x}Si{sub x} alloys. The observed discontinuity of the slope dT{sub N}/dx at the triple point and the bending of the CI phase boundary are also natural consequences of this model

Fishman, R.S.

Jiang, X.W.

UNT (University of North Texas) Digital Library

1 I Annual Conference on Magnetism and Magnetic Materials I I Atlanta, Georgia, November 11,-1996 I “The submitted manuscript has been authored by a contractor of the U.S. Government under contract No.  D E - A C 0 5 - 9 6 0 R 2 4 4 6 4 .  Accordingly, the U.S. Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for U.S. Government purposes.” RECEIVED CHARGE-DENSITY WAVE AND MAGNETIC PHASE DIAGRAM OF CHROMIUM ALLOYS R. S. Fishman, X. W. Jiang SOLID STATE DIVISION OAK RIDGE NATIONAL LABORATORY Managed by LOCKHEED MARTIN ENERGY RESEARCH COW. Under Contract No. DE-AC05-960R22464 With the U. S. DEPARTMENT OF ENERGY OAK RIDGE, TENNESSEE August 1996 ER , CHARGE-DENSITY WAVE AND MAGNETIC PHASE DIAGRAM OF CHROMIUM ALLOYS R.S. Fishman Solid Slate Division, P.O. Box 2008, Oak Ridge National Lab, Oak Ridge, TN 37831-60.32 X.W. Jiang* Physics Department, North Dakota State University, Faqo, ND 58105-5566 (August 8, 1996) Abstract The magnetic phase diagrams of all dilute Cr alloys can be explained by a simple theoretical model with coupled spin- and charge-density waves and a finite electron reservoir. If the chargedensity wave and electron reservoir are sufficiently large, the paramagnetic to commensurate spin-density wave transition becomes strongly first order, as found in Crl,,Fe, and CrI-& alloys. The observed discontinuity of the slope d T ~ / d z  at the triple point and the bending of the CI phase boundary are also natural consequences of this model. DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsi- bility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Refer- ence herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom- mendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. 1 DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. Since the spin-density wave1*' (SDW) of Cr alloys is produced by the Coulomb attraction between electrons and holes on almost perfectly nested Fenni surfaces, the wavevector and amplitude of the SDW can be controlled by changing the relative sizes of those Fermi surfaces with doping. For pure Cr,3 the SDW is incommensurate (I) with the bcc lattice and the paramagnetic (P) to I phase transition at 310 K is weakly first order. When the electron Fermi surface is sufficiently close in size' to the hole Fermi surface, the SDW becomes commensurate (C) with the lattice. In the simplest model of Cr alloys, dopants affect the nesting free energy solely by altering the mismatch between the Fermi surfaces. But it is well-known3 that different dopants have dramatically different effects on the magnetic ordering and phase boundaries of Cr alloys. Whereas the PC transition in other alloys is second order, the PC transition is strongly first order in CrFe and CrSi alloys. While a second-order CI phase boundary falls to the C side of the triple point for CrMn, CrRe, and CrRu alloys, a first-order CI boundary falls to the I side for CrFe alloys. This diversity has puzzled theorists and delighted experimentalists for man>- years. Here, we argue that the magnetic phase diagrams of all dilute Cr alloys can be qualitatively explained by a simple model with a coupled chargedensity wave (CDW) and a finite reservoir of electrons. Both x-ray5p6 m d  neutron scattering'*' measurements revealed the existence of a CDW in pure Cr over 20 years ago. Recently, Hill et uL9 performed an exhaustive set of x-ray measurements on the temperature dependence of the CDV*. If the wavevectors of the ISDW are Q; = (G/2)(1 & @), on either side of G/2 = 27ri/u, then the wavevectors of the CDW are 24; = G(l f q), on either side of the reciprocal latrice vector G. In the I phase, a strain couples to the CDW with the same wavevectors 24;. Young and Sokoloff'* suggested that the CDW was responsible for the weak h - o r d e r  PI transition of pure Cr. However, its effect on the phase diagram was never considered. The SDW consists of spin-triplet electron-hole pairs formed by the Coulomb attraction U > 0 between electrons and holes on Fermi surfaces a and b. To minimize the condensation free energy' on both sides of the two Fermi surfaces, the ordering wavevectors Qk of the SDW lie closer to G/2 than the nesting wavevectors Q* = (G/2)(1 k a). The mismatch 2 0 between the a and b Fermi surfaces can be controlled by doping with another transition metal: adding Mn, Fe, Re, or Ru raises the chemical potential and decreases a; adding V lowers the chemical potential and increases the mismatch a. For pure Cr, 8 M 0.05 so the hole Fermi surface is only slightly larger than the electron surface. As a decreases and the nesting improves, the SDW wavevector parameter a’ < 8 also decreases until, for a :mall enough d > 0, the SDW becomes commensurate with 8 = 0. The energy mismatch between the Fermi surfaces is given by zo = 47rav~/&u, which increases linearly with the V concentration and decreases linearly with the concentration of Mn, Fe, Re, or Ru. For pure Cr, zo x 475 meV. Another quantity which will appear shortly is the N&l temperature 57; a 80 meV of a perfectly nested Cr alloy with zo = 0. Since ZO > 0, the hole Fermi surface is larger than the electron Fermi surface and not all of the b holes can be paired to a electrons in the SDW. The density of unpaired holed3 on the b Fermi surface is given by pehto/4. As first shown by Young and Sokoloff,” the Coulomb attraction U’ > 0 between unpaired holes and paired electrons on the b Fermi surface produces the CDW in the I phase with order parameter b cc -U’(g/T’)2 < 0. Starting with the Green’s function of Cr alloys, we have shown14 that the spin and charge distributions are given by where X = Upeh/8  and A’ = U’p,h/8 are dimensionless coupling constants and u(r) is a periodic Bloch function normalized to one in volume V .  These relations imply that the SDW and CDW are in phase: the electron number is a maximum whenever the spin is a maximum or a minimum, Because the Bloch functions of the d-band electrons are strongly peaked at the atomic sites, the amplitudes of the spin and electron number in the I phase are SO = -(hg/2A) cos((d+ + 4-)/2)(V/N)p,h and eo = -(6/2X’)(V/N)p,h. Of course, the CDW e(r) carries a net charge only in the C phase with Q’+ = QL. But in order to conserve 3 electron number, the CCDW must vanish and the phase difference d+ - 4- must equal 7r/2 plus some multiple of A. Besides the nesting electron jack and hole octahedron centered at the I? and H points in reciprocal space, the band s t r ~ c t u r e ' ~  of Cr alloys also contains electron balls midway between the r and H points and hole pockets at the N points. These other bands may be lumped together into an electron reser~oir '~ . '~  with density-of-states pr. If the density-of- states of the a and b bands is P e h ,  then the power of the reservoir is defined by p = Pr/Peh. Below T', the SDW removes electrons from the a and b Fermi surfaces. So for a b i t e  reservoir with p < 00, the chemical potential will decrease and the wavevector mismatch will increase as the SDW grows. Since jumps in the order parameters require sudden changes in the chemical potential, a finite reservoir suppresses first-order phase transitions. The free energy difference AF(p,T)  between the magnetic and paramagnetic states is derived e1~ewhere.l~ When p < 00, zO(T) increases with decreasing temperature and the phase diagram is evaluated in terms of the paramagnetic mismatch to = &(T > TN). Both order parameters g and d as well as the SDW wavevector parameter 3 are obtained by minimizing AF(p,  T). Although the self-consistent equations for those parameters are not affected by the finite reservoir, solutions which are saddle points or even maxima of A F ( p  = 00,Tf may become the minima of AF(p, 2') when p < 00. Our results for a,n infinite reservoir are presented in Fig.1, Surprisingly, the PI transition is always second order. By equating the SDW ordering and nesting wavevectors (equident to setting 8 = a), Young and Sokolof€'* obtained a first-order PI transition above a threshold value of A'. However, when the free energy AF(p,  2') is minimized with respect to 8. the I solutions near the first-order phase boundary are always unstable. Instead, a nonzero CDW drives a first-order PC transition and pushes the triple point towards higher values of a. As A' -+ 1/2, the CDW becomes unstable and the I phase disappears. For A' > 1/2, the free energy is minimized when 6 = -00 and no physical solutions exist. A natural consequence of this model is the discontinuity at the triple point in the deriva- tive of the N6el temperature with respect to either the energy mismatch zofz) or the dopant 4 concentration x. While this discontinuity has been observed in all Crl-,A, alloys with a triple point,3 it has never been previously explained. The normalized and dimensionless latent heat L' = L/&hT;' of the first-order PC transition is plotted in the inset of Fig.1. As observed in CrFe and CrSi the latent heat peaks to the left of the triple point due to the rapid drop in the first-order N k l  temperature TN near the triple point. If Peh = 2.4 states/Ry atom," then the observed maximum latent heat of 12.6 J/moI for CrFe alloys is fitted by A' a 0.135. Correspondingly, the CDW has an amplitude of x 0.005 electrons per site at zero temperature on the I side of the CI phase boundary. Although it correctly predicts the strongly first-order PC transition in CrFe and CrSi alloys, the free energy AF(p = m,T) cannot explain the weak first-order PI transition in pure Cr or the second-order CI transition in CrMn. These deficiencies are remedied by including the effects of a finite electron reservoir. Since the effective mismatch zo(T) increases with decreasing temperature, a finite elec- tron reservoir favors the I phase of the SDW. When p is sufficiently small, the full free energy AF(p ,  T )  stabilizes the formerly unstable I solutions near the first-order phase boundary. Fixing A' = 0.4, we plot the ph&e diagram for three powers of the reservoir in Fig.2. With decreasing reservoir power, the triple point shifts towards smaller values of zo and higher temperatures, the first-order N&l temperature is suppressed, and the CI phase boundary bends towards the C side of the triple point. For p = 0 and A' = 0.4, the first-order PC transition is replaced by a weak first-order PI transition, just as observed in CrMn alloys. For p < 2.4 and any A' < 1/2, the CI phase transition becomes second order, as observed in most Cr The inset to Fig.2 shows that the latent heat of the first-order PC or PI phase transition is suppressed as the reservoir power decreases. So the estimates given above for A' and eo in CrFe alloys are only lower bounds. At least qualitatively, the C and I phase boundaries of all dilute Cr alloys can be ex- plained by the dependences of the reservoir power p and the coupling A' on the impurity concentration z. While the originally proposed model12 for the weak first-order PI transi- 5 tion in pure Cr was flawed, our slightly more complex model with a finite reservoir produces such a phase transition. For CrMn alloys, a first-order PC transition may be prevented by the suppression of A’ with the Mn concentration. But unlike other impurity atoms: Fe impurities retain their localized magnetic  moment^'^*^^ within the Cr host below TN. Since fewer of its impurity electrons enter the conduction band, the triple point for CrFe alloys lies at  a higher concentration than for CrMn alloys: 2.4% Fe compared to 0.3% Mn. By enlarging the electron reservoir, the magnetostriction21 of CrFe alloys bends the first-order CI phase boundary to the I side of the triple point. By enhancing the coupling constant A’, the indirect exchange of b+ holes and b- electrons mediated by the paramagnetic Fe moments may generate a large CDW in the I phase and drives the strong first-order PC transition of CrFe alloys. Unfortunately, this picture cannot explain the strong first-order PC transition produced by nonmagnetic Si impurities. So the dramatic effect of Si on the magnetic phase boundary and reentrant C phase3 of CrSi alloys cannot be explained by our model. Nonetheless, we believe that the strong first-order PC transition in CrSi alloys is also preceeded by the development of a large CDW in the I phase. Several traits can now be associated with the magnitudes of A’ and p. Along with the strong first-order PC transition, large values of A‘ and p also bend the first-order CIphase boundary to the I side of the triple point and generate a large discontinuity in dTvJdz0 at the triple point. All of these traits appear in CrFe and CrSi  alloy^.^ Small values of A’ and p ,  on the other hand, are associated with a second-order PC transition, a second-order CI phase boundary pushing into the C phase, and a smaller discontinuity in dT’/dzo. These characteristics are shared by CrMn, CrRe, and CrRu  alloy^.^ The growth of the CDW on the I side of the phase boundary should be straightforward to verify. As first observed experimentally’ and confirmed by our model, the ratio is almost independent of temperature and provides a direct measure of the coupling constant 6 A’. So low-temperature x-ray or neutron-scattering measurements of b / S l  on a series of I Crl-,A, alloys can confirm that A’(z) grows with the Fe or Si concentration and decreases with the Mn concentration. The model presented in this paper is certainly not complete. Impurity scattering22 must be included to explain the linear decrease3 of the NCel temperature with the V concentration. Spin-orbit coupling is needed to explain the presence3 of both transversely (mLQi, T > 120 K) and longitudinally (mllQi, T < 120 K) polarized phases in pure Cr. But we believe that our basic model, with coupled spin- and charge-density waves in the presence of a finite electron reservoir, provides the simplest qualitative explanation for the magnetic phase diagrams of all dilute Cr alloys. One of us (RF) would like acknowledge support from the L.S. Department of Energy under Contract No. DEFG06-94ER45519 and under Contract So. DEAC05860R.22464 with Lockheed Martin Energy Research Corp. Useful conversations with B. Larson, V.S. Viswanath, and especially with S.H. Liu are also gratefully acknowledged. 7 Current 0909 REFERENCES ddress: Center for Ceramic Research, Rutgers University, Piscat wa NJ 08855- W.M. Lomer, Proc. Phys. SOC. London 80, 489 (1962). * P.A. Fedden and P.C. Martin, Phys. Rev. 143, 8245 (1966). The properties of Cr alloys are reviewed by E. Fawcett, H.L. Alberts, V. Yu. Galkin, D.R. Soakes, and J. V. Yakhmi, Rev. Mod. Phys. 66,26 (19%). In particular, this comprehen- sive work contains all the references on CrFe and CrSi alloys. ' R.S. Fishman and S.H. Liu, Phys. Rev. B 48, 3820 (1993). 5Y. Tsunoda, M. Mori, N. Kunitomi, Y. Teraoka, and J. Kanamori, Solid State Comrnun. 14, 287 (1974). A.F. Prekul and S.V. Sudareva, Phys. Met. Metall. 46. 46 (1979). 'R. Pynn, W. Press, S.M. Shapiro, and S.A. Werner, Phys. Reo. B 13, 295 (1976). S. Iida, M. Khono, Y. Tsunoda, -and N. Kunitomi, J.  Phys. SOC. Jap. 50, 2581 (1981). J,P. Hill, G. Hegelsen, and D. Gibbs, Phys. Rev. B 51. 10336 (1995). lo S. Nakajima and Y. Kurihara, J.  Phys. SOC. Japan 38, 330 (1975). '' M. Mori and Y. Tsunoda, J. Phys. C 5 ,  L77 (1993). C.Y. Young and J.B. Sokoloff, J. Phys. F 4, 1304 (1974). l3 R.S. Fishman and V.S. Viswanath, (unpublished). l4 X.W. Jiang and R.S. Fishman, (unpublished). l5 S. Asano and J. Yamashita, J .  Phys. SOC. Jap. 23, 714 (1967). "A. Shibatani, K. Motizuki, and T. Nagamiya, Phys. Reo. 177, 984 (1969). 8 l7 R.S. Fishman and S.H. Liu, fhys .  Rev. B 49,3308 (1994). "H.L. Skriver, J.  fhys. F: Metal Phys. 11, 97 (1981). l9 Y. Ishikawa, S. Hoshino, and Y .  Endoh, J. Phys. SOC. Japan 22,1221 (1967). 2o T. Suzuki, J.  Phys. SOC. Japan 43, 869 (1977). 21 K. Nakanishi and T. Kasuya, J. Phys. SOC. Jup. 42, 833 (1977). 22 J. Zittartz, Phys. Reo. 164, 575 (1967). 9 FIGURES FIG. 1. The phase diagram of Cr alloys with p = 00 and A' = 0.4 (solid), 0.35 (long dash), 0.30 (medium dash), and 0.20 (short dash). The triple points are labeled by the filled circles and the second order paramagnetic phase boundary is denoted by a thin solid line. Inset is the normalized latent heat L' = L/pehTrj2 for the same parameters. The three different magnetic phases are labeled. FIG. 2. The phase diagram of Cr alloys with A' = 0.4 and p = 00 (solid), 1 (long dash), or 0 (short dash). The normalized latent heat L' is inset for the same parameters with the triple points marked. 10 0 0 cc) N 0 ZIC r7 P 0 bo I cd 0 0 0 0 ru G) N 0 z +  27 P N 0 7 iu b bo 0 I I L' in I I i I /  / 

Charge-density wave and magnetic phase diagram of chromium alloys

https://digital.library.unt.edu/ark:/67531/metadc687206/m2/1/high_res_d/385590.pdf

Charge-density wave and magnetic phase diagram of chromium alloys

Abstract

Similar works

Full text

Available Versions

UNT (University of North Texas) Digital Library