- Email: [email protected]
Persistent currents in mesoscopic copper rings
Physica B 169 (1991) North-Holland
245-256
Persistent Laurent AT&T
currents in mesoscopic copper rings*
P. LCvy
Bell Laboratories,
The invited
talk was given
Murray Hill, NJ 07974, USA by the author
the total magnetization of an ensemble of 10’ isolated mesoscopic copper At sufficiently low temperature, oscillates as a function of the enclosed magnetic flux with a period of half a flux quantum, h/2e. The amplitude oscillatory moment is ~1.2 x lo- I5 A m2 and decreases exponentially with increasing temperature on the scale correlation energy E, = -rr’hDIL*. This is evidence for a flux periodic persistent current in each ring of average 3 x 10~~’ ev,/L. Single-particle effects only give rise to an h/2e periodicity in a closed system where the variations chemical potential are correlated to the variation of the density of states with magnetic flux. On the other electron-electron interactions always give rise to persistent currents periodic in h/2e whether the system is open or
rings of the of the value of the hand, closed.
1. Introduction The Aharanov-Bohm effect demonstrates that electromagnetic fields (E, B) are not sufficient to describe the quantum behavior of an electron in an external field. While the potentials (6, A) do give a complete description, they are not physically observable and can be arbitrarily changed by a gauge transformation. In fact the electron wavefunction is only determined by the phase factor R = exp[ielfi (J, A dl - $J dt)] and not the phase J, A dl- 4 dt along the accessible paths C for the electron [l]. The knowledge of the phase factors R is equivalent to the knowledge of the fields (E, B) in connected geometries since all paths can be deformed to a point. In a ring geometry, the paths C around the ring are unshrinkable and the phase factor R is observable. The presence of scattering potentials due to impurities in a solid can randomize this phase if inelastic transitions between distinguishable quantum states occur. This condition for phase coherence is the only requirement which must be met to observe the Aharanov-Bohm effect. The conductance of a metal can under conditions appropriate to mesoscopic systems be viewed as a transmission probability for an electron from one lead into another [2]. This quantity is modulated by the enclosed flux 4 in a ring geometry. Quantum oscillations in the conductance of a metal ring are therefore a direct consequence of the Aharanov-Bohm effect. [3]. This paper considers other effects of electomagnetic flux on thermodynamic properties and more specifically the effect on the equilibrium current carried by an isolated metal ring at finite temperature
141, dF 1=-g
where F is the ring free energy. When the electron phase coherence is maintained around the ring, Buttiker, Imry and Landauer pointed out in a seminal paper [5], that the ground state energy of a ring oscillates periodically as a function of the enclosed flux $J with a period & = h/e, the flux quantum [6]. *Work
done
in collaboration
0921-4526/91/$03.50
with G. Dolan,
@?I 1991 - Elsevier
Science
J. Dunsmuir Publishers
and H. Bouchiat. B.V. (North-Holland1
246
L. P. L&y
I Persistent currents in mesoscopic
copper rings
The random potential caused by impurities repeats periodically after each revolution around the ring and the eigenstates can be labeled by a Bloch wavevector k,,. The vector potential can be eliminated from the Schrodinger equation through the gauge transformation A’ = A - 4/L, where L is the perimeter of the ring. This is equivalent to a flux dependent boundary condition
IcI,,(x+ L) = wO~4/4,)4,,(x) As the flux is changed from 0 to $ +,, the boundary condition (2) changes The wavevectors k,, are flux-dependent and quantized according to
(2) from periodic
to anti-periodic.
This condition is similar to the quantization of the fluxoid in superconductors [7] and imposes a flux periodicity of the energy spectrum F, with periodicity 4,). In particular, each state has a velocity v,, = ae,,/R dk,, and carries a current j, = -ev,,lL = -d&,,/d4. Since the total current and all thermodynamic quantities are flux periodic, they can be expanded in Fourier series of the flux. For example, the free energy
F=
c f,,co@ +/4(r)
n =o
.
(4)
The order of magnitude for the first harmonic of the current j, = -27rf,/4,, at low temperature may be obtained as follows. Let E, = 2f, be the change in total energy between periodic and anti-periodic boundary conditions. Following Thouless’s uncertainty principle argument [S], this energy sensitivity to a change in boundary conditions may be estimated as E, i= h lr,, , where rn == L’lh, v, is the time it takes an electron to diffuse around the ring and A, is the electron mean free path. While there are difficulties associated with this argument [9], it gives the correct [lo, 111 order of magnitude for the current amplitude j, = e/7, carried by one ring. The statistical properties of persistent currents are determined by the correlations between the velocities v,, of each level. They can be inferred from the energy level statistics, and more specifically, from the correlations in the density of states. In the diffusive regime, the level structure and the density of states depend on the details of the disorder. When averaging over disorder, it is expected that the average density of states is constant and flux independent. At fixed chemical potential, this leads to an average persistent current which is exponentially small in A,/ L [12]. On the other hand, other effects alter this conclusion. Electron-electron interactions introduce a correction to the density of states which is pinned at the Fermi level since er is unchanged to first order. Its sign is determined by the repulsive (the screened Coulomb potential) or attractive (phonon mediated) nature of the electron-electron interaction [13]. Since the level structure is no longer invariant with respect to a shift of energy close to the Fermi level, the density of states becomes, on average, flux dependent. The associated persistent currents are then determined by the nature of the interaction through its sign and magnitude. The flux periodicity is hi2e (and not h/e) since they are induced by interacting pairs [14, 151. This behavior, reminiscent of superconductivity, is not fortuitous: superconducting fluctuations are also known to induce persistent currents well above the transition temperature of a superconductor [16]. Since this effect arises from fluctuations it is present independent of the sign of the interaction, but real physical difference exists between attractive and repulsive forces: persistent currents diverge at the transition temperature of a superconductor and have an anomalous flux dependence [ 171. Bouchiat and Montambaux [18] first pointed out that an averaged persistent current exists in systems with fixed number of particles even in the absence of interactions. The canonical ensemble describing
L. P. L&y
I Persistent
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in mesoscopic
copper
241
rings
such a system gives a different average than the grand canonical ensemble describing such a system gives a different average than the grand canonical ensemble which connects the system to a particle in the averaged energy is only of order of the energy of reservoir. This difference 6E = ( E)N - (E), one particle and is negligible if the system volume is large. However, a thermodynamic limit is reached in mesoscopic systems only after averaging over many systems: the small difference 6E then becomes extensive and in principle observable. Numerical studies [18] of non-interacting electrons showed this averaged persistent current had half-flux quantum periodicity. To explain the physics, lmry [19] noted that the constraint on the number of particles induces a flux dependence of the Fermi energy (or chemical potential) which is correlated with the variation of the density of states. This correlation induces an average current which is proportional to the square of the density of state fluctuations [20] in agreement with numerical studies [lS]. This quadratic dependence halves the flux periodicity from & to &,,. We now present the available experimental observations [21] of ensemble currents prior to giving a more quantitative discussion of these effects.
2. Ensemble
averaged
persistent
currents
averaged
persistent
in copper
To get an idea of the magnitude of the persistent currents expected in copper, we use the estimate = e/TD = ev,h,lL’. We obtain A, from the measured residual resistivity p = 2.1 p,fl cm of mesoscopic i tYP copper wires [22] at 4.2 K. Using
for copper, we get A, = 300 A indicating that the mean and the value k, = 1.36 X 10’ cm-’ appropriate free path is limited by boundary scattering between the Cu and its oxide layer. The actual copper ‘rings’ studied were squares 0.55 km on a side with a cross section of 350 x 450 A’. With these parameters the a magnetic moment p = 5 X 1O-22 A m* = 54 pg. The phase typical current i,,, = 1.9 nA induces from the conductance oscillation of a wire network at coherence length I, = 2 km was obtained T = 1.7 K made with the same copper material and similar aspect ratio as the ring samples. Below 3 K, 1, is limited by low-energy electron-electron scattering and scales with temperature as T-l” [23]. Below 0.8 K, the phase coherence length 1, measured in transport experiments saturates perhaps because of magnetic impurities. We have studied 10’ isolated copper rings fabricated by electron-beam lithography and liftoff techniques on a sapphire substrate [24]. An electron micrograph of this sample is shown in fig. 1. Such a sample realizes an ensemble average where both the disorder and the number of electrons change from ring to ring, but are given and independent of flux in each ring (canonical ensemble). We probe persistent currents through the average oscillatory magnetic moment [25] with period 4p = +,, or $&,. Its size is so small that magnetic impurities present in the measuring system are expected to dominate. However, the expected signal is a periodic and therefore nonlinear function of the flux, while the background susceptibility is highly linear. An effective way to reject this background is to measure the curvature of the induced moment with flux by modulating 4 = 4bc + @ACsin(nt). The induced response is p.(t) = c ,,=I,
/I+~(?.) L
cos(2nfir)
+ P~,~+,(T)
sin(2n
+ l)nt
1
A
L. P. Ltvy
248
Fig. 1. Scanning electron linewidth is under SO0 A.
micrograph
I Persistent currenfs in mesoscopic copper rings
of the copper
rings
used
in the experiment.
The squares
are 0.55 km on a side.
The
The Fourier coefficients pH can be computed only if the dependence of ( j($))(or( F(4))) with flux is known. We assume that at the temperatures of interest (0.01 K < T < 0.3 K)( j($)) - j’ sin 21~d,/@, and that the higher harmonics j,, = -2nnf,/+,, [eqs. (1) and (4)] with flux (which are not to be confused with the harmonics of the time dependent response tc sin(nL?nt) to the driving flux) decrease rapidly with n(section 3). If we only retain the lowest non-zero harmonic (j, or j2) the Fourier amplitudes CL, are
I+~(T) = %(T)JZn(~)
sin0 ,
IL~~+~(~)=~~CL,(T)J~~+,(~)COS~
(7)
where 6 = 21~4,,./4, and 0 = 2rr&,,/$,. When the AC amplitude is sufficiently large (-$4,), higher harmonics appear as modulation sidebands of the measured signal as illustrated in fig. 2. On the other hand, paramagnetic impurities contribute predominantly to the fundamental (0) response provided that the Zeeman splitting pBB,, remains small compared to the temperature k,T. The harmonic amplitudes p,, are measured by synchronous digitization of the signal during an integral number of periods (~20) followed by Fourier analysis. The odd harmonics are in phase with the applied flux while even harmonics are 90” out of phase since the persistent currents are odd functions of the magnetic flux. In absence of a DC Aux, only the third harmonic p3 is present and grows as the temperature is lowered as shown in fig. 3(b) (for comparison, fig. 3(a) shows no third harmonic response without sample). Before discussing this temperature dependence, we qualitatively describe the oscillatory behavior of the second and third harmonics at T = 0 with static magnetic field. Traces (c), (d) and (e) of
L.P.
L&y
I Persistenf currents in mesoscopic
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249
t-El E
z
3R
-@p/2
flux
@p/2
Fig. 2. Fourier transform of the in-phase magnetization signal modulated at 0.3 Hz by a sine wave of 15 G amplitude. The sample temperature is T = 25 mK, while the magnetometer is held at T = 450 mK. Inset: schematic dependence of the average current with flux. A flux modulation of _’ a$~, (arrows and highlighted region) produces the desired nonlinear response.
~a@)
temperature
t+(T)
(mK)
(plotted in arbitrary Fig. 3. Temperature dependence of the second (k, right column) and third ( ps, left column) harmonics units) of the magnetization response using a 15 G AC drive. (a) Empty magnetometer, (b) sample with no DC field, (c) 15 G DC field, (d) 30 G DC field, (e) 60 G DC field. The solid lines are one parameter fits to /+(O) exp(-kTIE,) adjusting only ~~,~(0).
250
L. P. L&y
i Persistent
currents
in mesoscopic
copper
rings
Fig. 3 show the temperature dependence of the second and third harmonics when a DC field of 15 G ($M> 30G (a+,,)> and 60 G (i&,) applied. At i &,, the T = 0 third harmonic has changed sign (0 = n) while the zero-field signal repeats at j&,, (0 = 2rr). This oscillatory behavior on the scale of half a flux quantum is consistent with the ensemble averaged properties of persistent currents. We now turn to the observed temperature dependence. In a disordered metal, two states separated by an energy E are spatially correlated over a distance L, = -rr(hD/s)“’ [26]. It follows that energy averaging (temperature) smears out wavefunctions only at sufficiently large distance. We therefore expect to lose sensitivity to a change in boundary conditions when this length L,. is less than L. For i+,, periodicity, one expects the condition to become L, s 2L since quasi-classical paths must enclose the flux twice to contribute to this periodicity. Persistent currents are therefore expected to decrease on a temperature scale kT = E, = ~‘fiLl/L”. The actual temperature dependence is determined by the detailed energy correlations in the density of states. Calculations [14] show that for T of order of E,, ( j) decreases as exp(-n’T/3E,) while at higher temperatures ( ;) decays more slowly as exp(-2v L/L,). For simplicity we fit our data to an exponential exp(- TIT”), although other temperature dependences are consistent with the data. From the mean free path, we estimate D = 4 A, V, = 150 cm2/s and T* = 80 mK. Once T* is known, the only parameters to be fit become the zero-temperature amplitudes of the second and third harmonics, p2,3( T) = P~,~(O) exp(- T/T*). For each value of the DC magnetic flux, this one-parameter fit is plotted in fig. 3 as a solid line. Within the experimental noise, this exponential behavior is nicely verified. Since persistent currents probe the sensitivity to a change in boundary conditions, this is the most direct determination of a correlation energy. With the procedure just described, we can quantitatively study the dependence of the fitted zero-temperature signal as a function of the applied DC field. The measured dependences of the second and third harmonics are plotted in fig. 4 over a full flux quantum. Both harmonics oscillate with magnetic field on a half flux-quantum scale. Furthermore, as we hinted earlier, their field dependences
0
20
40
60
60
100
120
B (Gauss) Fig. 4. Dependence of the fits to the second and third harmonics amplitudes amplitude. 130 G correspond to one &,,. The horizontal error bars arise from bars are statistical.
at T = 0 with DC field as detected with a 15 G uncertainties in the magnetic field. Vertical error
L. P. LCvy I Persistent currents in mesoscopic
251
copper rings
are 90” out of phase; i.e., the second harmonic is maximum when the third harmonic goes through zero [eq. (7)]. Based on a calibration against a CuMn spin-glass film, the magnitude of the oscillatory moment is estimated to be 1.2 x lo-l5 A m2 within factors of two. Dividing by the number of rings, the average moment corresponds to a current amplitude of 0.24 ev,A,l L 2 in each ring. The observed signal appear to be too large to be attributed to random fluctuations of j which would be of order j,,,fl. With one and a half oscillations, we can only state that the h/2e harmonic is sizable, but other harmonics may also be present. In particular, the lowest approximation ( j) CCsin(4n+/+,) may be questioned because of the low ratio in eq. (2) of the second to third harmonic compared to the ratio of Bessel functions J2(1.7)lJ,(1.7) = 3.3 expected from eq. (7). The modulation technique used, makes it inherently difficult to determine the absolute sign of the averaged persistent current which at this stage remains undetermined. 3. From density of states to persistent
currents
In the diffusive regime, a comprehensive description of the statistics of persistent currents can be given in terms of the properties of the one-particle density of states. In the absence of interactions, the existence of a strong spectral rigidity due to level repulsion dominates the properties of the density of states [27],
where s is the spin degeneracy, with energy E,($) and lifetime on magnetic flux is given by
A(k,, E) = y,(r[(.s - E,,)~ + -Y:])~ ’ is the spectral function of the state 12 h/2-y, (due to inelastic processes). The dependence of the total energy
E(4) = j- cf(& - P.(~))~(~,~1de where f is the Fermi
N=
function
(9)
3
and /_Lthe chemical
potential
Ixf(& - /44)b(d+ &)de
determined
as usual
by
(10)
In particular, the disordered average energy (E(4)) I* at fixed chemical potential only depends on the average density of states (n(4, E)). Th us in absence of interactions, it suffices to evaluate the flux dependence of the average density of states to determine the average current ( j) = -a( E(4)),la4 at zero temperature. Since the disorder mixes randomly states in an energy band 2y, = h/r,, the average spectral weight (A) is obtained by setting y, = ye = hh,/2v, and ck = h2(k: + ki) /2m where k, is given by eq. (3) and k, is the transverse channel wavevector [28]. The harmonics of the averaged density of states (n) = C IZ! cos(27~l+/&,) are then
n,(z) = 5
F
1 dx cos(2nxl)((e t --oc
- E, - c0~2)2 + yf)-’
(11)
252
L. P. L&y
I Persistent
currents
in mesoscopic
copper
rings
where E(, = A2/2mL'and E, = (fik,)'/2m. In the diffusive regime, the Fourier amplitudes ~1, become exponentially small when I> O[n, 0~ exp(-ly,L/2hv,) = exp(-lL/A,)] and the average density of states is constant. It follows [12,29] that the harmonics of the average current (i,) I” for fixed chemical potential are proportional to exp(-/L/2&). (This is consistent with the intuitive notion that a path must enclose the flux I times to contribute to the Ith harmonic of the flux.) In this case, the averaged persistent current is unobservably small. In spite of these arguments, an average persistent current exists in closed systems even in the absence of interactions, as first demonstrated by Bouchiat and Montanbaux [IS]. To explain the physics, we follow here the discussion of Imry [19]. Defining the flux dependence of the chemical potential as ~(4) = cF + A(4), the density of states is expanded close to .sF as n(4, a) = n(+, er) + (e - F~) an(4, &,)/de [30] and eq. (10) may be rewritten at T = 0 as
(12) Upon ensemble average, the left hand side of eq. (12) is a flux independent constant which we set to zero by an appropriate choice of or. Then, the chemical potential and the density of states are on average correlated by the constraint on the number of particles. Similarly, the total energy may be written as
E(4) - E, = A(4)&, + 4
A’(&(+,eF)+ an(;;e,)) ,
where E,, = ];’ n($, c)e de becomes flux independent and (13), the average energy at fixed N reduces to (E(4))
- Et, = 4 (A’(4)n(+,
(13)
eF
+))
^I + (n(eF))(A2(@))
after ensemble
>
averaging.
Combining
eqs. (12)
(14)
since the average density of states is much greater than its fluctuation. The quadratic dependence on A halves the flux periodicity from do to $4”. Because of level repulsion, h’ is expected to fluctuate rapidly with flux. Large fluctuations imply aA2/8+ + a2/$,, where 6 is the level spacing. Similarly, this suggests that A’(+) contains many harmonics of the flux. Very recently, Altshuler, Gefen, Imry [31] and A. Schmid have been able to relate (A’(4)) to the fluctuations in the density of states studied by Altshuler and Schlovskii [27]. At T = 0,they find that the average current
is independent of disorder and inversely proportional to the number of channels M = 4rrkiA. Montambaux et al. have carried large scale simulations and conclude that the 14, harmonic of the average current depends weakly on the disorder and the number of channels approximately as =ev,lL(h,IML)"2.A similar result has been obtained by Von Oppen and Riedel [31] in the limit where the mean free path A, is comparable to the transverse dimensions. How the average current depends on disorder as the ballistic regime is approached remains an open question. Ambegaokar and Eckern have also shown that interaction effects give rise to a flux periodic persistent current whether the system is open or closed [14]. This conclusion was also reached by the presence of electron-electron interactions Altshuler et al. [15] in the context of SNS junctions: shifts the energy of the system. This energy shift depends on the flux dependent quasi-momentum
L. P. Ltvy
I Persistent currents in mesoscopic
253
copper rings
carried by the electrons along the ring, hence leads to a persistent current in the ground state. At the simplest level, this can be analyzed as a self-energy shift of the quasi-particles (a mean field description) which is largest close to the Fermi level: then this interacting electron problem reduces to a change &Z(E) in the one-electron density of states at the Fermi level. In this sense, the associated persistent currents can still be described at the one-electron level. To wit, this change in the density of states averaged over disorder has been calculated by Altshuler and Aronov [13] in terms of the diffusion coefficient D and the Fourier transform of the interaction potential v averaged over the Fermi surface,
&z(e)= -7
sn( eF)V
hDk2 C k e* + [hDk2 + y]’ ’
where h/2y is the inelastic Cooperon equation, (-is
- D(V-
2ieAlfi)2
(15)
In eq. (15), the sum over k is extended
lifetime.
- y)P(r)
over the eigenvalues
= 6(r),
(16)
with periodic boundary conditions around the ring. The quantization of the corresponding wavevector is k, = 2TlL(n + 24/&,), and not eq. (3) which is appropriate to the eigenstates Schrodinger equation. The harmonics 6n, are calculated after Fourier transform as
sn(e,)V 6n, = -___4aE
c
where
#&EC,
E, is the correlation
iYE,) energy
of the
Bloch of the
(17)
>
and
-r
Z(u, v) =
I -cx
dx
x2 cos(2nlx) U2 + (X2 + r.J)’
= 2,:~~~2;~,~
exp(-T[[2[
y + (v’ + u*)“~]]“*)
,
(18)
of the where R = cos(lr1[2U]“2 + arr) for small y. From the density of states, the flux dependence current is obtained as the flux derivative of the energy using eqs. (1) and (9). One thus easily recovers the results of Ambegaokar and Eckern [14]: the average current is periodic in h/2e and of order 0.42nVevFA,lL2. They also find that ( j) decreases approximately exponentially on a scale T* = 3hDl L*. Higher harmonics of the current scale as I-* with a characteristic temperature T* /12 [32]. In other words, the Ith harmonic of the average current carried by a ring of length L is equal to the first harmonic of the current carried by a ring of length L’ = lL, in agreement with the quasi-classical picture of a diffusing particle. Only eq. (1.5) is consistent with this scaling. Measurements of the harmonic content of the average persistent current would therefore determine the corrections to the density of states. Finally, Eckern [33] and Altshuler et al. [15] analyzed the reduction of the electron-electron coupling constant due to higher order contributions. They conclude that the first order contribution v is reduced by approximatively a factor between 4 and 10. On the other hand, spin-orbit scattering and Zeeman splitting have negligible effects [33] on the averaged current. To conclude this section, it is instructive to relate the fluctuations of the persistent current characterized by the typical current j,,, = ( j2)l’* [lo] to th e fl UCt ua t’ions of the density of states [27]. Since j:,, = @W)E(~‘)W W’),=,, 9 we first relate the energy fluctuations
254
L.P. L&y
I Persistrnt currents in mesoscopic
copper rings
(E(~)E(m’))-(E(~))(E(S’))=JI&E’f(&-I*)f7E’-P) 0
0
x ((n(e)n(d)) to the level density fluctuations. Altshuler fluctuations in the level density as
K,, = (4GW)
= K(e,
-
E2,
and
~ (P(E)) (n(d)))
Shlovskii
have
dE ds’
calculated
the
(19) ensemble
averaged
- (+,>)bW) 4 - $‘I + KC&,-
~2,
4 + 4’) ,
(20a)
where K(E, qf~-+ 4’) =
-$
Re 7
(F + i(y + hDk’(b,
i 4’))))?
.
WJb)
Here the Bloch momenta k(+ * 4’) are quantized by k,, = 2rr/L.(rz + (4 L +‘)/$I,,). the level density fluctuation in harmonics of the flux follows as K12($, 4’) = 2 K/ cos(2n@i&,) i-0
I t dx
_Z I-
u2 - (x3 + v)’
(U’ + (xZ + V)‘]’ J
~Q(u, ~1 2(u’ + v?)3s4exp(Ir1[2(v”
SEE,,
of
(21)
cos(2nl+‘/&,)
The Fourier amplitudes K, = -(s/(h~)E,)~~,( energy E, and the dimensionless function
J,(u, v) =
The expansion
i-y/Z,)
are expressed
in terms
of the
correlation
cos(2?Tlx)
+ u’)“‘]‘~‘),
(22)
where Q(u, V) is an oscillatory function. The fluctuations in the energy [obtained from eq. (19)] have the same flux dependence as the density of states. Their harmonic amplitudes (ey) are proportional to Efil”. This leads to a typical current jtvp = ( j:)“2 = EC/&, in agreement with the work of Cheung et al. [lo]. We also recover the peculiar scaling (l-“’ ) of the harmonics of the typical current derived by Altshuler and Spivak [27]. This scaling law is not consistent with semi-classical path arguments. This presumably reflects the non-trivial fluctuations in the density of states (eq. 20) associated with the spectral rigidity. Persistent currents are equilibrium thermodynamic properties of mesoscopic rings. We have emphasized their relation to the flux dependence of the density of states to place such effects in the usual context of orbital magnetism.
4. Discussion
and conclusion
Two possible interpretations for the observed quantum oscillations of the magnetization of copper rings have so far been proposed. Both are based on the existence of persistent currents in mesoscopic rings [4]. However, they rely on different physical mechanisms to explain the ensemble averaged properties.
L. P. Ltvy
I Persistent
currents
in mesoscopic
copper
rings
255
Bouchiat and Montambaux [18] have emphasized that canonical and grand-canonical ensemble give an extensive difference in the thermodynamic averages of mesoscopic systems. They find an averaged persistent current with half-flux quantum periodicity in the canonical ensemble. The magnitude of the current observed in their numerical simulations can only be compared to experiment through the mean-free path. This quantity is poorly determined by simulations making quantitative comparison difficult. The current observed in simulations is paramagnetic. In the diffusive regime, the sign of the current is not expected to depend on spin-orbit interactions, as the magnetic field always increases the rigidity of the spectrum. On the other hand Ambegaokar and Eckern [14], and Altshuler et al. [15] emphasize the thermodynamic corrections associated with interactions. They have demonstrated the relevance of fluctuation effects more familiar in the context of superconductivity to mesoscopic structures. While the observed behavior can also be qualitatively explained with this theory, a quantitative comparison is also difficult because the electron-electron coupling constant is not known in copper. Based on simple estimates, one would expect the screened Coulomb repulsion to dominate the phonon mediated electron-electron interaction in copper. However, from studies of SNS junctions [35], there is indirect evidence that the electron-electron interaction in bulk copper may be attractive with a coupling constant n(c,)V= 0.11, On the other hand, the corresponding T, = 60 mK obtained from BCS theory is considerably higher than the current experimental limit of T, < 5 x lo-’ K (361. How can one differentiate between these mechanisms? Aside from the experimental difficulties associated with the determination of the sign and exact magnitude of the effect, there are other problems. First, interaction effects depend on the sign and the magnitude of the electron-electron in bulk copper, and neither of them are unambiguously known. Secondly, the sign of single particle effects also depends on the strength of spin-orbit, interactions. The only clear-cut test is to measure the magnetization of a network held at a fixed chemical potential. In this case only interaction effects should survive. The comparison with theory would be unambiguous if this experiment is carried out on a material with a well-known electron-electron interaction, i.e., a superconductor above its critical temperature.
Acknowledgements G. Dolan and of the original V. Ambegaokar, Finally, I have Montambaux and
J. Dunsmuir have made the splendid samples shown in fig. 1. H. Bouchiat gave much motivation and made many suggestions throughout this work. B. Altshuler, Y. Imry and B. Spivak have clarified many important issues related to this work. benefited from many discussions with A. Benoit, Y. Gefen, R. Landauer, G. E.K. Riedel.
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[ll
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L. P. LPvy I Persistent
currents
in mesoscopic
copper
rings
[9] AS stated, the argument should apply to the current carried by one level. However, this current is in fact mr times smaller than the total current, and the correlations between the slopes of the levels within an energy band E, add up to give the Thouless result. M,,, z MAJL is the effective number of channels. [lOI H.F. Cheung, E.K. Riedel and Y. Gefen, Phys. Rev. Lett. 62 (1989) 582. [ll] It is also possible to formally map a ring threaded by a flux 4 to an SNS junction with a phase difference (Y = 2n4/&, between superconductors; B.L. Altshuler and B.Z. Spivak, Zh. Eksp. Teor. Fiz. 92 (1987) 609 [JETP 65 (1987) 3431. [12] 0. Entin-Wohlman and Y. Gefen, Europhys. Lett. 8 (1989) 477. [13] B.L. Altshuler and A.G. Aronov, in: Electron-Electron Interactions in Disordered Systems A.L. Efros and M. Pollack eds. (North-Holland, Amsterdam, 1985). [14] V. Ambegaokar and U. Eckern, Phys. Rev. Lett. 65 (1990) 381. [15] B. Altshuler, D.E. Kmelnitskii and B.Z. Spivak, Solid State Commun. 48 (1983) 841. [16] L. Gunter and Y. Imry, Solid State Commun. B 7 (1969) 1391; J. Kurjijarvi, V. Ambegaokar and G. Eilenberger, Phys. Rev. B 5 (1972) 868; Y. Fu and C. Park, preprint. [ 171 V. Ambegaokar, private communication. [IS] H. Bouchiat and G. Montambaux, J. Phys. (Paris) 50 (1989) 2695; G. Montambaux, H. Bouchiat, D. Siegeti and R. Firesner, Phys. Rev. B (1990). (191 Y. Imry, preprint NATO AS1 les Arcs France (1990). [21] L.P. Levy, G. Dolan, J. Dunsmuir and H. Bouchiat, Phys. Rev. Lett. 64 (1990) 2074. [22] The wires used were 400 x 600 A’ in cross section and 2 pm in length. [23] S. Wind, M.J. Rooks, V. Chandrasekar and D.E. Prober, Phys. Rev. Lett. 57 (1986) 633. [24] G.J. Dolan and J. Dunsmuir, Physica B 152 (1988) 7. [25] A more complete description of the techniques used can be found in L.P. Levy et al., Localization 1990 IOP publishing ltd. P.A. Lee and T.V. Ramakrishnan Phys. Rev. B 24 (1981) 6783; the disorder averaged [26] E. Abrahams, P.W. Anderson, density-density correlation function A is explicitly calculated: A(k, E) = n(e,)ehDk’i(e2 + (CD/?)*). [27] B.L. Altshuler and B.I. Shlovskii, Zh. Eksp. Teor. Fiz. 91 (1986) 220 [JETP 64 (1986) 1271. [28] G. Rickayzen, Green’s functions and condensed matter, Ch. 4 (Academic, New York, 1980). [29] H.F. Cheung, Y. Gefen and E.K. Riedel, IBM Res. & Dev. 32 (1988) 359. [30] Higher-order corrections are of order 6/y. where 6 is the level spacing and y the level width. In semiconductors where 6 > y this expansion is meaningless. [31] B. Altshuler, Y. Gefen and Y. Imry. (1990) preprint; A. Schmid, (lYYO), preprint; F. Von Oppen and E. Riedel. (1990) preprint. [32] The effective damping y, also scales as yl’ where y is the level width. [33] U. Eckern, preprint. [34] Y. Meir, Y. Gefen and 0. Entin-Wohlman, Phys. Rev. Lett. 63 (1989) 798. [35] J. Clarke, Proc. R. Sot. London A 308 (1969) 447. [36] Copper is the coolant used in adiabatic demagnetization stages.
245-256
Persistent Laurent AT&T
currents in mesoscopic copper rings*
P. LCvy
Bell Laboratories,
The invited
talk was given
Murray Hill, NJ 07974, USA by the author
the total magnetization of an ensemble of 10’ isolated mesoscopic copper At sufficiently low temperature, oscillates as a function of the enclosed magnetic flux with a period of half a flux quantum, h/2e. The amplitude oscillatory moment is ~1.2 x lo- I5 A m2 and decreases exponentially with increasing temperature on the scale correlation energy E, = -rr’hDIL*. This is evidence for a flux periodic persistent current in each ring of average 3 x 10~~’ ev,/L. Single-particle effects only give rise to an h/2e periodicity in a closed system where the variations chemical potential are correlated to the variation of the density of states with magnetic flux. On the other electron-electron interactions always give rise to persistent currents periodic in h/2e whether the system is open or
rings of the of the value of the hand, closed.
1. Introduction The Aharanov-Bohm effect demonstrates that electromagnetic fields (E, B) are not sufficient to describe the quantum behavior of an electron in an external field. While the potentials (6, A) do give a complete description, they are not physically observable and can be arbitrarily changed by a gauge transformation. In fact the electron wavefunction is only determined by the phase factor R = exp[ielfi (J, A dl - $J dt)] and not the phase J, A dl- 4 dt along the accessible paths C for the electron [l]. The knowledge of the phase factors R is equivalent to the knowledge of the fields (E, B) in connected geometries since all paths can be deformed to a point. In a ring geometry, the paths C around the ring are unshrinkable and the phase factor R is observable. The presence of scattering potentials due to impurities in a solid can randomize this phase if inelastic transitions between distinguishable quantum states occur. This condition for phase coherence is the only requirement which must be met to observe the Aharanov-Bohm effect. The conductance of a metal can under conditions appropriate to mesoscopic systems be viewed as a transmission probability for an electron from one lead into another [2]. This quantity is modulated by the enclosed flux 4 in a ring geometry. Quantum oscillations in the conductance of a metal ring are therefore a direct consequence of the Aharanov-Bohm effect. [3]. This paper considers other effects of electomagnetic flux on thermodynamic properties and more specifically the effect on the equilibrium current carried by an isolated metal ring at finite temperature
141, dF 1=-g
where F is the ring free energy. When the electron phase coherence is maintained around the ring, Buttiker, Imry and Landauer pointed out in a seminal paper [5], that the ground state energy of a ring oscillates periodically as a function of the enclosed flux $J with a period & = h/e, the flux quantum [6]. *Work
done
in collaboration
0921-4526/91/$03.50
with G. Dolan,
@?I 1991 - Elsevier
Science
J. Dunsmuir Publishers
and H. Bouchiat. B.V. (North-Holland1
246
L. P. L&y
I Persistent currents in mesoscopic
copper rings
The random potential caused by impurities repeats periodically after each revolution around the ring and the eigenstates can be labeled by a Bloch wavevector k,,. The vector potential can be eliminated from the Schrodinger equation through the gauge transformation A’ = A - 4/L, where L is the perimeter of the ring. This is equivalent to a flux dependent boundary condition
IcI,,(x+ L) = wO~4/4,)4,,(x) As the flux is changed from 0 to $ +,, the boundary condition (2) changes The wavevectors k,, are flux-dependent and quantized according to
(2) from periodic
to anti-periodic.
This condition is similar to the quantization of the fluxoid in superconductors [7] and imposes a flux periodicity of the energy spectrum F, with periodicity 4,). In particular, each state has a velocity v,, = ae,,/R dk,, and carries a current j, = -ev,,lL = -d&,,/d4. Since the total current and all thermodynamic quantities are flux periodic, they can be expanded in Fourier series of the flux. For example, the free energy
F=
c f,,co@ +/4(r)
n =o
.
(4)
The order of magnitude for the first harmonic of the current j, = -27rf,/4,, at low temperature may be obtained as follows. Let E, = 2f, be the change in total energy between periodic and anti-periodic boundary conditions. Following Thouless’s uncertainty principle argument [S], this energy sensitivity to a change in boundary conditions may be estimated as E, i= h lr,, , where rn == L’lh, v, is the time it takes an electron to diffuse around the ring and A, is the electron mean free path. While there are difficulties associated with this argument [9], it gives the correct [lo, 111 order of magnitude for the current amplitude j, = e/7, carried by one ring. The statistical properties of persistent currents are determined by the correlations between the velocities v,, of each level. They can be inferred from the energy level statistics, and more specifically, from the correlations in the density of states. In the diffusive regime, the level structure and the density of states depend on the details of the disorder. When averaging over disorder, it is expected that the average density of states is constant and flux independent. At fixed chemical potential, this leads to an average persistent current which is exponentially small in A,/ L [12]. On the other hand, other effects alter this conclusion. Electron-electron interactions introduce a correction to the density of states which is pinned at the Fermi level since er is unchanged to first order. Its sign is determined by the repulsive (the screened Coulomb potential) or attractive (phonon mediated) nature of the electron-electron interaction [13]. Since the level structure is no longer invariant with respect to a shift of energy close to the Fermi level, the density of states becomes, on average, flux dependent. The associated persistent currents are then determined by the nature of the interaction through its sign and magnitude. The flux periodicity is hi2e (and not h/e) since they are induced by interacting pairs [14, 151. This behavior, reminiscent of superconductivity, is not fortuitous: superconducting fluctuations are also known to induce persistent currents well above the transition temperature of a superconductor [16]. Since this effect arises from fluctuations it is present independent of the sign of the interaction, but real physical difference exists between attractive and repulsive forces: persistent currents diverge at the transition temperature of a superconductor and have an anomalous flux dependence [ 171. Bouchiat and Montambaux [18] first pointed out that an averaged persistent current exists in systems with fixed number of particles even in the absence of interactions. The canonical ensemble describing
L. P. L&y
I Persistent
currents
in mesoscopic
copper
241
rings
such a system gives a different average than the grand canonical ensemble describing such a system gives a different average than the grand canonical ensemble which connects the system to a particle in the averaged energy is only of order of the energy of reservoir. This difference 6E = ( E)N - (E), one particle and is negligible if the system volume is large. However, a thermodynamic limit is reached in mesoscopic systems only after averaging over many systems: the small difference 6E then becomes extensive and in principle observable. Numerical studies [18] of non-interacting electrons showed this averaged persistent current had half-flux quantum periodicity. To explain the physics, lmry [19] noted that the constraint on the number of particles induces a flux dependence of the Fermi energy (or chemical potential) which is correlated with the variation of the density of states. This correlation induces an average current which is proportional to the square of the density of state fluctuations [20] in agreement with numerical studies [lS]. This quadratic dependence halves the flux periodicity from & to &,,. We now present the available experimental observations [21] of ensemble currents prior to giving a more quantitative discussion of these effects.
2. Ensemble
averaged
persistent
currents
averaged
persistent
in copper
To get an idea of the magnitude of the persistent currents expected in copper, we use the estimate = e/TD = ev,h,lL’. We obtain A, from the measured residual resistivity p = 2.1 p,fl cm of mesoscopic i tYP copper wires [22] at 4.2 K. Using
for copper, we get A, = 300 A indicating that the mean and the value k, = 1.36 X 10’ cm-’ appropriate free path is limited by boundary scattering between the Cu and its oxide layer. The actual copper ‘rings’ studied were squares 0.55 km on a side with a cross section of 350 x 450 A’. With these parameters the a magnetic moment p = 5 X 1O-22 A m* = 54 pg. The phase typical current i,,, = 1.9 nA induces from the conductance oscillation of a wire network at coherence length I, = 2 km was obtained T = 1.7 K made with the same copper material and similar aspect ratio as the ring samples. Below 3 K, 1, is limited by low-energy electron-electron scattering and scales with temperature as T-l” [23]. Below 0.8 K, the phase coherence length 1, measured in transport experiments saturates perhaps because of magnetic impurities. We have studied 10’ isolated copper rings fabricated by electron-beam lithography and liftoff techniques on a sapphire substrate [24]. An electron micrograph of this sample is shown in fig. 1. Such a sample realizes an ensemble average where both the disorder and the number of electrons change from ring to ring, but are given and independent of flux in each ring (canonical ensemble). We probe persistent currents through the average oscillatory magnetic moment [25] with period 4p = +,, or $&,. Its size is so small that magnetic impurities present in the measuring system are expected to dominate. However, the expected signal is a periodic and therefore nonlinear function of the flux, while the background susceptibility is highly linear. An effective way to reject this background is to measure the curvature of the induced moment with flux by modulating 4 = 4bc + @ACsin(nt). The induced response is p.(t) = c ,,=I,
/I+~(?.) L
cos(2nfir)
+ P~,~+,(T)
sin(2n
+ l)nt
1
A
L. P. Ltvy
248
Fig. 1. Scanning electron linewidth is under SO0 A.
micrograph
I Persistent currenfs in mesoscopic copper rings
of the copper
rings
used
in the experiment.
The squares
are 0.55 km on a side.
The
The Fourier coefficients pH can be computed only if the dependence of ( j($))(or( F(4))) with flux is known. We assume that at the temperatures of interest (0.01 K < T < 0.3 K)( j($)) - j’ sin 21~d,/@, and that the higher harmonics j,, = -2nnf,/+,, [eqs. (1) and (4)] with flux (which are not to be confused with the harmonics of the time dependent response tc sin(nL?nt) to the driving flux) decrease rapidly with n(section 3). If we only retain the lowest non-zero harmonic (j, or j2) the Fourier amplitudes CL, are
I+~(T) = %(T)JZn(~)
sin0 ,
IL~~+~(~)=~~CL,(T)J~~+,(~)COS~
(7)
where 6 = 21~4,,./4, and 0 = 2rr&,,/$,. When the AC amplitude is sufficiently large (-$4,), higher harmonics appear as modulation sidebands of the measured signal as illustrated in fig. 2. On the other hand, paramagnetic impurities contribute predominantly to the fundamental (0) response provided that the Zeeman splitting pBB,, remains small compared to the temperature k,T. The harmonic amplitudes p,, are measured by synchronous digitization of the signal during an integral number of periods (~20) followed by Fourier analysis. The odd harmonics are in phase with the applied flux while even harmonics are 90” out of phase since the persistent currents are odd functions of the magnetic flux. In absence of a DC Aux, only the third harmonic p3 is present and grows as the temperature is lowered as shown in fig. 3(b) (for comparison, fig. 3(a) shows no third harmonic response without sample). Before discussing this temperature dependence, we qualitatively describe the oscillatory behavior of the second and third harmonics at T = 0 with static magnetic field. Traces (c), (d) and (e) of
L.P.
L&y
I Persistenf currents in mesoscopic
copper rings
249
t-El E
z
3R
-@p/2
flux
@p/2
Fig. 2. Fourier transform of the in-phase magnetization signal modulated at 0.3 Hz by a sine wave of 15 G amplitude. The sample temperature is T = 25 mK, while the magnetometer is held at T = 450 mK. Inset: schematic dependence of the average current with flux. A flux modulation of _’ a$~, (arrows and highlighted region) produces the desired nonlinear response.
~a@)
temperature
t+(T)
(mK)
(plotted in arbitrary Fig. 3. Temperature dependence of the second (k, right column) and third ( ps, left column) harmonics units) of the magnetization response using a 15 G AC drive. (a) Empty magnetometer, (b) sample with no DC field, (c) 15 G DC field, (d) 30 G DC field, (e) 60 G DC field. The solid lines are one parameter fits to /+(O) exp(-kTIE,) adjusting only ~~,~(0).
250
L. P. L&y
i Persistent
currents
in mesoscopic
copper
rings
Fig. 3 show the temperature dependence of the second and third harmonics when a DC field of 15 G ($M> 30G (a+,,)> and 60 G (i&,) applied. At i &,, the T = 0 third harmonic has changed sign (0 = n) while the zero-field signal repeats at j&,, (0 = 2rr). This oscillatory behavior on the scale of half a flux quantum is consistent with the ensemble averaged properties of persistent currents. We now turn to the observed temperature dependence. In a disordered metal, two states separated by an energy E are spatially correlated over a distance L, = -rr(hD/s)“’ [26]. It follows that energy averaging (temperature) smears out wavefunctions only at sufficiently large distance. We therefore expect to lose sensitivity to a change in boundary conditions when this length L,. is less than L. For i+,, periodicity, one expects the condition to become L, s 2L since quasi-classical paths must enclose the flux twice to contribute to this periodicity. Persistent currents are therefore expected to decrease on a temperature scale kT = E, = ~‘fiLl/L”. The actual temperature dependence is determined by the detailed energy correlations in the density of states. Calculations [14] show that for T of order of E,, ( j) decreases as exp(-n’T/3E,) while at higher temperatures ( ;) decays more slowly as exp(-2v L/L,). For simplicity we fit our data to an exponential exp(- TIT”), although other temperature dependences are consistent with the data. From the mean free path, we estimate D = 4 A, V, = 150 cm2/s and T* = 80 mK. Once T* is known, the only parameters to be fit become the zero-temperature amplitudes of the second and third harmonics, p2,3( T) = P~,~(O) exp(- T/T*). For each value of the DC magnetic flux, this one-parameter fit is plotted in fig. 3 as a solid line. Within the experimental noise, this exponential behavior is nicely verified. Since persistent currents probe the sensitivity to a change in boundary conditions, this is the most direct determination of a correlation energy. With the procedure just described, we can quantitatively study the dependence of the fitted zero-temperature signal as a function of the applied DC field. The measured dependences of the second and third harmonics are plotted in fig. 4 over a full flux quantum. Both harmonics oscillate with magnetic field on a half flux-quantum scale. Furthermore, as we hinted earlier, their field dependences
0
20
40
60
60
100
120
B (Gauss) Fig. 4. Dependence of the fits to the second and third harmonics amplitudes amplitude. 130 G correspond to one &,,. The horizontal error bars arise from bars are statistical.
at T = 0 with DC field as detected with a 15 G uncertainties in the magnetic field. Vertical error
L. P. LCvy I Persistent currents in mesoscopic
251
copper rings
are 90” out of phase; i.e., the second harmonic is maximum when the third harmonic goes through zero [eq. (7)]. Based on a calibration against a CuMn spin-glass film, the magnitude of the oscillatory moment is estimated to be 1.2 x lo-l5 A m2 within factors of two. Dividing by the number of rings, the average moment corresponds to a current amplitude of 0.24 ev,A,l L 2 in each ring. The observed signal appear to be too large to be attributed to random fluctuations of j which would be of order j,,,fl. With one and a half oscillations, we can only state that the h/2e harmonic is sizable, but other harmonics may also be present. In particular, the lowest approximation ( j) CCsin(4n+/+,) may be questioned because of the low ratio in eq. (2) of the second to third harmonic compared to the ratio of Bessel functions J2(1.7)lJ,(1.7) = 3.3 expected from eq. (7). The modulation technique used, makes it inherently difficult to determine the absolute sign of the averaged persistent current which at this stage remains undetermined. 3. From density of states to persistent
currents
In the diffusive regime, a comprehensive description of the statistics of persistent currents can be given in terms of the properties of the one-particle density of states. In the absence of interactions, the existence of a strong spectral rigidity due to level repulsion dominates the properties of the density of states [27],
where s is the spin degeneracy, with energy E,($) and lifetime on magnetic flux is given by
A(k,, E) = y,(r[(.s - E,,)~ + -Y:])~ ’ is the spectral function of the state 12 h/2-y, (due to inelastic processes). The dependence of the total energy
E(4) = j- cf(& - P.(~))~(~,~1de where f is the Fermi
N=
function
(9)
3
and /_Lthe chemical
potential
Ixf(& - /44)b(d+ &)de
determined
as usual
by
(10)
In particular, the disordered average energy (E(4)) I* at fixed chemical potential only depends on the average density of states (n(4, E)). Th us in absence of interactions, it suffices to evaluate the flux dependence of the average density of states to determine the average current ( j) = -a( E(4)),la4 at zero temperature. Since the disorder mixes randomly states in an energy band 2y, = h/r,, the average spectral weight (A) is obtained by setting y, = ye = hh,/2v, and ck = h2(k: + ki) /2m where k, is given by eq. (3) and k, is the transverse channel wavevector [28]. The harmonics of the averaged density of states (n) = C IZ! cos(27~l+/&,) are then
n,(z) = 5
F
1 dx cos(2nxl)((e t --oc
- E, - c0~2)2 + yf)-’
(11)
252
L. P. L&y
I Persistent
currents
in mesoscopic
copper
rings
where E(, = A2/2mL'and E, = (fik,)'/2m. In the diffusive regime, the Fourier amplitudes ~1, become exponentially small when I> O[n, 0~ exp(-ly,L/2hv,) = exp(-lL/A,)] and the average density of states is constant. It follows [12,29] that the harmonics of the average current (i,) I” for fixed chemical potential are proportional to exp(-/L/2&). (This is consistent with the intuitive notion that a path must enclose the flux I times to contribute to the Ith harmonic of the flux.) In this case, the averaged persistent current is unobservably small. In spite of these arguments, an average persistent current exists in closed systems even in the absence of interactions, as first demonstrated by Bouchiat and Montanbaux [IS]. To explain the physics, we follow here the discussion of Imry [19]. Defining the flux dependence of the chemical potential as ~(4) = cF + A(4), the density of states is expanded close to .sF as n(4, a) = n(+, er) + (e - F~) an(4, &,)/de [30] and eq. (10) may be rewritten at T = 0 as
(12) Upon ensemble average, the left hand side of eq. (12) is a flux independent constant which we set to zero by an appropriate choice of or. Then, the chemical potential and the density of states are on average correlated by the constraint on the number of particles. Similarly, the total energy may be written as
E(4) - E, = A(4)&, + 4
A’(&(+,eF)+ an(;;e,)) ,
where E,, = ];’ n($, c)e de becomes flux independent and (13), the average energy at fixed N reduces to (E(4))
- Et, = 4 (A’(4)n(+,
(13)
eF
+))
^I + (n(eF))(A2(@))
after ensemble
>
averaging.
Combining
eqs. (12)
(14)
since the average density of states is much greater than its fluctuation. The quadratic dependence on A halves the flux periodicity from do to $4”. Because of level repulsion, h’ is expected to fluctuate rapidly with flux. Large fluctuations imply aA2/8+ + a2/$,, where 6 is the level spacing. Similarly, this suggests that A’(+) contains many harmonics of the flux. Very recently, Altshuler, Gefen, Imry [31] and A. Schmid have been able to relate (A’(4)) to the fluctuations in the density of states studied by Altshuler and Schlovskii [27]. At T = 0,they find that the average current
is independent of disorder and inversely proportional to the number of channels M = 4rrkiA. Montambaux et al. have carried large scale simulations and conclude that the 14, harmonic of the average current depends weakly on the disorder and the number of channels approximately as =ev,lL(h,IML)"2.A similar result has been obtained by Von Oppen and Riedel [31] in the limit where the mean free path A, is comparable to the transverse dimensions. How the average current depends on disorder as the ballistic regime is approached remains an open question. Ambegaokar and Eckern have also shown that interaction effects give rise to a flux periodic persistent current whether the system is open or closed [14]. This conclusion was also reached by the presence of electron-electron interactions Altshuler et al. [15] in the context of SNS junctions: shifts the energy of the system. This energy shift depends on the flux dependent quasi-momentum
L. P. Ltvy
I Persistent currents in mesoscopic
253
copper rings
carried by the electrons along the ring, hence leads to a persistent current in the ground state. At the simplest level, this can be analyzed as a self-energy shift of the quasi-particles (a mean field description) which is largest close to the Fermi level: then this interacting electron problem reduces to a change &Z(E) in the one-electron density of states at the Fermi level. In this sense, the associated persistent currents can still be described at the one-electron level. To wit, this change in the density of states averaged over disorder has been calculated by Altshuler and Aronov [13] in terms of the diffusion coefficient D and the Fourier transform of the interaction potential v averaged over the Fermi surface,
&z(e)= -7
sn( eF)V
hDk2 C k e* + [hDk2 + y]’ ’
where h/2y is the inelastic Cooperon equation, (-is
- D(V-
2ieAlfi)2
(15)
In eq. (15), the sum over k is extended
lifetime.
- y)P(r)
over the eigenvalues
= 6(r),
(16)
with periodic boundary conditions around the ring. The quantization of the corresponding wavevector is k, = 2TlL(n + 24/&,), and not eq. (3) which is appropriate to the eigenstates Schrodinger equation. The harmonics 6n, are calculated after Fourier transform as
sn(e,)V 6n, = -___4aE
c
where
#&EC,
E, is the correlation
iYE,) energy
of the
Bloch of the
(17)
>
and
-r
Z(u, v) =
I -cx
dx
x2 cos(2nlx) U2 + (X2 + r.J)’
= 2,:~~~2;~,~
exp(-T[[2[
y + (v’ + u*)“~]]“*)
,
(18)
of the where R = cos(lr1[2U]“2 + arr) for small y. From the density of states, the flux dependence current is obtained as the flux derivative of the energy using eqs. (1) and (9). One thus easily recovers the results of Ambegaokar and Eckern [14]: the average current is periodic in h/2e and of order 0.42nVevFA,lL2. They also find that ( j) decreases approximately exponentially on a scale T* = 3hDl L*. Higher harmonics of the current scale as I-* with a characteristic temperature T* /12 [32]. In other words, the Ith harmonic of the average current carried by a ring of length L is equal to the first harmonic of the current carried by a ring of length L’ = lL, in agreement with the quasi-classical picture of a diffusing particle. Only eq. (1.5) is consistent with this scaling. Measurements of the harmonic content of the average persistent current would therefore determine the corrections to the density of states. Finally, Eckern [33] and Altshuler et al. [15] analyzed the reduction of the electron-electron coupling constant due to higher order contributions. They conclude that the first order contribution v is reduced by approximatively a factor between 4 and 10. On the other hand, spin-orbit scattering and Zeeman splitting have negligible effects [33] on the averaged current. To conclude this section, it is instructive to relate the fluctuations of the persistent current characterized by the typical current j,,, = ( j2)l’* [lo] to th e fl UCt ua t’ions of the density of states [27]. Since j:,, = @W)E(~‘)W W’),=,, 9 we first relate the energy fluctuations
254
L.P. L&y
I Persistrnt currents in mesoscopic
copper rings
(E(~)E(m’))-(E(~))(E(S’))=JI&E’f(&-I*)f7E’-P) 0
0
x ((n(e)n(d)) to the level density fluctuations. Altshuler fluctuations in the level density as
K,, = (4GW)
= K(e,
-
E2,
and
~ (P(E)) (n(d)))
Shlovskii
have
dE ds’
calculated
the
(19) ensemble
averaged
- (+,>)bW) 4 - $‘I + KC&,-
~2,
4 + 4’) ,
(20a)
where K(E, qf~-+ 4’) =
-$
Re 7
(F + i(y + hDk’(b,
i 4’))))?
.
WJb)
Here the Bloch momenta k(+ * 4’) are quantized by k,, = 2rr/L.(rz + (4 L +‘)/$I,,). the level density fluctuation in harmonics of the flux follows as K12($, 4’) = 2 K/ cos(2n@i&,) i-0
I t dx
_Z I-
u2 - (x3 + v)’
(U’ + (xZ + V)‘]’ J
~Q(u, ~1 2(u’ + v?)3s4exp(Ir1[2(v”
SEE,,
of
(21)
cos(2nl+‘/&,)
The Fourier amplitudes K, = -(s/(h~)E,)~~,( energy E, and the dimensionless function
J,(u, v) =
The expansion
i-y/Z,)
are expressed
in terms
of the
correlation
cos(2?Tlx)
+ u’)“‘]‘~‘),
(22)
where Q(u, V) is an oscillatory function. The fluctuations in the energy [obtained from eq. (19)] have the same flux dependence as the density of states. Their harmonic amplitudes (ey) are proportional to Efil”. This leads to a typical current jtvp = ( j:)“2 = EC/&, in agreement with the work of Cheung et al. [lo]. We also recover the peculiar scaling (l-“’ ) of the harmonics of the typical current derived by Altshuler and Spivak [27]. This scaling law is not consistent with semi-classical path arguments. This presumably reflects the non-trivial fluctuations in the density of states (eq. 20) associated with the spectral rigidity. Persistent currents are equilibrium thermodynamic properties of mesoscopic rings. We have emphasized their relation to the flux dependence of the density of states to place such effects in the usual context of orbital magnetism.
4. Discussion
and conclusion
Two possible interpretations for the observed quantum oscillations of the magnetization of copper rings have so far been proposed. Both are based on the existence of persistent currents in mesoscopic rings [4]. However, they rely on different physical mechanisms to explain the ensemble averaged properties.
L. P. Ltvy
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Bouchiat and Montambaux [18] have emphasized that canonical and grand-canonical ensemble give an extensive difference in the thermodynamic averages of mesoscopic systems. They find an averaged persistent current with half-flux quantum periodicity in the canonical ensemble. The magnitude of the current observed in their numerical simulations can only be compared to experiment through the mean-free path. This quantity is poorly determined by simulations making quantitative comparison difficult. The current observed in simulations is paramagnetic. In the diffusive regime, the sign of the current is not expected to depend on spin-orbit interactions, as the magnetic field always increases the rigidity of the spectrum. On the other hand Ambegaokar and Eckern [14], and Altshuler et al. [15] emphasize the thermodynamic corrections associated with interactions. They have demonstrated the relevance of fluctuation effects more familiar in the context of superconductivity to mesoscopic structures. While the observed behavior can also be qualitatively explained with this theory, a quantitative comparison is also difficult because the electron-electron coupling constant is not known in copper. Based on simple estimates, one would expect the screened Coulomb repulsion to dominate the phonon mediated electron-electron interaction in copper. However, from studies of SNS junctions [35], there is indirect evidence that the electron-electron interaction in bulk copper may be attractive with a coupling constant n(c,)V= 0.11, On the other hand, the corresponding T, = 60 mK obtained from BCS theory is considerably higher than the current experimental limit of T, < 5 x lo-’ K (361. How can one differentiate between these mechanisms? Aside from the experimental difficulties associated with the determination of the sign and exact magnitude of the effect, there are other problems. First, interaction effects depend on the sign and the magnitude of the electron-electron in bulk copper, and neither of them are unambiguously known. Secondly, the sign of single particle effects also depends on the strength of spin-orbit, interactions. The only clear-cut test is to measure the magnetization of a network held at a fixed chemical potential. In this case only interaction effects should survive. The comparison with theory would be unambiguous if this experiment is carried out on a material with a well-known electron-electron interaction, i.e., a superconductor above its critical temperature.
Acknowledgements G. Dolan and of the original V. Ambegaokar, Finally, I have Montambaux and
J. Dunsmuir have made the splendid samples shown in fig. 1. H. Bouchiat gave much motivation and made many suggestions throughout this work. B. Altshuler, Y. Imry and B. Spivak have clarified many important issues related to this work. benefited from many discussions with A. Benoit, Y. Gefen, R. Landauer, G. E.K. Riedel.
References
[ll
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L. P. LPvy I Persistent
currents
in mesoscopic
copper
rings
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