Ferromagnetic order of nuclear spins coupled to conduction electrons: A combined effect of electron-electron and spin-orbit interactions
Spontaneous nuclear spin polarization in semiconductor heterostructures at finite but low temperatures has recently attracted a considerable attention both on the theoretical 1–4 and experimental 5 sides. Apart from a fundamental interest in the new type of a ferromagnetic phase transition, the interest is also motivated by an expectation that spontaneous polarization of nuclear spins should suppress decoherence in single-electron spin qubits caused by the hyperfine interaction with the surrounding nuclear spins,1,2 and ultimately facilitate quantum computing with single-electron spins.6,7
Improvements in experimental techniques have lead to extending the longitudinal spin relaxation times in semiconductor quantum dots (QDs) to as long as 1 s. 8–10 The decoherence time in single electron GaAs QDs has been reported to exceed s in experiments using spin-echo techniques at magnetic fields below 100 mT,11,12 whereas a dephasing time of GaAs electron-spin qubits coupled to a nuclear bath has lately been measured to be above s. 13 Still, even state-of-the-art dynamical nuclear polarization methods 14–18 allow for merely up to polarization of nuclear spins, 18 whereas polarization of above is required in order to extend the electron spin decay time only by one order of magnitude. 17 Full magnetization of nuclear spins by virtue of a ferromagnetic nuclear spin phase transition (FNSPT), if achieved in practice, promises a drastic improvement over other decoherence reduction techniques.
The main mechanism of the interaction between nuclear spins in the presence of conduction electrons is the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. 19 The effective Hamiltonian of the RKKY interaction between on-site nuclear spins of magnitude ,
is parameterized by an effective exchange coupling
where is the hyperfine coupling constant, is the number density of nuclear spins, and
is the (static) correlation function of electron spins. [Hereafter, we will refer to —and to its momentum-space Fourier transform—as to “spin susceptibility,” although it is to be understood that this quantity differs from the thermodynamic susceptibility, defined as a correlation function of electron magnetization, by a factor of where is the Bohr magneton.] It is worth emphasizing that contains all the effects of the electron-electron interaction;1,2 this circumstance has two important consequences for the RKKY coupling. First, the electron-electron interaction increases the uniform spin susceptibility, which should lead to an enhancement of the critical temperature of the FNSPT, at least at the mean-field level. Second, stability of the nuclear-spin ferromagnetic order is controlled by the long-wavelength behavior of the magnon dispersion which, in its turn, is determined by at . In a spin-isotropic and translationally invariant system,
with , while the magnetization is given by
where is the nuclear-spin magneton (we set throughout the paper). The second term in Eq. (1.5) describes a reduction in the magnetization due to thermally excited magnons. In a free two-dimensional electron gas (2DEG), is constant for , and thus the magnon contribution to diverges in the limit, which means that long-range order (LRO) is unstable. However, residual interactions among the Fermi-liquid quasiparticles lead to a nonanalytic behavior of the spin susceptibility; for , , where both the magnitude and the sign of depend on the strength of the electron-electron interaction.20,21 In two opposite limits, i.e., at weak coupling and near the Stoner instability, 22 the prefactor is positive which, according to Eqs. (1.4) and (1.5), means that LRO is unstable. However, is negative [and thus the integral in Eq. (1.5) is convergent] near a Kohn-Luttinger superconducting instability;3,23 also, in a generic Fermi liquid with neither strong nor weak interactions, is likely to be negative due to higher-order scattering processes in the particle-hole channel. 24–26
The spin-wave-theory argument presented above is supported by the analysis of the RKKY kernel in real space. A linear-in- term in corresponds to a dipole-dipole like, term in (see Sec. III). If , the dipole-dipole interaction is repulsive, and the ferromagnetic ground state is unstable and vice versa, if , the dipole-dipole attraction stabilizes the ferromagnetic state.
It is worth noting here that even finiteness of the magnon contribution to the magnetization does not guarantee the existence of LRO. Although the Mermin-Wagner theorem 27 in its original formulation is valid only for sufficiently short-range forces and thus not applicable to the RKKY interaction, it has recently been proven 28 that magnetic LRO is impossible even for the RKKY interaction in . From the practical point of view, however, the absence of LRO in 2D is not really detrimental for suppression of nuclear-spin induced decoherence. Indeed, nuclear spins need to be ordered within the size of the electron qubit (a double QD system formed by gating a 2DEG) as well as its immediate surrounding such that there is no flow of magnetization. Since fluctuations grow only as a logarithm of the system size in 2D, it is always possible to achieve a quasi-LRO at low enough temperatures and on a scale smaller that the thermal correlation length. In addition, spin-orbit interaction (SOI)—which is the main subject of this paper, see below—makes a long-range order possible even in 2D. 28
The electron spin susceptibility in Eq. (1.4) was assumed to be at zero temperature. First, since the nuclear spin temperature is finite, the system as a whole is not in equilibrium. However, a time scale associated with “equilibration” is sufficiently long to assume that there is no energy transfer from the nuclear- to electron-spin system. Second, if the electron temperature is finite, the linear scaling of is cut off at the momentum of order . For , such that and, according to Eq. (1.5), spin waves would destroy LRO. However, at low enough temperatures, the thermal length is much larger than a typical size of the electron qubit . (For example, mm at mK.) Therefore and, indeed, the electron temperature can be assumed to be zero.
In practically all nuclear-spin systems of current interest, such as GaAs or carbon-13 nanotubes, spin-orbit interaction (SOI) plays a vital role. The main focus of the paper is the combined effect of the electron-electron and spin-orbit (SO) interactions on the spin susceptibility of 2DEG and, in particular, on its dependence, and thus on the existence/stability of the nuclear-spin ferromagnetic order.
The interplay between the electron-electron and SOIs is of crucial importance here. Although the SOI breaks spin-rotational invariance and thus may be expected to result in an anisotropic spin response, this does not happen for the Rashba and Dresselhaus SOIs alone: the spin susceptibility of free electrons is isotropic [up to terms] as long as both spin-orbit-split subbands remain occupied. 4 The electron-electron interaction breaks isotropy, which can be proven within a Fermi-liquid formalism generalized for systems with SOI. 29 Specific models adhere to this general statement. In particular, for a dense electron gas with the Coulomb interaction. 30
In this paper, we analyze the dependence of the spin susceptibility in the presence of the SOI. The natural momentum-space scale introduced by a (weak) Rashba SOI with coupling constant ( ) is the difference of the Fermi momenta in two Rashba subbands:
where is the band mass of 2DEG. Accordingly, the dependence of on is different for above and below ; in the latter case, it is also different for the out-of-plane and in-plane components. To second order in electron-electron interaction with potential , the out-of-plane component is independent of for :
On the other hand, the in-plane component scales linearly with even for :
In Eqs. (1.7a) and (1.7b), , is the Fermi momentum, is the Fermi energy, is the spin susceptibility of a free 2DEG, and denotes a nonanalytic part of . For , the spin susceptibility goes back to the result of Ref. 20 valid in the absence of the SOI:
Note that the subleading term in in Eq. (1.7b) differs by a factor of from the leading term in in Eq. (1.8). There is no contradiction, however, because Eqs. (1.8) and (1.7b) correspond to the regions of and , respectively.
Equations (1.7a) and (1.7b) show that the uniform spin susceptibility is anisotropic: . This implies that the RKKY coupling is stronger if nuclear spins are aligned along the normal to the 2DEG plane, and thus the nuclear-spin order is of the Ising type. In general, a 2D Heisenberg system with anisotropic exchange interaction is expected to have a finite-temperature phase transition. 31 In an anisotropic case, the dispersion of the out-of-plane spin-wave mode is given by2,32
with . Ising-like anisotropy implies a finite gap in the magnon spectrum. In our case, however, the situation is complicated by the positive slope of the linear dependence of the second-order result for , which according to Eq. (1.9) translates into decreasing with . Combining the asymptotic forms of from Eqs. (1.7a), (1.7b), and (1.8) together, as shown in Fig. 1, we see that is necessarily negative in the interval , and thus LRO is unstable. Therefore anisotropy alone is not sufficient to ensure the stability of LRO: in order to reverse the sign of the dependence, one also needs to invoke other mechanisms, arising from higher orders in the electron-electron interaction. We show that at least one of these mechanisms—renormalization in the Cooper channel—is still operational even for and capable of reversing the sign of the dependence is the system is close to (but not necessarily in the immediate vicinity of) the Kohn-Luttinger instability.
We note that the dependences of on in the presence of the SOI is similar to the dependences on the temperature and magnetic field, 4 presented below for completeness:and . As Eqs. (1.7a), (1.7b), and (1.10a) demonstrate, while nonanalytic scaling of with all three variables ( , , ) is cut off by the scale introduced by SOI, scaling of continues below the SOI scale. This difference was shown in Ref. 4 to arise from the differences in the dependence of the energies of particle-hole pairs with zero total momentum on the magnetic field: while the energy of such a pair depends on the SO energy for , this energy drops out for .
In addition to modifying the behavior of for , SOI leads to a new type of the Kohn anomaly arising due to interband transitions: a nonanalyticity of at . The nonanalyticity is stronger in than in : , while , where is the step function. Consequently, the real-space RKKY interaction exhibits long-wavelength oscillations and in addition to conventional Friedel oscillations behaving as . It is worth noting that the long-wavelength Friedel oscillations occur only in the presence of both electron-electron and SO interactions.
This paper is organized as follows. In Sec. II, we derive perturbatively the electron spin susceptibility of interacting 2DEG with the SOI as a function of momentum; in particular, Secs. II A,II B,II C,II D outline the derivation of all relevant second-order diagrams, Sec. II E is devoted to Cooper renormalization of the second order result, and in Sec. II F, we show that, in contrast to the spin susceptibility, the charge susceptibility is analytic at small (as it is also the case in the absence of SOI). In Sec. III, we derive the real-space form of the RKKY interaction and show that it exhibits long-wavelength oscillations with a period given by the SO length . Details of the calculations are delegated to Appendices A—D. In particular, the free energy in the presence of the SOI is derived beyond the random phase approximation in Appendix D. The summary and discussion of the main results are provided in Sec. IV.
II. SPIN SUSCEPTIBILITY OF INTERACTING ELECTRON GAS
Dynamics of a free electron in a two-dimensional electron gas (2DEG) in the presence of the Rashba spin-orbit interaction with a coupling strength is described by the following Hamiltonian:
where is the electron momentum of an electron and is a vector of Pauli matrices. The interaction between electrons will be treated perturbatively. For this purpose, we introduce a Green's function:
where with being a fermionic Matsubara frequency, , and is a Rashba index.
The nonanalytic part of a spin susceptibility tensor to second order in electron-electron interaction is given by seven linear response diagrams depicted in Figs. 2–7. Due to symmetry of the Rashba SOI, and .
In the following sections, we calculate all diagrams that contribute to nonanalytic behavior of the out-of-plane, , and in-plane, , components of the spin susceptibility tensor for small external momenta ( ) and at In the absence of SOI, the nonanalytic contributions to the spin susceptibility from individual diagrams are determined by “backscattering” or “Cooper-channel” processes,20,24 in which two fermions with initial momenta and move in almost opposite directions, such that . Backscattering processes are further subdivided into those with small momentum transfer, such that , and those with momentum transfers near , such that . In the net result, all contributions cancel out and only contributions survive. We will show that this is also the case in the presence of the SOI. In what follows, all “ ” diagrams are to be understood as the channel of the backscattering process.
A. Diagram 1
1. General formulation
The first diagram is a self-energy insertion into the free-electron spin susceptibility, see Fig. 2. There are two contributions to the nonanalytic behavior: (i) from the region of small momentum transfers, i.e., ,
and (ii) from the region of momentum transfers close to , i.e., and ,
Here, and (and the same for other momenta). The time component of is equal to zero throughout the paper. Since the calculation is performed at there is no difference between the fermionic and bosonic Matsubara frequencies. A factor of 2 appears because the self-energy can be inserted either into the upper or the lower arm of the free-electron susceptibility. As subsequent analysis will show, a typical value of the momentum transfer is on the order of either the external momentum or the “Rashba momentum” [cf. Eq. (1.6)], whichever is larger. In both cases, while the momenta of both fermions are near , thus we neglect in the angular dependences of the Rashba vertices: with . [The origin of the axis is arbitrary and can be chosen along ] Also, we impose the backscattering correlation between the fermionic momenta: in the part of the diagram. With these simplifications, we obtain
where summation over the Rashba indices is implied,
and, finally, the partial components of the dynamic particle-hole bubble are given by
The main difference between the out-of-plane and in-plane components is in the structure of the “quaternion,” defined by Eq. (2.7e) and calculated explicitly in Appendix A [cf. Eq. (A3)]. The dependence of on the external momentum enters only in a combination with the SOI coupling as , where . Combinations of indices are determined by the spin vertices and are, therefore, different for the out-of-plane and in-plane components. The out-of-plane component contains only such combinations for which the coefficient is finite. Therefore the SOI energy scale is always present and, for , one can expand in . The leading term in this expansion is proportional to but any finite-order correction in vanishes. In fact, one can calculate the entire dependence of on (which is done in Appendix B) and show that is indeed independent of for (and similar for the remaining diagrams). On the other hand, some of the quaternions that enter the in-plane component have and thus do not contain the SOI, which means that one cannot expand in anymore. These quaternions provide linear-in- dependence of even for , where the slope of this dependence is of that in the absence of the SOI. This is the origin of the difference in the dependences of and , as presented by Eqs. (1.7a) and (1.7b).
The evaluation of the out-of-plane and in-plane part of diagram 1 is a subject of the next two sections.
2. Diagram 1: out-of-plane component
We begin with the out-of-plane component of the spin susceptibility, in which case and . Summation over the Rashba indices yields
As we explained in Sec. II A 1, the quaternions in Eqs. (2.8a) and (2.8b) contain only in combination with . Therefore, for , the leading term is obtained by simply setting , upon which the remaining integrals can be readily calculated. The results are given by Eqs. (A7) and (A8), so that
3. Diagram 1: in-plane component
The in-plane component of the spin susceptibility differs substantially from its out-of-plane counterpart due the angular dependence of the traces and which, for the in-plane case, read
(For the sake of convenience, we choose the axis to be perpendicular to when calculating all diagrams for .) Summing over the Rashba indices, one arrives at
Details of the calculation are given in Appendix. A2b; here, we present only the results for :
Notice that the linear-in- dependence survives in the in-plane component of the spin susceptibility even for . Similar behavior was found in Ref. 4 for the temperature dependence of the uniform spin susceptibility in the presence of the SOI.
B. Diagram 2
Diagram 2, shown in Fig. 3, is a vertex correction to the spin susceptibility. As in the previous case, there are two regions of momentum transfers relevant for the nonanalytic behavior of the spin susceptibility: the region, where
and the -region, where
where(2.15c) and (2.15d) are derived in Appendix A.
Traces entering the part of the out-of-plane and in-plane components are evaluated as
Summing over the Rashba indices and using the symmetry properties of and , it can be shown that the parts of diagrams 1 and 2 cancel each other
which is also the case in the absence of the SOI. 20 Therefore we only need to calculate the part of diagram 2.
1. Diagram 2: out-of-plane component
Summation over the Rashba indices with the coefficient for the out-of-plane part gives
where stands for the preceding terms with an opposite sign of momentum. Integrating over and at , yields [cf. Eq. (A9)]
Again, an exact calculation at finite proves that this results holds for any .
2. Diagram 2: in-plane component
The in-plane component comes with a Rashba coefficient , such that
The first part, proportional to , contains the SOI coupling . In this part, can be set to zero, and the resulting linear-in- part equals half of that for the out-of-plane component due to the integral over . On the other hand, in the term proportional to , the dependence on drops out upon integration over , and the final result for reads as [cf. see Eq. (A12)]
C. Diagrams 3 and 4
We now turn to “Aslamazov-Larkin” diagrams, Fig. 4, which represent interaction via fluctuational particle-hole pairs. Without SOI, these diagrams are identically equal to zero because the spin vertices are averaged independently and thus vanish. With SOI, this argument does not hold because the Green's functions now also contain Pauli matrices and, in general, diagrams 3 and 4 do not vanish identically. Nevertheless, we show here that the nonanalytic parts of diagrams 2 and 3 are still equal to zero.
Diagrams 3 and 4 correspond to the following analytical expressions:
Note that the second trace is the same in both diagrams. In what follows, we prove that
for both small and large momentum transfers .
1. Diagrams 3 and 4: out-of-plane components
2. Diagrams 3 and 4: in-plane components
For the in-plane part of the spin susceptibility, the proof is more complicated as the traces do not vanish on their own. To calculate the part, we need the following two objects:
The prime over denotes that integration over the angle is not yet performed as compared to defined by Eq. (2.15d).
Summing over the Rashba indices, one finds
and, therefore, the in-plane component at small momentum transfers vanishes.
The trace for the case turns out to be the same as for the one because
However, in order to see the vanishing of the part, the integral over has to be evaluated explicitly with , i.e.,
where we used an expansion of around : . Summing over the Rashba indices, we obtain
and, therefore, the part of the in-plane components of diagrams 3 and 4 is also equal to zero.
D. Remaining diagrams and the final result for the spin susceptibility
The remaining diagrams can be expressed in terms of the those we have already calculated.
Diagram 5 in Fig. 5 reads
and . A factor of 4 appears because the “sunrise” self-energy can be inserted into either the lower or the upper arm of the bubble while each of the interaction lines can carry momentum of either or . A minus sign is due to an odd number of fermionic loops. Upon summation over the Rashba indices, we obtain
For diagram , we obtain
In both cases, . Using the symmetry property in , summing over the Rashba indices, and noticing that , we arrive at
Finally, diagram shown in Fig. 7 is related to diagram 2 at small momentum transfers. Indeed,
where again . After summation over the Rashba indices, this diagram proves related to the small-momentum part of diagram 2 as
The results of this section along with Eq. (2.17) show that the sum of all diagrams proportional to cancel each other:
Therefore, as in the absence of SOI, the nonanalytic part of the spin susceptibility is determined only by the Kohn anomaly at .
E. Cooper-channel renormalization to higher orders in the electron-electron interaction
An important question is how the second-order results, obtained earlier in this section, are modified by higher-order effects. In the absence of SOI, the most important effect—at least within the weak-coupling approach—is logarithmic renormalization of the second-order result by the interaction in the Cooper channel. As it was shown in Ref. 3, this effect reverses the sign of the dependence due to proximity to the Kohn-Luttinger superconducting instability; the sign reversal occurs at , where is the Kohn-Luttinger critical temperature. For momenta below the SO scale ( ), ceases to depend on but still scales linearly with . What is necessary to understand now is whether the linear-in- term in is renormalized in the Cooper channel. The answer to this question is quite natural. The and terms in the second-order result for [see Eq. (1.7b)] come from different parts of diagram: the term comes from independent part and vice versa. Starting from the third order and beyond, these two terms acquire logarithmic renormalizations but the main logarithm of these renormalizations contains only one energy scale. In other words, the term is renormalized via while the is renormalized via . For example, the third-order result for the part of diagram 1 (see Fig. 2) reads [for simplicity, we assume here a contact interaction with ]
where and is the ultraviolet cutoff. Details of this calculation are given in Appendix C. It is clear already from this result that the logarithmic renormalization of the term in remains operational even for , with consequences similar to those in Ref. 3.
F. Charge susceptibility
In the absence of SOI, nonanalytic behavior as a function of external parameters , , is present only in the spin but not charge susceptibility.20,33,34 An interesting question is whether the charge susceptibility also becomes nonanalytic in the presence of SOI. We answer this question in the negative: the charge susceptibility remains analytic. To show this, we consider all seven diagrams replacing both spin vertices by unities. The calculation goes along the same lines as before, thereby we only list the results for specific diagrams; for ,
whereas on its own ( ). First, we immediately notice that SOI drops out from every diagram even in the limit . Second, the sum of the nonanalytic parts of all the charge susceptibility diagrams is zero, , as in the case of no SOI.
III. RKKY INTERACTION IN REAL SPACE
A nonanalytic behavior of the spin susceptibility in the momentum space leads to a power-law decrease of the RKKY interaction with distance. In this section, we discuss the relation between various nonanalyticities in and the real-space behavior of the RKKY interaction. We show that, in addition to conventional Friedel oscillations, a combination of the electron-electron and SO interactions leads to a new effect: long-range Friedel-like oscillations with the period given by the SO length.
A. No spin-orbit interaction
First, we discuss the case of no SOI, when the spin susceptibility is isotropic: . For free electrons, the only nonanalyticity in is the Kohn anomaly at which translates into Friedel oscillations of the RKKY kernel; in 2D and for , 35
One effect of the electron-electron interaction is a logarithmic amplification of the Kohn anomaly (which also becomes symmetric about the point): . 36 Consequently, is also enhanced by a logarithmic factor compared to the free-electron case: .
however, as we explained in Sec. I, both the magnitude and sign of can be changed due to higher-order effects. (Cooper channel renormalization leads also to multiplicative corrections to the linear-in- term; those correspond to multiplicative renormalization of the real-space result and are ignored here.)
In 2D, is related to via
Power-counting suggests that the term in translates into a dipole-dipole-like term in . To see if this is indeed the case, we calculate the integral
with an arbitrary cutoff , and search for a universal, -independent term in the result. If such a term exists, it corresponds to a long-range component of the RKKY interaction. Using an identity and integrating by parts, we obtain
where is the Struve function. The asymptotic expansion of the last term in the preceding equation indeed contains a universal term
where stands for nonuniversal terms. A corresponding term in reads
As a check, we also calculate the Fourier transform of the -independent term in . The corresponding integral
does not contain a -independent term and, therefore, a constant term in does not produce a long-range component of the RKKY interaction, which is indeed the case for free electrons.
Equation (3.7) describes a dipole-dipole-like part of the RKKY interaction that falls off faster than Friedel oscillations but is not oscillatory. (Incidentally, it is the same behavior as that of a screened Coulomb potential in 2D, which also has a nonanalyticity at small . 37)
In a translationally invariant system, . Therefore, if , i.e., increases with , the dipole-dipole interaction is repulsive for parallel nuclear spins and attractive for antiparallel ones. Since the behavior sets in only at large distances, the resulting phase is a helimagnet rather than an antiferromagnet. Vice versa, if the dipole-dipole interaction is attractive for parallel spins. This corresponds precisely to the conclusions drawn from the spin-wave theory: a stable FM phase requires that , which is the case if .
B. With spin-orbit interaction
C. Free electrons
In a free-electron system, the SOI splits the Fermi surface into two surfaces corresponding to two branches of the Rashba spectrum with opposite helicities. Consequently, both components of the spin susceptibility in the momentum space have two Kohn anomalies located at the momenta with To see this explicitly, we evaluate the diagonal components of for ,
For , the matrix elements of the spin operators in the helical basis reduce to
Therefore contains only contributions from intraband transitions,
Each of the two terms in Eq. (3.11) has its own Kohn anomaly at , . In real space, this corresponds to beating of Friedel oscillations with a period
This behavior needs to be contrasted with that of Friedel oscillations in the charge susceptibility, where, to leading order in , the Kohn anomaly is present only at twice the Fermi momenta in the absence of SOI. 38 Consequently, the period of Friedel oscillations is the same as in the absence of SOI. (Beating occurs in the presence of both Rashba and Dresselhaus interactions. 39) This is so because, for near the matrix element entering reduces to
which implies that contains only contributions from interband transitions:
The Kohn anomaly in corresponds to the nesting condition , which is satisfied only for
1. Interacting electrons
The electron-electron interaction is expected to affect the -Kohn anomalies in and in a way similar to that in the absence of SOI. However, a combination of the electron-electron and SO interaction leads to a new effect: a Kohn anomaly at the momentum . Consequently, the RKKY interaction contains a component that oscillates with a long period given by the SO length rather than the half of the Fermi wavelength.
To second order in the electron-electron interaction, the full dependence of the electron spin susceptibility on the momentum is shown in Appendix B to be given by
Equations (3.13a) and (3.13b) are valid for an arbitrary value of the ratio (but for ). For , both and scale as . For , continues to scale as (but with a smaller slope compared to the opposite case), while is independent. The crossover between the two regimes is not continuous, however, because certain derivatives of both and diverge at . Expanding around the singularity at , one finds
where is the step function, and . The dependences of and are shown in Fig. 8.
The singularity is stronger in whose second derivative diverges at , whereas it is only third derivative of that diverges at this point. Both divergences are weaker than the free-electron Kohn anomaly
We now derive the real-space form of the RRKY interaction, starting from . Substituting Eq. (3.14b) into Eq. (3.3) and noting that only the part proportional to contributes, we arrive at the following integral:
where is an arbitrarily chosen cutoff that does not affect the long-range behavior of . Replacing by its large- asymptotic form and by in all nonsingular and nonoscillatory parts of the integrand, we simplify the previous expression to
Integrating by parts twice and dropping the high-energy contribution, we arrive at an integral that converges at the upper limit. The final result reads
Equation (3.17) describes long-wavelength Friedel-like oscillations that fall off with faster than the usual oscillations. Notice that Eq. (3.17), while valid formally only for , reproduces correctly the dipole-dipole term [see Eq. (3.7) with ] in the opposite limit of . Therefore Eq. (3.17) can be used as an extrapolation formula applicable for any value of .
In addition to the Kohn anomaly at , the in-plane component also contains a nonoscillatory but nonanalytic term proportional to . As it was also the case in the absence of SOI, this term translates into a dipole-dipole part of the RKKY interaction. Analysis of Sec. III fully applies here; we just need to replace the prefactor in Eq. (3.7) by , where is defined by Eq. (3.2). The role of the cutoff in Eq. (3.4) is now being played by , therefore, for . For , the prefactor is the same as in the absence of SOI. Summarizing, the dipole-dipole part of the in-plane RKKY interaction is
The oscillatory part of is obtained by the same method as for with the only difference that one needs to integrate by parts three times in order to obtain a convergent integral. Consequently, falls off with as . The dependence of , resulting from the SOI, is given by a sum of the nonoscillatory and oscillatory parts:
Finally, the conventional, Friedel oscillations should be added to Eqs. (3.17) and (3.19) to get a complete dependence. The dipole-dipole part and long-wavelength Friedel oscillations fall off faster then conventional Friedel oscillations. In order to extract the long-wavelength part from the data, one needs to average the measured over many Fermi wavelengths. Recently, oscillations in the RKKY interaction between magnetic adatoms on metallic surfaces have been observed directly via scanning tunneling microscopy. 40 Hopefully, improvements in spatial resolution would allow for an experimental verification of our prediction for the long-wavelength component of the RKKY interaction.
As a final remark, we showed in Sec. II F that the charge susceptibility does not exhibit small- nonanalyticities. This result also implies that the long-wavelength oscillations are absent in the charge susceptibility; therefore Friedel oscillations produced by nonmagnetic impurities contain only a conventional, component.
IV. SUMMARY AND DISCUSSION
We have studied the nonanalytic behavior of the electron spin susceptibility of a two-dimensional electron gas (2DEG) with SOI as a function of momentum in the context of a ferromagnetic nuclear-spin phase transition (FNSPT). Similarly to the dependence on temperature and magnetic field, 4 the combined effect of the electro-electron and spin-orbit interactions affects two distinct components of the spin susceptibility tensor differently. For , where is the effective electron mass and is the spin-orbit coupling, the out-of-plane component of the spin susceptibility does not depend on momentum (in other words, the momentum-dependence is cut off by the SOI), [cf. Eq. (1.7a)], whereas its in-plane counterparts, , scale linearly with even below the energy scale given by the SOI [cf. Eq. (1.7b)].
Beyond second order in electron-electron interaction, renormalization effects in the Cooper channel, being the most relevant channel in the weak-coupling regime, start to play a dominant role. As we have shown in Sec. II E, the leading linear-in- term becomes renormalized by , while the subleading linear-in- term acquires additional dependence. This behavior is a natural consequence of the separation of energy scales in each of the diagrams and suggests that, in general, , where stands for a generic energy scale (in our case but temperature or the magnetic field could be included as well).
Our analysis of the spin susceptibility gives important insights into the nature of a FNSPT. First, the SOI-induced anisotropy of the spin susceptibility implies that the ordered phase is of an Ising type with nuclear spins aligned along the axis since . Second, the ferromagnetic phase cannot be stable as long as the higher-order effects of the electron-electron interaction are not taken into account. In this paper, we focused only on one type of those effects, i.e., renormalization in the Cooper channel. Without Cooper renormalization, the slope of the magnon dispersion is negative, even though the magnon spectrum is gapped at zero momentum, cf. Fig. 1. This implies that spin-wave excitations destroy the ferromagnetic order. Only inclusion of higher-order processes in the Cooper channel, similarly to the mechanism proposed in Ref. 3, leads to the reversal of the slope of the spin susceptibility in the (not necessarily immediate) vicinity of the Kohn-Luttinger instability, and allows for the spin-wave dispersion to become positive at all values of the momentum. This ensures stability of the ordered phase at sufficiently low temperatures.1,2
We have also shown that a combination of the electron-electron and SO interactions leads to a new effect: a Kohn anomaly at the momentum splitting of the two Rashba subbands. Consequently, the real-space RKKY interaction has a long-wavelength component with a period determined by the SO rather than the Fermi wavelength.
Another issue is whether the SOI modifies the behavior of the charge susceptibility which is known to be analytic in the absence of the SOI.20,33,34 As our calculation shows, the answer to this question is negative.
One more comment on the spin and charge susceptibilities is in order: despite the fact that we considered only the Rashba SOI, all our results are applicable to systems where the Dresselhaus SOI with coupling strength takes place of Rashba SOI, i.e., , ; in this case, the Rashba SOI should be simply replaced by the Dresselhaus SOI ( ).
Finally, we analyzed the nonanalytic dependence of the free energy, , in the presence of the SOI and at zero temperature beyond the RPA. This analysis is important in the context of interacting helical Fermi liquids that have recently attracted considerable attention. 41–43 In contrast to the RPA result, 42 which predicts that the free energy scales with as , our result shows that the renormalization is stronger, namely, , where and .
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