Presentation Groupmeeting June 3 rd, sorry 10 th, 2009 by Jacques Klaasse

Transcription

1 Presentation Groupmeeting June 3 rd, sorry 10 th, 2009 by Jacques Klaasse

2 Spin Density Waves This talk is based on a book-chapter on antiferromagnetism, written by Anthony Arrott in Rado-Suhl, Volume IIB, Contents: - Exchange interactions - Spin Density Waves - Neutron diffraction - Chromium - Conclusions

3 Exchange interactions Starting point of the (historical) discussion: - atoms with intrinsic localised magnetic moment. - exchange interaction (Heisenberg) because of direct overlap. - Weiss molecular field. The interaction can be ferromagnetic or antiferromagnetic, dependent on the sign of the exchange parameter. Kramers introduced also superexchange mediated by electrons on intervening non-magnetic atoms (oxygen!). In 1946 Stoner questioned the picture in case of metals. He presented the collective electron picture : itinerant (ferro)magnetism by mutual exchange of d-band electrons. Stoner Criterion: ferromagnetism occurs if D(ε F ) * I S > 1 where D(ε F ) is the density of states at the Fermi level, and I S is the Stoner exchange parameter.

4 Exchange interactions The two subbands are shifted in energy because of the exchange interaction. The shift is based on the Hubbard Hamiltonian Un n, which can be rewritten, with n = n + n, as (U/4) { n 2 (n -n ) 2 }. The exchange potential is not a fixed potential but is governed by the other electrons. Moment and potential are both given by the same medium. This results in a non-zero ferromagnetic moment even in zero applied field, as long as the gain in exchange energy is larger than the loss in kinetic energy. The Coulomb interaction in metals seems to favour ferromagnetic coupling!!

5 Exchange interactions Let D be the average DOS per spin direction around ε F and let the splitting be ΔE. Then (n -n ) = D.ΔE Let s = ( n -n ) / N, and Let I s be the exchange parameter. We calculate now the splitting energy. We do E kin first. E kin = 0 ΔE/2 (2ε) D dε = (D/4) ΔE 2 = (D/4) (Ns/D) 2 = (Ns/2) 2 (1/D). For the exchange we found E exch = -(I s /4) (Ns) 2, so, for the total splitting energy we find: E s = (N 2 s 2 /4) { 1/D I s } = (N 2 s 2 /4D) { 1 DI s }. From this formula follows the Stoner Criterion. This shows to work for ferromagnets like Fe and Ni, but not for much more. Obviously, a high D (or low density in real space) favours ferromagnetism. Above Tc the material should be a Pauliparamagnet. This is not seen!

6 Help screen for calculating E kin ε ε dn = D dε ΔE/2 n -n = D ΔE = N s D(ε) ΔE / 2 0 (2ε ) Ddε

7 Exchange interactions Another class of materials contain localised moments apart from conduction electrons. It is shown by Ruderman, Kittel, Kasuya, and Yosida that the interaction can be formulated in a way that a Heisenberg-like picture is simulated without direct overlap (RKKY interaction, published between 1954 and 1957). This makes the problem similar to the non-metallic problem. The coupling can be FM as well as AFM, and is strongly oscillating with distance. This work followed upon a discussion by Zener on the role of the conduction electrons in providing (ferro)magnetic interactions. Zener also revived the old suggestion of Néel that Cr and Mn as metals were antiferromagnets, where so far antiferromagnetism seemed to be only a property of non-metals. What happens here. Is Cr metal an RKKY magnet, or do we have something special? The moments of Cr ++ and Cr +++ are 4.9 and 3.8 Bohrmagnetons per atom respectively and the Cr ++ saturation moment should be 4 μ B.

8 Exchange interactions Stimulated by this discussion Shull and Wilkinson proved that these metals are weakly antiferromagnetic. See Rev. Mod. Phys 25 (1953), p100. Moreover, the wavelength was not equal to the length of the cubic unit cell (Corliss et al., PRL 3 (1959), p211). These results came on the moment that neutron diffraction techniques became a suitable tool for determining magnetic structures. We come to that later. Anyhow, there was a problem how to explain these results.

9 SDW s It was Overhauser (around 1960) who showed that in the onedimensional Hartree-Fock approximation the antiferromagnetic state could exist and may have lower energy. The AF periodicity is not given by the lattice but by the wave vector equal to the diameter of the volume of occupied states in k-space. With the help of neutron diffraction techniques, it is shown that these Spin Density Waves indeed exist. Main conclusions from Overhauser s work: a) Spin density waves are allowed states. b) SDW s may be ground state. c) SDW s with wave vector q = 2k F are most likely to minimise energy.

10 SDW s Source: A.W. Overhauser, PRL 3,9 (1959) 414 Source: [Overhauser (1962)]

11 SDW s Here, we will not give all the mathematical details of the HF procedure. We only give some flavour of what was going on here, in particular we will show some pictures to elucidate the situation. For detailed information we recommend the following paper: A. W. Overhauser, Phys. Rev. 128 (1962) p You need a reasonable starting wave function (for fermions this is a Slater determinant) and a smart trial potential. Then you have to solve the Schrödinger equation by a variational method until you find an internally consistent solution where your potential is stable under continued iterations. With a proper starting set, the procedure is generally convergent, but it is not sure your solution is the real ground state.

12 SDW s Some citations on the HF method: far from being straightforward coupled integral equations are thoroughly nonlinear and require an iteration technique for their solution. repeated until a self consistent set of solutions is obtained convergence dependent on initial guess of the one-particle states. Overhauser started his calculations with a helical polarization ( this leads to an off-diagonal contribution to the oneelectron exchange potential. )

13 SDW s Overhauser showed that for spin up and spin down a gap opens at the Fermi wavevector, but for the two at a different sign of k F (=q/2). The two waves at k F and -k F give together two charge density waves at q=2k F with a certain spin polarization, resulting in a static spin density wave with constant charge density. Source: A.W. Overhauser, PRL 4,9 (1960) 462 It has some resemblance with the opening of the gap at the Brillouin zone, but there the potential is fixed, here the potential is determined by the electron gas, with largest effect near the Fermi wave vector.

14 Source: [Overhauser (1962)] SDW s

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16 SDW s The spin susceptibility shows for SDW s to diverge at 2k F. Source: [Overhauser (1962)]

17 SDW s In 3 dimensions the problem may not yet be solved but here we give an artists impression of the situation. The gap causes a lowering of the D(ε F ), and thus of the linear term, γ, in the specific heat. From this effect the gapped surface fraction of the Fermi Surface can be derived. In order to conserve entropy, the entropy loss by the lower γ is recovered at the transition point to the paramagnetic state, resulting over there in a peak in the observed heat capacity. In the resistance, a jump should be expected on opening the gap, caused by a lowering of the number of available carriers, which is proportional to the opened Fermi Surface fraction.

18 SDW s In order to see whether in reality SDW s occur, we have to minimise the total energy, being the sum of the total kinetic energy including the effects of the SDW s and the total potential energy including the total exchange energy. The algebra necessary to do this, and calculate the correct parameters, is considerable even in the one dimensional case.

19 SDW s In three dimensions, the so called nesting vectors play the role of the 2k F from the one dimensional case. What is told is that SDW vectors should be in directions where the density of states is high. This sounds reasonable. Source: Wikipedia

20 Neutron diffraction It is not surprising that the discovery of Spin Density Waves (Shull and Wilkinson, 1953) goes parallel with the development of neutron diffraction techniques. However, also working with neutrons has its restrictions. Longitudinal waves cannot be detected: neutrons see no spins but only magnetic field. A longitudinally polarized magnetic field wave is incompatible with the Maxwell equations. Further: domains!! Domains add ambiguity to the interpretation of results. Source: [Shull & Wilkinson (1953)] ( Normal bcc structure: h + k + l = even. )

21 Arrott p333: It is not possible to decide from the diffraction experiments between the existence of a helical spin density wave state and the presence of two types of domains each with transverse linear spin density waves but with mutually perpendicular polarizations. A magnetic field may unravel the problem. Other problems: the intensities of the SDW reflections are very weak, and you don t know where they are. (needle in hayloft). Neutron diffraction With K = 2πG ± q, only the G=0 reflections give sufficient intensity. So, K is, in reciprocal space, not known In magnitude, nor in direction. Source: [Shull & Wilkinson (1953)]

22 Neutron diffraction Spin configurations are generally described in terms of Helical Spin Density Waves: p(z) = p (e x cos qz ± e y sin qz ) p(z) = p (e y cos qz ± e z sin qz ) p(z) = p (e z cos qz ± e x sin qz ), for q // z. The first are called normal helical waves, the last two lines describe end-over-end helical waves. This set of functions form a complete set to describe any spin wave with q = ± qe z. The use of this set instead of plane waves gives some profit in the analysis of neutron diffraction patterns. The normal and the end-over-end behave differently (Arrott, p 300).

23 Chromium A review article by E Fawcett, Rev. Mod. Phys. 60 (1988) p209, gives 75 pages of material on chromium properties. Too much to handle here. There is agreement on the AF low temperature state with T N =311K and μ 0.5μ B. In the picture here you see the behaviour of the thermal expansion, resistivity, specific heat, and thermo electric power.

24 Chromium From neutron results it follows that there is a second spin-flip transition at about 115K. Below 115K the SDW s are longitudinal, and above it the SDW s are transverse. Source: [Fawcett (1988)]

25 Chromium My interest is in particular the heat capacity. This here, on this picture, looks like a second order transition. However this is cold rolled material. The results showed to be strongly dependent on the strain situation. In the next slide we show results on a strain-free single crystal.

26 Chromium This is clearly a First-order SDW-PM Transition. I. S. Williams, E. S. R. Gopal, and R. Street, J. Phys F: Metal Phys, 9 (1979) P 431.

27 Chromium There is a similarity in the structure of the SDW HF equations and those of the BCS model for superconductivity. The temperature dependence of the SDW gap, which is proportional to the amplitude of the SDW s is a lookalike of the BCS curve. Maybe the similarity is not only mathematical. Source: [Overhauser, 1962] Maybe SDW s and SC are two sides of the same coin.

28 Chromium The entropy around the transition (obtained from heat capacity measurements) amounts to about somewhat less than 0.02 J/K.mol. This should be the recovery of the effect of a lower γ. From γ =1.4 mj/k.mol we can derive the total entropy of the electron gas at T N, being about 0.42 J/K.mol. This should point in the direction that 4.5% of the FS is gapped. This is in agreement with the resistivity jump of about 5% at T N. If Cr should have a permanent moment on-site, the entropy in the peak should be of the order of R.ln(2) = 5.76 J/K.mol. The observed entropy is about two orders of magnitude lower, indicating no permanent moment is present on the Cr sites.

29 Chromium From the BCS theory follows that the gap is (3.5 * k B T N ) 0.1 ev. For this energy holds roughly ħ 2 R 0 2 /2m* where R 0 is (in reciprocal space!!) the radius of the truncated part of FS and m* the effective mass. If p is the number of truncated faces (here 6), then for the total truncated fraction, t, holds t = p R 0 2 / 4 k F 2. With m* 1.5m, a value for R 0 can be derived: 0.2*10 8 cm -1. From q 2k F follows k F 1.1*10 8 cm -1. Result: t 0.05, in good agreement with the earlier estimates.

30 Conclusions Spin Density Waves are possible solutions for the free electron state. Overwhelming evidence exists that in Cr this SDW solution is ground state. In an SDW state a part of the FS is gapped. The wave vector of the SDW is determined not by the lattice but by a nesting vector, being about 2k F for a simple Fermi sphere. It is not clear whether, in a real situation, these nesting vectors can be calculated, or simply follow from experiment.

31 Thank you for your attention. Help, a gap, I ve to flip over