Quantum criticality in ferromagnetic single-electron transistors
- *Department of Physics and Astronomy, Rice University, Houston, TX 77251-1892; and ‡Department of Physics, University of California, Riverside, CA 92521
-
Communicated by Elihu Abrahams, Rutgers, The State University of New Jersey, Piscataway, NJ, November 4, 2005 (received for review August 31, 2005)
Abstract
Considerable evidence exists for the failure of the traditional theory of quantum critical points, pointing to the need to incorporate novel excitations. The destruction of Kondo entanglement and the concomitant critical Kondo effect may underlie these emergent excitations in heavy fermion metals (a prototype system for quantum criticality), but the effect remains poorly understood. Here, we show how ferromagnetic single-electron transistors can be used to study this effect. We theoretically demonstrate a gate-voltage-induced quantum phase transition. The critical Kondo effect is manifested in a fractional-power-law dependence of the conductance on temperature (T). The AC conductance and thermal noise spectrum have related power-law dependences on frequency (ω) and, in addition, show an ω/T scaling. Our results imply that the ferromagnetic nanostructure constitutes a realistic model system to elucidate magnetic quantum criticality that is central to the heavy fermions and other bulk materials with non-Fermi liquid behavior.
Aquantum critical point (QCP) occurs at zero temperature as a system changes from one ground state to another, and it controls physical properties over a wide region in the phase diagram at finite temperatures (1–6). Electrons in condensed matter are traditionally described as a Fermi liquid, a collection of essentially independent particles. Near a QCP, however, electrons are coupled to each other in a singular fashion; how such electron correlations lead to non-Fermi liquid states is an open issue that is central to a variety of strongly correlated systems, including high-temperature superconductors and heavy fermion metals (5). Single-electron devices play a unique role in the study of correlated electronic states, in addition to their potential application to quantum electronics and quantum information processing. The Fermi liquid state has been studied systematically in semiconductor quantum dots and single-molecule transistors (7–14). Here, the low-energy excitations of the normal metal leads are electrons near their respective Fermi energies. These itinerant electrons are entangled with the magnetic moment localized on the dot, producing a Kondo resonance. The manifestation of the Kondo resonance in the conductance spectrum was predicted (7, 8) and was subsequently observed in semiconductor quantum dots (9–11) and in single-molecule transistors (12–14). Ingenious proposals and structures (15–17) have been put forth to model the non-Fermi liquid states associated with the multichannel Kondo systems and the two-impurity Kondo effect (5). The experimental observation of the non-Fermi liquid states is, however, still lacking, in part because such states of quantum-impurity models are not robust, requiring special symmetries that are difficult to realize (5).
Recently, it has become possible to fabricate single-electron transistors with leads made of ferromagnetic metals (18). Fig. 1a provides a schematic illustration. The dot, which denotes a quantum-mechanical spin (local moment) of a molecule or that of a semiconductor quantum dot in the Coulomb-blockade regime, is also capacitively coupled to a nonmagnetic gate electrode. Here, we propose this structure as a model system to study how magnetism interplays with the Kondo effect, leading to a QCP with non-Fermi liquid behavior. Our key observation is that the ferromagnetic leads contain not only conduction electrons but also spin waves, collective low-energy excitations arising because of the spontaneously broken spin symmetry of a ferromagnet. We are then led to a description in terms of a Bose–Fermi Kondo model (2, 19–22), which couples the local moment to both a fermionic bath of conduction electrons and a bosonic one of spin waves.
Single-electron transistor with ferromagnetic leads. (a) A schematic set up. The opposing arrows denote AP alignment of the magnetizations of the ferromagnetic source and drain. V g is the voltage of the normal-metallic gate. QD, quantum-dot island of a semiconductor quantum dot or a molecule of a single-molecular transistor. (b) The majority and minority electron bands of the left (L) and right (R) leads. E F, Fermi energy. Also shown is the local moment on the dot (blue arrow). (c) Transverse spin excitations of the ferromagnetic leads. The red line describes spin waves whose energy (ωq) depends on the wave vector (q) quadratically. The shaded area denotes the continuum associated with particle-hole excitations. The energy of the spin wave at the point it merges into the continuum [Λ, not shown] is of the same order of magnitude as Λ0, the energy cost to create a particle-hole excitation at q = 0. (d) The phase diagram of the Bose–Fermi Kondo model (see Eq. 9). The parameters J and g are the couplings of the local moment to the fermions and spin waves, respectively. Lines with arrows denote the RG flow. There are three RG fixed points. Kondo and LM, Kondo screened fixed point, corresponding to a Fermi liquid, and the critical local-moment fixed point, describing a quantum-critical phase. QC, quantum-critical fixed point, characterizing the critical Kondo screening on the entire separatrix (red line, corresponding to the critical coupling g c as a function of J). Varying the gate voltage, V g, tunes both J and g, along the blue-dashed line.
We focus on the antiparallel (AP) configuration of the magnetizations of the two ferromagnetic leads, and consider symmetric
couplings between the dot and the left/right leads. The fermionic bath (c
k
σ) describes a linear combination of the conduction electron bands of the two leads. The bosonic bath contains two components,
one each for the two orthogonal spin polarization directions that are perpendicular to the lead magnetizations; each component
is a linear combination of the corresponding spin waves in the two leads. We note on a number of features of the resulting
Bose–Fermi Kondo model (see Eq. 9) specific to the system. First, while the energy bands of the two spin components of the conduction electrons in each metallic
lead is (Zeeman) split by the magnetization, the linear combination that forms the fermionic bath contains no Zeeman splitting
(ref. 23 and 24 and references therein) as can be readily seen from Fig. 1b. Second, in addition to fringing fields from the ferromagnetic leads, an effective local magnetic field (h
loc) for the local moment is usually generated due to dot-lead exchange coupling. In our AP case, however, the contributions
from the two leads compensate with each other and h
loc vanishes. Third, the two-component nature of the bosonic bath means that our model has an easy-plane spin anisotropy. Finally,
the dispersion of the ferromagnetic spin waves (ωq) is such that the spectral density of the bosonic bath has a square-root dependence on the frequency, 
for 0 < ω <Λ, the cutoff frequency below which spin waves exist (cf. Fig. 1c). The sublinear dependence on frequency is commonly referred to as “subohmic” dissipation. We note that the gap of spin waves
in real ferromagnets is nonzero; its effect will be discussed shortly.
We find that a quantum phase transition can be induced by varying the gate voltage, V
g. This voltage determines Δ, the valence-fluctuation energy cost of the dot: Δ=Δ0 – cV
g, where Δ0 is the corresponding energy cost in the absence of the gate voltage, and the proportionality factor c = eC
gate/C
tot is determined by the gate and total capacitances, C
gate and C
tot, respectively. The parameter Δ in turn determines the strength of the interactions between the local moment and the bosonic
and fermionic baths. The local-moment regime arises when Δ≫Γ≡ π ρ0
t
2, the noninteracting resonance width; here, t is the dot-lead hybridization matrix, and ρ0 is the fermion density of states at the chemical potential. The resulting Bose–Fermi Kondo model is given in Eq. 9. The Kondo exchange coupling with the fermions, and the bare Kondo scale (at g = 0), have the standard forms: J ∼ Γ/ρ0Δ and 
∼ (1/ρ0)exp(–π Δ/Γ), respectively. The coupling with the bosonic bath is found (see Formal Details) to be g ∼Γ/(ρ0Δ)2. Hence, the ratio g/
is exponentially large, and it decreases as Δ is reduced. When Δ becomes comparable with Γ, the system reaches the mixed-valence
regime, and g/
becomes of order unity. Changing V
g then takes the system through second-order quantum phase transition (Fig. 1d).
More specifically, for gate voltages such that the system (dot + leads) is either in the mixed valence regime or in the local-moment
regime but with moderate ratios of Δ/Γ, the usual Kondo effect takes place. At low temperatures, the system is in a Fermi
liquid ground state. As Δ is increased by varying V
g, interactions with the spin waves destroy the Kondo effect. Kondo resonances no longer form, and the system is in a non-Fermi
liquid ground state. In between the system passes through a QCP, which is also a non-Fermi liquid. These phases and the phase
transition are expressed in terms of the renormalization group (RG) flows and RG fixed points in Fig. 1d. At relatively small g/
, the RG flow is toward the Kondo fixed point. For sufficiently large g, the flow is instead toward a fixed point located on the J = 0 axis (the horizontal line of Fig. 1d) where the local moment is completely decoupled from the fermions but is still strongly fluctuating (13) through the coupling to the spin waves. We will call this phase in the entire g > g
c region a “critical local-moment phase.” In between, the system passes through g = g
c where the RG flow is toward a quantum-critical fixed point. Here, the local moment fluctuates in a critical fashion (at time
scales beyond the short-time cutoff, –h/
), and the electronic excitations have a non-Fermi liquid form. The subohmic nature of the dissipative bosonic bath (Eq. 1) is necessary for the existence of quantum criticality.
Results and Discussion
To systematically study the transport properties across the quantum phase transition, we apply a dynamical large-N approach (27–29) to the Bose–Fermi Kondo model (see Formal Details). We first consider the single-electron spectral function on the dot, expressed in terms of the scattering T-matrix. The
results are shown in Fig. 2. Because the universal properties are insensitive to the way the quantum-critical regime is accessed, we have, for convenience,
chosen to work with fixed 
and varying g. The different curves correspond to different ratios of g/
and, correspondingly, different values of the gate-voltage V
g. The spectral weight of the Kondo resonance is seen to decrease as g increases, and vanishes continuously when g reaches g
c. At the QCP, the Kondo screening is critical: a Kondo resonance is no longer fully developed, but vestiges of it remain.
For g > g
c, the Kondo effect is completely destroyed: the spectral density vanishes at ω = 0. At finite energies, however, the spectral
density is still finite reflecting on a finite effective (scale-dependent, or renormalized) Kondo coupling.
Evolution of the Kondo resonance. The scattering T-matrix, 𝒯(ω), is defined as the Fourier transform of 
(where τ is the imaginary time) of the Bose–Fermi Anderson model; 
is the imaginary part of 𝒯, and is equal to π times the single-electron spectral function on the dot. The main plot uses
the fermionic half-bandwidth, D = 1/2ρ0, as the normalization for energies. The peak near ω = 0, arising for g < g
c, corresponds to the Kondo resonance. The Inset, limited to the energy range of about 
, highlights the vicinity of zero energy. Different 
corresponds to different gate voltage, V
g. The parameters adopted are εd = –0.3D, U = ∞, t = 0.1D, corresponding to 
D; the cut-off energy for the bosonic bath (Eq. 1) is Λ= 0.05 D, and the proportionality factor on the right side of Eq. 1 is 1/Γ(3/2). The scaling exponents are universal, but 
is not, depending on ρs, Λ, and other microscopic parameters of the ferromagnets. The same parameters apply to Figs. 3 and 4.
The linear-response DC conductance (G) directly manifests the critical nature of the Kondo effect. Its temperature dependence is given in Fig. 3. On the Kondo side (g < g c), the conductance increases as the temperature is lowered, reflecting the development of a Kondo resonance (7, 8). Inside the critical local-moment phase (g > g c), the conductance vanishes in a power-law form as temperature goes to zero. It vanishes at T = 0 because Kondo screening has been completely destroyed. At finite temperatures, charge transport can still occur because of the finite effective Kondo coupling at nonzero energies. Hence, the way the conductance goes to zero characterizes the critical behavior of the Kondo coupling at the critical local-moment fixed point. At the QCP (g = g c), the conductance behaves in a way intermediate between the two sides. The zero-temperature conductance is finite but is reduced from its counterpart on the Kondo side. The temperature-dependent part has a power-law form that is distinct from both the local-moment and Kondo sides, characterizing the critical exponents unique to the QCP.
DC conductance. At g < g c, the Kondo assisted conductance increases as temperature is lowered. At the QCP (g = g c) and inside the critical local-moment phase (g > g c), the temperature dependence has the forms of Eqs. 2 and 3, respectively. G 0 is e 2/h.
To understand the detailed temperature dependences, we have analytically determined the zero-temperature T-matrix. At the
QCP (g = g
c), its imaginary part is 
. On the local-moment side (g > g
c), the asymptotic low-energy behavior is 
. The exponents here reflect the critical exponents of the QCP and the critical local moment phase with a dissipative spectrum
given in Eq. 1. The results suggest that the linear response DC conductance is, at the QCP, 
and, inside the critical local-moment phase (g > g
c), 
Indeed, these fractional-power-law forms fit the numerical results shown in Fig. 3. The success of this fitting suggests that the T-matrix satisfies an ω/T scaling. This is verified by the scaling plots shown in Fig. 4a for g = g
c and in Fig. 4b for g > g
c.
ω/T scaling. (a) The T-matrix as a function of ω/T, at the QCP. For each temperature, the T-matrix falls on the universal scaling curve until ω reaches the order of 
. Notice that 
, T → 0, ω/T → 0) is different from 
, T → 0, ω/T → ∞). (b) Similar result inside the critical local-moment phase, at a g > g
c; the critical exponent is 0.5.
AC transport and noise properties provide the means to probe the ω/T scaling. At the QCP, the real part of the AC conductance is 
whereas, inside the local-moment phase (g > g
c), 
Here, 
, 
, and 
are scaling functions. The thermal current fluctuations (Johnson noise) spectrum, S(ω), is simply Eqs. 4 and 5 multiplied by 2–hω[1 + n
B(ω)], where the Bose factor n
B(ω) ≡ 1/[e–h
ω/
k
B
T–1]. S(ω)/ω, too, satisfies ω/T scaling at the QCP and inside the critical local-moment phase.
We turn next to the feasibility of the experimental measurements. The non-Fermi liquid behavior we have derived applies to
the scaling regime, the region in which universal properties arise. We have so far assumed a spin-wave spectrum that is gapless.
Real ferromagnets contain interactions (such as spin–orbit coupling) that break spin-rotational symmetry, which will cut off
the scaling regime at low energies. The bulk spin–anisotropic interaction results in a spin wave gap of about 0.25 K for Ni
films (31) and even smaller values in the case of permalloys. The surface spin–anisotropic interactions yield a finite de-pinning energy
that, however, is negligibly small (of the sub-mK range; see Formal Details). The upper cutoff energy of the scaling regime is given by the Kondo scale, 
, which, for single-molecular transistors, can be of the order of 50–100 K (14, 32). Hence, we can expect more than two decades of both temperature and frequency over which the non-Fermi liquid behavior occurs.
To study the AC noise spectrum, the frequencies needed fall in the range of about a few gigahertz to a terahertz. Much of
this frequency range is already feasible experimentally: noise measurements up to 90 gigahertz have been reported (33).
We have also assumed symmetric couplings of the dot to the two leads, which ensure a zero local effective magnetic field for the local moment due to exchange (h loc = 0). [The fringing fields from the ferromagnetic leads are expected to be small (≪0.6 K for Ni) (18).] Significantly different left/right couplings will yield a finite h loc [which, on the Kondo side, in turn results in a splitting of the Kondo resonances even in the AP configuration (18, 23, 24)]. Some of the single-molecule transistors reported in ref. 18 indeed have approximately symmetric couplings, as evidenced by the smallness of the splitting in the Kondo peaks. Even if it is nonzero, h loc can be compensated by an external local magnetic field (23). The latter can be generated by direct currents through nearby conductors, as in magnetic random access memory applications (34).
So far, we have considered ferromagnetic metallic leads. A variation is to use metallic leads that are paramagnetic, but are nearly ferromagnetic. The absence of static ordering implies that the local effective field (from dot-lead exchange) vanishes under all circumstances. Spin waves are no longer sharply defined; they do, nonetheless, appear in the form of paramagnons, which provide a dissipative bosonic bath with a fluctuation spectrum similar to that of Eq. 1, but with an exponent of 1/3, instead of 1/2. This is still subohmic, so with appropriate modifications to the fractional exponent and scaling functions, essentially all of our results will still apply. [The almost magnetic nature of the leads here can generate multichannels of effective conduction electrons (35, 36), which can influence the finite temperature properties on the Kondo side. This effect, however, is not expected to modify the quantum critical behavior.] For instance, the conductance exponent at g = g c is now 1/3 (instead of 1/4), and that for g > g c becomes 2/3 (instead of 1/2). Palladium would be one candidate material in this category.
Our results are insensitive to the presence or absence of particle-hole symmetry. In addition, there is no need to achieve multiple channels of fermionic couplings. These make the non-Fermi liquid behavior of a ferromagnetic single-electron transistor more robust than that of either the multichannel or multiimpurity Kondo systems. There is a simple reason behind this robustness. The phases of the system we have discussed (the Kondo phase on the one hand, and the critical local-moment phase on the other) are genuinely distinct and must be separated by a quantum phase transition. Because the transition is second order, a distinctive QCP must arise, and quantum criticality in turn makes the system a non-Fermi liquid. In the multichannel or multiimpurity Kondo systems (5), however, all of the stable phases are Fermi liquids, making special symmetries (channel degeneracy or particle-hole symmetry) necessary to reach any non-Fermi liquid state.
It is instructive to discuss the broader implications of our results. Orthodox theory of any QCP (4, 6) describes it in terms of order-parameter fluctuations and maps it to a classical counterpart in elevated dimensions. For
the Bose–Fermi Kondo model with a bosonic fluctuation spectrum given by Eq. 1, the corresponding critical point is noninteracting (27, 37, 38). On the other hand, the ω/T scaling implies (4) that the critical point we are dealing with is a fully interacting one. A further support for the interacting nature of
the QCP comes from Fig. 4a. It is seen that the T-matrix at the limit of zero temperature and ω → 0 [corresponding to ω/T ≫ 1 and yielding 
; Eq. 4] is different from that at the limit of ω = 0 and T → 0 [so that ω/T ≪ 1, giving rise to 
]. This is in contrast to noninteracting critical points of quantum impurity models in which the two limits are believed to
be interchangeable (39). An experimental observation of either ω/T scaling or 
would provide a direct signature of the inherent quantum nature of the QCP.
In addition to their intrinsic interest, our results will establish how the Kondo effect influences quantum criticality. The latter not only is directly relevant to the quantum-critical heavy fermions (1–3), whose properties strongly deviate from the expectations of the order-parameter-fluctuation picture, but also illustrates the central role of quantum entanglement effects (beyond the fluctuations of order parameter) in quantum criticality in general.
Conclusions
To summarize, we propose the ferromagnetic single-electron transistor as a model system to study non-Fermi liquid states and quantum criticality. We have shown that the QCP is robust theoretically and feasible in experimental implementation. Finally, our results imply that the system, readily measurable beyond the equilibrium regime, will serve as an ideal setting for the much-needed explorations of the nonequilibrium properties at quantum criticality. All these features make it a tunable spintronic system to study issues that are important to a broad array of strongly correlated electron materials.
Formal Details
Derivation of the Bose–Fermi Kondo Model. Consider the regime where the dot has an odd number of electrons and the valence-fluctuation energy Δ is large compared with
the resonance width Γ. At temperatures well below Δ/k
B, valence fluctuations can occur only virtually, and the dot is effectively a localized moment. Hybridization processes between
the dot and electrodes give rise to an antiferromagnetic coupling between this moment and the spins of the low-energy conduction
electrons in the lead metal. The conduction-electron spin in a direction perpendicular to the ordered moments of the ferromagnets
is a linear combination of the transverse part of the triplet quasiparticle-hole excitations and the spin waves. So we expect
the local moment to be coupled not only to the spins of the quasiparticles but also to the spin waves. To determine the specific
value of the bosonic coupling constant, we go beyond these general arguments and carry out a generalized Schrieffer–Wolff
transformation followed by a projection (𝒫) onto the local-moment subspace: 
where H
0 is the charge-conserving part of the dot Hamiltonian. The generator of the canonical transformation (S) follows from the requirement that [H
0 + H
lead,S] = H
hyb, and can be formally expressed as (40), 
where the Liouville operators are defined as Lx A = [Hx, A]. We model the ferromagnetic metal of a lead in terms of a one-band Hamiltonian, H
lead = ΣiH
kin,
i – 2u/3 Σr
,
i(s
i(r))2, where 
, with i = L, R being the lead index. The interaction term is decoupled in terms of the vector boson field 
as follows, 
where 
. Within the ferromagnetic-metal phase, the z-component φz
,
i acquires a static value, mi, and the transverse components φβ,
i, β = x, y, describe the spin waves. Inserting Eqs. 7 and 8 in Eq. 6 leads to a Bose–Fermi Kondo model, 
with the coupling constant g ∼ ut
2/Δ2, which gives rise to the expression for g stated earlier once we recognize that, in itinerant ferromagnets, ρ0
u ∼ 1. The conduction electron dispersion in each lead is Zeeman split: 
. However, only one linear combination of the conduction electrons from the two leads is coupled to the dot spin (7, 8): 
. The energy dispersion for this linear combination is spin-independent and will be denoted as E
k. A similar linear combination arises for the spin waves from the two leads: 
. The local effective field, h
loc = gΣimi, vanishes in the AP case (mL = –mR). The Hamiltonian (9) has an easy-plane anisotropy. Taking advantage of the observation (26, 27) that the scaling properties of both the QCP and the critical local-moment phase in this case are similar to those of its
SU(2) and SU(N) counterparts, we will generalize the local moment to a form with SU(N) symmetry, and the conduction electrons
to a form with SU(N) × SU(N/2) symmetry (28, 29). After a rescaling of the Kondo coupling to order 1/N, and the coupling to the spin waves to order 
, we arrive at the Hamiltonian: 
Here, S, c
k
ασ, and Φq denote the local moment, fermionic bath, and bosonic bath, respectively, 
and 
, where 
is the vector of Pauli matrices. σ = 1,..., N labels spin indices and α = 1,..., N/2 the channel indices for conduction electrons.
The spin-wave spectrum determines the dissipative bosonic bath spectrum. We end up with Eq. 1 using the spin-wave dispersion ωq = ρs q 2, where ρs is the spin-wave stiffness.
Dynamical Large-N Method. The dynamical saddle-point equations for the large-N limit of the Bose–Fermi Kondo Hamiltonian are described in ref. 27. In addition to allowing a controlled solution in general, this limit allows us to calculate the conduction electron T-matrix.
The saddle-point equations are expressed in terms of Gf(τ) and GB(τ), the propagators for the slave-f-fermion of the dot spin and for the B-field representing the f
†
c combination (27–29), respectively. Once these quantities are determined, the T-matrix is given by the Fourier transform of 
. The large-N limit for the Kondo phase (g < g
c) is described by the fixed point at g = 0, which is a multichannel fixed-point (28, 29). For the critical local moment phase (g > g
c) and the QCP (g = g
c), on the other hand, both the large-N and the N = 2 cases describe non-Fermi liquid states, and the two cases have similar critical behaviors (26, 27).
At T = 0, the large-N limit is solved analytically, by making the scaling ansatz for both the leading and subleading terms, 
and inserting these into the saddle-point equations. Consider first the leading order. For the critical local-moment phase,
α1 = 1/4 and β1 = 5/4. As a result, 
, corresponding to an ω1/2 dependence. (In the more general case with ω1–ε replacing 
in Eq. 1, the result is ωε.) For the QCP (g = g
c), α1 = 1/4 and β1 = 3/4, yielding 
, corresponding to a constant 
[the critical exponent at the leading order (being equal to 1), although neither the critical amplitude (the constant a given below) nor the finite-temperature behavior (Fig. 4a) was anticipated by the general argument of ref. 30.]. The ω-dependence comes from the subleading terms of Gf(τ) and GB(τ). We find α2 = 1/2 and β2 = 1, making the dominant subleading contribution to 
to be 1/τ5/4, which in turn corresponds to an ω1/4 dependence. (For general ε, this becomes ωε/2.) The final results, for g = g
c and g > g
c, are given in Results and Discussion, just before Eq. 2, with the numerical factors a = tan(π /8)/(8Nρ0), b = π [5 tan(π /8)]1/2/[Nρ0
K
0
g
cΓ3(1/4)], and c = (1/8)Γ(5/4)[3 tan(π /8)/2]1/2
J/[NK
0
gΛ1/4]. Here, Γ(x) is the gamma function and K
0 = [Γ(3/2)/4π 2ρ3/2
s]1/2 is related to the proportionality factor on the right side of Eq. 1.
At finite temperatures, the saddle-point equations in the large-N limit are solved numerically in terms of real frequencies (27). To check the robustness of our results with respect to particle-hole symmetry breaking, we have also solved the extension
of the above saddle-point equations to the Bose–Fermi Anderson model in the local-moment regime without particlehole symmetry
(setting N = 2): 
with 
, and εd ≠ –U/2. The scaling behavior is found to be the same as that of either the numerical results in the large-N limit we have considered, which has a particle-hole symmetry, or (where a comparison can be made) the analytical results
in the same limit.
Calculation of the Transport Quantities. The linear-response DC conductance is 
where f(ε) is the Fermi–Dirac distribution function. Likewise, the real part of the linear-response AC conductance (the current response
to an oscillating bias voltage) can also be related to the scattering T-matrix (41, 42) under the condition of vanishing charge accumulation: 
The thermal current-fluctuation noise spectrum is simply related to G′(ω) through the fluctuation-dissipation theorem. The result is given in Eqs. 4 and 5, with the following limiting values: 
, 
, and 
.
De-Pinning Energy of Spin Waves at the Surface. The form of the coupling between the local moment and spin waves, given in the third term on the right side of Eq. 9, has been derived in the absence of any surface spin–anisotropic interaction. The latter introduces a pinning of the spin
waves of sufficiently long wavelengths (43). Below the pinning wave vector, q < q
pin, the wave function at the surface is linearly (43, 44) proportional to q; in turn, the coupling constant between S
β and φβ,
q
,
i is proportional to q, changing the spectrum of the dissipative bath from Eq. 1 to ∼ω3/2. From the known result (43, 44) q
pin
a ≈ E
sa/E
ex, where a is the lattice constant and E
sa and E
ex are, respectively, the surface anisotropic and bulk isotropic exchange energies, we can estimate the corresponding pinning
energy scale 
. (Here, we have used the mean-field expression ρs ≈ E
ex
a
2.) Being smaller than E
sa by a factor E
sa/E
ex, the pinning energy is expected to be rather small. For Fe-Co films, for instance, E
sa ∼ 1 K [corresponding to a typical surface anisotropy (45) of the order 0.1 erg/cm2 (1 erg = 10–7 J)] and E
ex > ≈1,000 K (based on the Curie temperature), giving rise to an E
pin < 1 mK.
Acknowledgments
We thank E. Abrahams, K. Ingersent, C. M. Varma, and G. Zaránd for useful discussions. This work was supported by the Deutsche Forschungsgemeinschaft (S.K.), the Robert A. Welch Foundation, National Science Foundation Grant DMR-0424125 (to S.K., L.Z., and Q.S.), the Alfred P. Sloan Foundation, the David and Lucille Packard Foundation, and National Science Foundation Grant DMR-0347253 (to D.N.).
Footnotes
-
↵ § To whom correspondence should be addressed. E-mail: qmsi{at}rice.edu.
-
Author contributions: S.B.K., L.Z., Q.S., and D.N. designed research; S.B.K., L.Z., and Q.S. performed research; and S.B.K. and Q.S. wrote the paper.
-
Conflict of interest statement: No conflicts declared.
-
Abbreviations: AP, antiparallel; QCP, quantum critical point; RG, renormalization group.
- Copyright © 2005, The National Academy of Sciences
