Current-flux characteristics in mesoscopic nonsuperconducting rings

Abstract

We propose four different mechanisms responsible for paramagnetic or diamagnetic persistent currents in normal metal rings and determine the circumstances for change of the current from paramagnetic to diamagnetic ones and {\it vice versa}. It might qualitatively reproduce the experimental results of Bluhm et al. (Phys. Rev. Lett. 102, 136802 (2009)). Comment: 8 pages, 1 figure

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Current-flux characteristics in mesoscopic
nonsuperconducting rings
L Machura1, Sz Rogozinski1and J ? Luczka1
1Institute of Physics, University of Silesia, Katowice, Poland
E-mail: lukasz.machura@us.edu.pl
Abstract.
diamagnetic persistent currents in normal metal rings and determine the circumstances
for change of the current from paramagnetic to diamagnetic ones and vice versa. It
might qualitatively reproduce the experimental results of Bluhm et al. (Phys. Rev.
Lett. 102, 136802 (2009)).
We propose four different mechanisms responsible for paramagnetic or
PACS numbers: 64.60.Cn, 05.10.Gg, 73.23.-b
Submitted to: J. Phys.: Condens. Matter
1. Introduction
In the absence of an applied voltage, an induced electrical current rapidly decays due
to dissipation processes. In normal (not superconducting) metal rings it typically dies
out within the relaxation time of order 10−13s. However, if a radius of the ring is small
enough (below microns) and temperature of the system is below 1 K, quantum effects
start to play a distinct role. Under right circumstances electrons in the ring are able
to preserve its coherence which in turn results in the persistent (dissipationless) current
induced by the static applied magnetic field.
The existence of persistent currents in metallic rings was predicted by Hund in 1938
[1]. More than 30 years later Bloch [2] and Kulik [3] confirmed this prediction by means
of quantum–mechanical models. The strong interest in physics of mesoscopic rings arose
after the 1983 paper [4] where the authors showed that persistent currents could flow
even in the presence of disorder. Experiments on persistent currents have produced a
number of confusing results in apparent contradiction with theory and even amongst the
experiments themselves (e.g. the response of rings was 10-200 larger than theoretically
predicted) [5]. We could observe persistent controversy on persistent currents for nearly
twenty years. Recently, two groups [6, 7] have developed new different techniques which
allow to make measurements a full order of magnitude more precise than any previous
attempts. Both experiments confirm to a high degree the physics theory regarding the
behavior of persistent currents. The group of K. Moler [6] has employed a scanning
arXiv:1008.2781v1 [cond-mat.mes-hall] 16 Aug 2010

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technique (a SQUID microscope) and measured the magnetic response of 33 individual
mesoscopic gold rings. Each ring was scanned individually, unlike past experiments
on persistent currents conducted by other groups.
approximately 10 million times. The team of J. Harris [7] has developed an alternative
measurement scheme. The aluminum rings have been deposited on a cantilever used as
a torque magnetometer whose vibration frequency can be precisely monitored. From the
frequency shift caused by the magnetic flux, the researchers could deduce the current
with a precision of two orders of magnitude greater than it was possible in the past.
They have studied several different cantilevers decorated with a single aluminium ring
or arrays of hundreds or thousands of identical aluminium rings. The rings on different
cantilevers had radiuses of 308-793 nanometers.
Persistent currents are highly sensitive to a variety of factors. It is clearly visible
in experimental data shown in panel (b) of Fig. 2 in Ref. [6]: for nominally identical
samples the observed response is paramagnetic (e.g. for the ring 1) or diamagnetic
(e.g. for the ring 2). Here, we propose several possible mechanisms for controlling the
response of the metallic mesoscopic rings and determine the conditions for which the
transition from paramagnetic to diamagnetic current is able to occur.
The paper is arranged as follows. In the section 2, we start with the presentation
of the model for the flux dynamics in the rings. Next, in the section 3, we identify the
operating conditions on which similar rings can exhibit opposite responses. We finalize
the paper with conclusions in the section 4.
In total the rings were scanned
2. Model for flux dynamics of mesoscopic rings
At low temperature, small normal metal rings threaded by a magnetic flux can
display persistent and non-dissipative currents carried by phase-coherent electrons. The
circumference of the ring should be smaller than the electrons phase coherence length.
This typically limits the sample size to below micrometers and the temperature to below
1 K. However, at temperature T > 0, a part of electrons looses its phase-coherence due
to thermal fluctuations and constitutes a dissipative Ohmic current associated with the
resistance R. The actual magnetic flux φ induced by the current flowing in the ring is
given by the relation
φ = φe+ LI, (1)
where φe is the magnetic flux generated by the external constant magnetic field, I
denotes the total current flowing in the ring and L stands for the self–inductance of the
ring. Dynamics of the magnetic flux φ in such a system is modeled by the dimensionless
Langevin-type equation [8]
?
where x = φ/φ0is the rescaled magnetic flux and φ0= h/2e is the flux quantum. The
rescaled time s = t/τ0, where the characteristic time τ0= L/R is the inductive time of
dx
ds= −dV (x)
dx
+2Dλ(x) Γ(t), (2)

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the ring. For a typical mesoscopic ring, L/R is in the picosecond range. The function
V (x) =1
2(x − xe)2+ B(x),
where xe= φe/φ0and [8]
∞
?
where α = LI0/φ0 and I0 is the maximal persistent current at zero temperature.
In the mechanical context, V (x) could be interpreted as an effective potential. The
temperature dependent amplitudes An(T0) are determined by the relation [9]
An(T0) =4T0
π1 − exp(−2nT0)
with the dimensionless temperature T0= T/T∗, where the characteristic temperature
T∗is proportional to the energy gap ∆F at the Fermi surface. The persistent current
strongly depends on the parity of the number of coherent electrons. It is taken into
account by assigning the probability p of an even number of the coherent electrons.
Then the corresponding probability of an odd number of the coherent electrons is given
by 1 − p.
Thermal equilibrium fluctuations are modeled by δ–correlated Gaussian white noise
Γ(t) of zero mean. Classically this situation holds true in many cases. When temperature
is lowered, however, the quantum nature of thermal fluctuations becomes important
and starts to play a role. Therefore the standard diffusion coefficient D0= kBT/R (kB
denotes the Boltzmann constant) is modified due to quantum effects like tunnelling,
quantum reflections and purely quantum fluctuations [10, 11, 12]. The modified diffusion
coefficient Dλassumes the form [11, 8]
β−1
1 − λβV??(x)
with β−1= kBT/2Em = k0T0, the elementary magnetic flux energy Em = φ2
and k0 = kBT∗/2Em is the ratio of two characteristic energies. The prime denotes
differentiation with respect to x. The dimensionless quantum correction parameter [10]
?
T0
where the psi function Ψ(z) is the logarithmic derivative of the Gamma function,
γ ? 0.5772 is the Euler gamma constant and C is capacitance of the system related
to charging effects. The parameter λ characterizes quantum corrections to classical
thermal fluctuations and can be formulated as the difference between quantum and
classical fluctuations of the dimensionless flux,
(3)
B(x) = α
n=1
An(T0)
2nπ
cos(2nπx)[p + (−1)n(1 − p)], (4)
exp(−nT0)
(5)
Dλ(x) =
(6)
0/2L
λ = λ0
γ + Ψ
?
1 +
?
??
,λ0=¯ hR
πφ2
0
,? =¯ h/2πCR
kBT∗
, (7)
λ = ?x2?q− ?x2?c
(8)
where ?·? denotes thermal equilibrium average, the subscripts q and c refer to quantum
and classical cases, respectively. Let us remember that the diffusion coefficient Dλ(x)
cannot be negative and therefore the parameter λ has to be chosen small enough to

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ensure its non-negativeness for any argument.
contains the multiplicative white noise Γ(t), it is important to stress that Eq. (2)
has to be interpreted in the Ito sense [13]. Therefore the corresponding Fokker–Planck
equation for the time evolution of the probability density P(x,t) reads [13]
?dV (x)
dx
Because in Eq. (2) the noise term
∂
∂tP(x,t) =
∂
∂x
P(x,t)
?
+
∂2
∂x2[Dλ(x)P(x,t)]. (9)
The average stationary dimensionless current i flowing in the ring can be calculated
from Eq. (1):
i = ?x? − xe,i = ?I?L/φ0
(10)
and the average stationary magnetic flux ?x? is calculated from the relation
?x? =
where P(x) = limt→∞P(x,t) is a stationary probability density being a solution of the
Fokker-Planck equation (9) for ∂P(x,t)/∂t = 0 and zero stationary probability current.
It has the form
?∞
−∞x P(x)dx,(11)
P(x) = N0D−1
λ(x)exp[−Ψλ(x)],(12)
where N0is the normalization constant and the generalized thermodynamic potential
Ψλ(x) reads
?dV (x)
dx
In the case when thermal fluctuations can be treated classically, i.e. when λ = 0,
the stationary state P(x) is described by the Boltzmann distribution. When thermal
fluctuations have to be considered as quantum fluctuations, the stationary state is still
a thermal equilibrium state but now described by the non-Boltzmann distribution (12)
with the x-dependence of the diffusion coefficient Dλ(x).
Ψλ(x) =
D−1
λ(x)dx = βV (x) −1
2λβ2[V?(x)]2. (13)
3. Current–flux characteristics
We are interested in the current-flux characteristics, i.e. in dependence of the stationary
current i = i(xe) on the applied magnetic flux xe. To this aim we exploit Eqs. (10)-
(13). Note that the external flux xeenters Eq. (10) and the remaining equations via
the potential (3) which occurs both in the diffusion function Dλ(x) and the generalized
thermodynamical potential Ψλ(x). Because the persistent current is a periodic function
of the magnetic flux, we consider the current-flux characteristics on the unit interval
of xe. We present four visually similar sets of the current-flux characteristics with
various set-ups to demonstrate sensitivity of the persistent currents to some subtle
effects. It could explain the different response of the nominally identical metal rings
to the magnetic flux presented in the recent experimental work [6]. The measured
persistent current of 15 nominally identical rings with radius R = 0.67µm exhibits both
the paramagnetic and diamagnetic response. In panel (b) of Fig. 2 in Ref.[6], one can

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0.50
6
(c)
0.25 0.00
xe
0.250.50
1
0
1
i·10−3
(a)
T0=0.6
T0=0.6423
T0=0.7
0.50 0.250.00
xe
0.25 0.50
2
1
0
1
2
i·10−3
(b)
k0=0.09
k0=0.1052
k0=0.12
0.50 0.250.00
xe
0.250.50
6
4
2
0
2
4
i·10−3
p=0.5
p=0.455
p=0.4
0.500.25 0.00
xe
0.250.50
1
0
1
i·10−3
(d)
λ0=0.0
λ0=0.00161
λ0=0.002
Figure 1. (color online) The stationary averaged velocity vs the external magnetic
flux xe. In the panel (a) we show current–flux characteristics in the classical regime
for three different temperatures T0 = 0.6,0.6423,0.7.
classical regime for three different structure constants k0= 0.09,0.1052,0.12. Panel
(c) exhibits again the persistent current for ring working in the classical regime for three
values of the probability p = 0.4,0.455,0.5 of an even number of coherent electrons
in the ring. Finally panel (d) reveals the influence of the quantum parameter λ0= 0
(indicating classical regime), 0.00161,0.002 (quantum regime). Blue (solid) lines mark
the paramagnetic response to the external stimulus, green (dashed) lines denote the
situation where the magnetic susceptibility is zero and finally red (dashed–dotted)
lines indicate the diamagnetic susceptibility of the normal metal mesoring around zero
external magnetic flux (see insets for details). The parameters not given explicitly are
set as follows: T = 0.5, p = 0.48, k = 0.08, α = 0.1.
Panel (b) presents also the
easily notice that in the linear response regime, i.e. for values of the external flux close
to zero, the susceptibility defined as the ratio of the average persistent current to the
external flux has sometimes positive slope indicating the paramagnetic current (see the
current for the rings 1, 3, 4, 10, 11, 12, 15 in Fig. 2(b) in [6]) and sometimes negative
slope exhibiting the diamagnetic response (see the current for the rings 2, 5, 6, 7, 8, 9,
13, 14 in Fig. 2(b) in [6]).
Figure 1 illustrate possible current-flux characteristics. All parameters are chosen
arbitrarily as we don’t aim to compare our results with the experimental data. The goal

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of this work is rather to constitute possible mechanisms responsible for diamagnetic
or paramagnetic responses. Neverthless, all curves corresponding to the paramagnetic
currents (blue solid lines) in all panels look very similar to the experimental curve shown
in figure S6(A) in the Supporting Online Material [14] of the paper [7].
In the panel (a) we depict the current–flux characteristics in the regime of classical
thermal fluctuations (i.e. when λ = 0) for three different temperatures. For the lowest
temperature T0= 0.6 (blue solid line) the situation reveals the paramagnetic response
of the ring to the applied magnetic flux. For the higher temperature T0 = 0.7 (red
dashed–dotted line) the response is diamagnetic. The cross–temperature below which
the response is paramagnetic and above which is diamagnetic was recognized numerically
at the value of TC
details).
Panel (b) presents the ring reaction to the external stimulus for three values of the
structure parameter k0= kBT∗/2Em. It stands for the ratio of the two characteristic
energies, the thermal energy kBT∗/2 to the magnetic energy Em = φ2
instructive to define k0in the alternative way as k0= LI0/φ0. At the value of k0= 0.09
the response is paramagnetic (blue solid line), for k0 = 0.12 it is diamagnetic (red
dashed-dotted line) and again we have numerically identified the cross–value at the
level of kC
rα = α/k0 = i0/I0 ∼ le/l describes the physical properties of the metal ring [15].
Here le is the elastic mean free path of the electron in the ring and l stands for the
circumference of the ring. For the multichannel rings the above ratio is greater than
one for the ballistic regime and smaller than one in the diffusive one. In the presented
collection of the parameters the cross–value of the para– to – diamagnetic response lies
slightly below the value of unity, i.e. at rC
in the diffusive regime and all rings designated by the higher values of k0showing the
diamagnetic susceptibility will also lie in the same diffusive regime.
It is well know that the most straightforward way of turning the susceptibility
from diamagnetic to paramagnetic is to change the probability p of an even number of
coherent electrons in the ring. In the panel (c) we present the current for three different
rings with probabilities p = 0.5 (blue solid line), 0.455 (green dashed line) and 0.5 (red
dashed–dotted line) exhibiting the paramagnetic, zero and diamagnetic susceptibility,
respectively.
In the last panel (d) we focus on the influence of quantumness of thermal
fluctuations on persistent currents. The comparison of the classical thermal fluctuations
with λ0= 0 and corresponding paramagnetic current (blue solid line) with two values
of the non–zero quantum noise parameter λC
susceptibility (green dashed line) and λ0 = 0.002 with diamagnetic response (red
dashed–dotted line) is presented. As shown, the sign of the current in the vicinity of zero
magnetic field can be easily affected by small perturbation of the quantum correction
parameter λ.
0 = 0.6423 (see green dashed line and the inset of the panel (a) for
0/2L. It is
0= 0.1052 (green dashed line). We note that the ratio of two parameters
α? 0.95. It means that this border value lies
0 = 0.00161 exhibiting zero magnetic

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4. Conclusions
We have proposed four various mechanisms that can lead to either diamagnetic or
paramagnetic currents in the metal rings. Two of them are related to the physical
properties of the metal rings. These are the structure parameter k0 together with
the probability p of an even number of coherent electrons in the ring. Another two
can be adjusted by tuning the temperature of the system. In the experiment [7], the
temperature uncertainty is 7% [14]. In all presented cases, the small change of the control
parameter causes the reversal of the susceptibility - from paramagnetic to diamagnetic
ones or vice versa. In the nowadays experiments on the mesoscopic rings it is impossible
to justify which system parameters are responsible for the observed responses. The fact
that the scientists are able to perform the measurements on single separated rings is a
great success and a milestone in the present state–of–the–art. In the near future the
experimental physicists will be able to prepare experiments in the desired and more
precise conditions and then our findings might be tested.
Acknowledgments
The work supported by the ESF Program Exploring the Physics of Small Devices.
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