Three-dimensional solids in the quantum limit of high magnetic fields.
James Hetherington Part II Experimental and Theoretical Physics, Cavendish Laboratory, Department of Physics, Cambridge University
This paper reviews descriptively work to date on the question of three-dimensional many-body systems subjected to a magnetic field sufficiently strong that the occupied single-electron states lie in only one Landau level. Relevant theory is reviewed, including electron-electron interactions, perturbation theory, and the theory of phase transitions. Basic properties of the quantum-limit state are examined, including the field-induced quasi-one-dimensionality of the Fermi surface. Various proposals for interaction-dominated states in the quantum limit are then considered, beginning with the formation of Density Waves, where some variable varies periodically along the direction of the field. The property in question might be a component of the mean electron spin (Spin Density Waves), the charge density (Charge Density Waves), or the density of carriers of different momenta (Valley Density Waves). Such states correspond to the formation of bound electron-hole pairs. The possibility of superconductivity at these very high fields is also reviewed. It is explained how diamagnetic and Pauli pair breaking are avoided, and why materials which are not superconducting at low temperature may be superconducting at high fields. The possibility of triplet superconductive pairing is considered. The suggested properties of these phases are discussed with regard to magnetic behaviour and d.c. conductivity. A radical proposal known as the Squeezing Solution, representing a completely new behaviour of many-electron systems, is also considered.
CONTENTS Interesting physics can be found in the case where the field applied to a solid is so large Abstract that the energy of only one Landau level is Contents below the Fermi energy- the “quantum limit”. The quantum limiting case has been 1. Introduction experimentally reached in the case of 2- 2. Groundwork dimensional systems, where phenomena such 2.1 Electron-electron interaction as the “quantum hall effect” are observed.3 2.2 Perturbation theory 2.3 Phases and phase transitions This limit is more difficult to achieve in 3- 3. Generalities dimensional systems, where the Fermi energy 3.1 Landau levels in solids is much greater. Experimental investigation of 3.2 The quantum limit this case has thus not been possible to date. 3.3 The fermi surface in the There has been, however, much theoretical quantum limit speculation regarding the behaviour of such 3.4 Spin systems. It is this work that is reviewed here. 3.5 Impurities 4. Proposals for the quantum limit state 4.1 Density waves 4.2 Superconductivity 2. GROUNDWORK 4.3 Yakovenko’s “Squeezing Solution” 2.1 Electron-electron interaction 5. Conclusion Electron-electron interactions, not included in References the simplest models of a solid in undergraduate physics, are significant in the quantum limit.
1. INTRODUCTION The effective electron-electron interaction includes, as well as the Coulomb force, a force In a magnetic field, single particle caused by the interaction of both electrons with wavefunctions take the form of so-called the ionic lattice, which may be viewed as caused by the exchange of phonons. This force “Landau levels”, where the momentum 2 perpendicular to the magnetic field is may be attractive, and tends to be short range. quantised.1 The effective range of the Coulomb force is reduced by the intervening sea of electrons (screening). † parameter, which is written <(r)(r)> , is One way of approaching interactions is nonzero only in that state, because such a state through the mean-field approximation2. The is not an eigenstate of the number of electrons. interaction of each electron with the Some of the electrons might be in Cooper expectation† distribution of the other electrons pairs, and so not be part of the Fermion is considered. If one uses single particle states system. (Action of (r)(r) reduces the to construct a trial solution, this is called the number of electrons by two.) Averages of field Hartree-Fock approximation. operators which are zero except in the presence of phase transitions are called “anomalous averages”. Mean field theory can be 2.2 Perturbation theory conveniently formulated using averages of field operators. We can start with the solution for the non- interacting system, and treat the interaction as To take into account the effect of a perturbation to the independent-particle “fluctuations” away from the most likely state solution. The Hartree-Fock theory with a trial is equivalent to including terms in perturbation solution constructed from independent-particle theory beyond first order. To do this in non- states is recovered in the first order of normal phases it is necessary to use a more perturbation theory expectation values. general starting point than the independent particle approximation. The rest of the perturbation series is equivalent to including “higher-order” processes: The 3. GENERALITIES total interaction must be expressed as a sum over various possible interactions. Such a 3.1 Landau levels in solids summation is aided by visualisation of the constituent processes in terms of Feynman The “Landau Levels”, the quantum form of the diagrams. (Figure 1) helical orbits of charged particles in a magnetic field, are characterised by three quantum 2.3 Phases and phase transitions2,5,6 numbers. These are n, related to the magnitude of momentum perpendicular to the field, p, the Where inter-particle interactions are present, a momentum parallel to the field, and x, which many-body system can show behaviour can be interpreted as one co-ordinate of the dominated by these. Such phases correspond to centre of the orbit. cases where the perturbation series for the interaction, starting from independent particles, In solids of finite volume the x,p spectrum is fails to converge. Such phases occur at low also discrete. The states correspond, for an temperatures, where thermal energies don’t isotropic dispersion relation, to a cylindrical dominate the energy of interaction. lattice in momentum space (Fig. 2) While this visualisation identifies x with the direction of 6 Landau describes transitions between phases the transverse momentum, the usual in terms of an “order parameter”, which is an wavefunctions chosen are not eigenstates of indication of the degree to which a phase this. transition has occurred. The number of lattice points for a given n, The order parameter has a microscopic spin-state, and range of p, p, depends1 upon interpretation. For example, that for H: superconductivity is connected to the expectation density of the bound electron- N = eHVp / 422c electron pairs called “Cooper pairs”, which n … Eq. 2 cause superconductivity. As the field increases, the momentum space One can often write order parameters in terms † separation of states in the radial direction of “field operators”, written (r) and (r), increases, but the angular density of states also which, acting on a state, respectively destroy increases, so that the total density of states and create particles in position eigenstates at r, remains constant. with spin . The superconducting order † Note the opposite spin labels. Conventional † Both quantum mechanical and thermal cooper pairs involve electrons in singlet spin averages can be involved in this process states. carriers to bind to impurities. This causes 3.2 The quantum limit serious experimental problems.
The quantum limit arises when the states with 4. PROPOSALS FOR THE n=0 become sufficient in number to accommodate all the electrons at energies QUANTUM LIMIT STATE below that of the lowest-energy state with n=1. 29 This is equivalent to the intuitive idea of the Abrikosov examined the properties of a solid quantum limit occurring when only one in the quantum limit, considering both Landau cylinder fits inside the zero-field Fermi isotropic and semimetal models. He shows that sphere, as can be proved from equation 2.26 an increase in impurity density causes the conductivity parallel to the field to decrease, The lower the density of carriers in the solid, and that perpendicular to the field to increase. the lower the Fermi energy and the easier it is This is in line with the picture of Landau levels to reach the quantum limit. For this reason, as helical orbits. some authors have considered the quantum limit in semimetals, where the number of Proposals for interaction-dominated states in charge carriers is very small. In the usual the quantum limit will now be considered (Fig model, these carriers occur in isolated pockets 4) of momentum space sometimes called “valleys”. (Fig 7) 4.1 Density Waves in the Quantum Limit.
7 3.3 The fermi surface in the quantum limit Celli and Mermin suggested that magnetic interactions favoured a state where there was a The fermi surface in the quantum limit, in the small component of mean spin perpendicular momentum-space visualisation described to the magnetic field, oscillating with circular 8 above, is a pair of “fermi rings” (with z- polarisation along the field direction . This is a “spin-density wave” (SDW). A charge-density- momenta pF(QL)). (Figure 3) This is important, because interaction effects are most significant wave (CDW), where the density of electric for states near the fermi surface.2 charge periodically varies, is also possible.
3.4 Spin Such density-wave (DW) states are much more likely when the fermi surface shows one- dimensional character: In the presence of a magnetic field, Zeeman splitting causes energy differences between The phase is formed by the mixing of states states of opposite spin. For free electrons, with opposite components of momentum along where the spin g-factor is 2, this results in the the direction of the wave, resulting in a energies of the nth spin-up and (n+1)th spin- standing wave. down Landau levels being equal. In a solid, however, the effective g, g*, can be different, To produce a coherent wave, we need many with some materials having very low g* (<0.1) states available with the same component of 4. This means that it is possible to have a momentum parallel to the wave direction. One quantum limit situation in which particles of state of each pair must be “initially” full and both spins are present, whenever g* < 2. the other empty, for otherwise we have done nothing to change the system. We thus seek a 3.5 Impurities Fermi surface with two parallel plane regions: one-dimensional character. Density-wave Scattering of electrons by impurities can states are favoured in the quantum limit, for swamp the interaction effects and remove the the two “Fermi rings” satisfy just this criterion. interaction-dominated phases suggested below. The wavevector of the density wave is Impurities will be significant if the mean free path of electrons with respect to collisions is 2pF(QL)/, parallel to the field. less than the length scale of the Landau wavefunctions. This result is found in many The DW state corresponds to the formation of different ways in the works considered below, bound electron-hole pairs, because it involves and there is not space here to discuss each superposition of filled (electron) and empty individual treatment. (hole) states.
There is also the effect known as “magnetic Spin density waves are not completely freezeout”, where the magnetic field causes theoretical; they have been used to explain the antiferromagnetism of chromium, which has diagram leads to a singularity, it is a sign that the appropriate parallel plane regions in its the electron-hole pair is not temporary; that is, Fermi surface. Density waves occur in other bound electron-hole states form. areas of physics, for example, explaining unexpected behaviour in the non- Abrikosov used his results to obtain anomalous superconducting state of high-temperature averages relating to the formation of density 10 superconductors . wave states. These are of the form < - + >
where + destroys an electron near the +ve
Celli and Mermin established that a spin- fermi surface and - a hole near the negative density wave is reasonable using Hartree-Fock energy fermi surface, and are found to be non- theory. The trial state was constructed from zero. Those corresponding to Landau-level single-particle wavefunctions, superconductivity (the formation of bound modified to include the possibility of mixing electron-electron or hole-hole pairs) are zero. expressed above. It is necessary to “hard-wire” the SDW into the trial solution in this way; the Abrikosov went on to show that when the Hartree-Fock theory with independent-electron coulomb interaction is not point-like at least wave-functions, as a subset of perturbation one singularity persists11. theory, can only produce normal phases. Since the SDW is explicitly assumed in the trial Brazovskii12 furthered Abrikosov’s work by solution, this method cannot show that it is the abandoning both the neglect of the transverse SDW that forms in the quantum limit. quantum number and the assumption of pointlike interaction. The relevant Feynman The CDW possibility was shown to be diagrams constitute figure 6. The resulting favoured in the Hartree-Fock approximation pattern has been compared to Parquet flooring, when the wave-vector has a small component so these equations are known as “Parquet perpendicular to the field13, because of the equations”. field-induced spatial confinement of the wavefunction. The SDW is not possible in the Brazovskii was able to determine the nature of g* 2 case, so this is the situation usually the vertex singularities resulting from these considered by those advocating a CDW. The equations, but not the behaviour of the vertices field-induced CDW was first mooted by away from the singularities. From this, he astronomers, because of the huge fields present obtained relevant anomalous averages, and in astronomical contexts.27 demonstrated that a DW state was possible. Without the detailed form of the vertices, the Abrikosov9 went beyond this, using behaviour of the system as the DW transition perturbation theory with a purely one- is approached cannot be determined, and so the dimensional model, assuming proper onset of the DW cannot be certain. consideration of x would produce only differences of order unity. Tesanovic and Halperin14, considered the quantum limit of a system with valleys. In Abrikosov assumed the interaction was addition to the SDW with g*2, they suggested effectively point-like, due to screening, and a state where the electrons in different valleys that all the electrons are in the same spin-state form out-of-phase CDWs. They called this in the quantum limit. The Pauli principle state a Valley-Density-Wave, and verified it means that the space-state is necessarily using Hartree-Fock theory. Such states have antisymmetric, and hence there can be no uniform charge density, and are thus have a pointlike interaction between electrons. smaller cost in electrostatic energy than simple Abrikosov includes, instead, a pointlike CDWs. interaction between electrons and holes. Two kinds of pointlike interaction, between like What will be the properties of the DW phase? particles and between unlike particles, are used By analogy with chromium, this state is to approximate the true interaction, with the expected to be antiferromagnetic; Brazovskii former being identically zero. estimates the magnitude of the jump in magnetic susceptibility at the phase transition. Abrikosov found that certain of the perturbed vertices diverge, and thus that an interaction- Tesanovic and Halperin, however, suggest dominated state is formed. A Feynman remarkable properties for the DW state, diagram such as figure 5 represents an including a highly anisotropic electrical interaction mediated through the temporary conductivity tensor, conducting perpendicular creation of an electron-hole pair. When this to the field, and insulating when the electric and magnetic fields are parallel. The states near the Fermi surface are removed into the It is postulated that the quantum-limiting field DW, and the gap causes insulation just like in enhances the attractive part of the interaction ordinary insulators. that gives rise to superconductivity16. The increased spatial confinement of the 4.2 Superconductivity in the Quantum wavefunctions with increased H enhances the Limit effect of the short-ranged attractive phonon interaction and reduces the effect of the long- The suggestion that the quantum limit be ranged repulsive coulomb interaction. This superconducting has received much attention means that a superconducting quantum limit is recently. Superconductivity is destroyed by a possible in materials that are not high magnetic field, for the supercurrents that superconducting at low fields. circulate in response to the field and provide the Meissner effect increase the energy of the Rasolt and Tesanovic look at what would be system. Above a certain field, the non- necessary take into account fluctuations but are superconducting state has a lower energy. This not able to solve the problem except in the so- is called “diamagnetic pair breaking”. called “ladder approximation”. Here, only electron-electron loops are considered, and a Gruenberg and Gunther15 were the first to superconducting instability is indeed found4. suggest that superconductivity may be Work in this area has been furthered by Goto enhanced by the action of a quantising and Natsume20 magnetic field. The properties of the quantum limit are not The majority of the work in this area, however, expected to include a gap in the excitation is due to Tesanovic and co-workers4. They spectrum at the Fermi energy, unlike in low- 21,22 argue that the nature of the Landau orbitals field superconductivity. The resistivity is means that the form of the local order almost zero parallel to the field, and very small 4 parameter (r) giving circulation of perpendicular to the field . There is no supercurrents is naturally satisfied in the Meissner effect in this state, because like in the quantum limit by the Landau orbitals without a so-called type II superconductors, the field cost in energy. They support this idea with penetrates the sample in a regular lattice of 4 mean field calculations. In this regime, the “vortex filaments”. superconducting transition temperature actually increases with the field. It has been suggested that the anomalous increase in the critical field of some type-II Their argument is initially conducted with superconductors at very low temperatures may g*=0. With g* nonzero the Zeeman effect be related to the presence of the quantum-limit 24 provides another way for the magnetic field to phase, but this remains controversial . destroy superconductivity, because the Cooper pairs are singlet spin states. This is called 4.3 Yakovenko’s “Squeezing Solution” “Pauli pair breaking”. Yakovenko23 put forward a new suggestion for Pauli pair breaking need not destroy the quantum limit. This is exciting, for the superconductivity provided g* < 2, because the number of fundamentally different behaviours singlet pairs can be formed from electrons in of many-electron systems, so called states with differing p15. This is only possible “universality classes”, already known is small. in the quantum limit, where the quasi-one- dimensionality allows a single p to stabilise He works with the Parquet equations as all the pairs. formulated by Brazovskii, (Figure 6) but is able, through use of mathematical tricks and Several authors17,18,19 look at the possibility of numerical calculations, to obtain a complete unusual forms of superconductivity where the solution. He assumes the field is small enough electrons in the pair are in a triplet spin-state. that the length scale of the wavefunction Such states depend upon the phonon-induced perpendicular to the field permits the inter-electron potential having an appropriate transverse range of the interaction to be form. The competition between the triplet assumed pointlike. This assumption is not pairing and the CDW with g*2 has been made for the direction parallel to the field, so considered18, and a mean-field analysis shows that despite assuming all the electrons are spin- that for some interactions the superconducting aligned, the electron-electron interaction is not state is favoured. ruled out. 1Landau and Lifshitz, “Quantum Mechanics By considering the relative strengths of the (Non-relativistic theory)”- Butterworth & singularities due to electron-electron and Heinmann 1977 electron-hole loops in the perturbation series, 2Ashcroft and Mermin “Solid State Physics” – he shows that for a “net” repulsive electron- Saunders College 1976 electron interaction the DW forms. For 3