Journal ArticleDOI

# Persistent current of one-dimensional perfect rings under the canonical ensemble.

15 Jan 1996-Physical Review B (Phys Rev B Condens Matter)-Vol. 53, Iss: 3, pp 1006-1009

AboutThis article is published in Physical Review B.The article was published on 1996-01-15 and is currently open access. It has received 4 citation(s) till now. The article focuses on the topic(s): Microcanonical ensemble & Canonical ensemble.

Topics: , Canonical ensemble (79%), Statistical ensemble (75%), Grand canonical ensemble (69%)

## Summary (1 min read)

### I. INTRODUCTION

• More detailed quantitative calculations have been given by Cheung et al. [2] [3] [4].
• The reason is that the canonical-ensemble occupation probability is complicated and sensitive to the details of the energy levels.
• The authors results show that replacing the canonical-ensemble occupation probability with the Fermi-Dirac distribution cannot lead to correct results for the persistent current.
• The authors results send warnings to those using the above approximation.

### III. PERSISTENT CURRENT IN THE RING

• It is possible to calculate the average magnetic moment from the occupation probability of the levels.
• As expected, the dominant term in the average magnetic moment is linear in magnetic field.
• The authors also noted that the amplitude of the harmonics is roughly exp(Ϫ2l 2 /␤⌬) in the grand-canonical ensemble.
• For systems with many rings, the electron number in each ring may not be the same.
• Analysis starting from Eq. ͑9͒ shows that this crossover of period is true over the whole range of temperatures.

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Persistent current of one-dimensional perfect rings under the canonical ensemble
Man-Kit Yip, Jiu-Ren Zheng,
*
and Ho-Fai Cheung
Department of Physics and Materials Science, City University of Hong Kong, Hong Kong
~Received 12 April 1995; revised manuscript received 18 August 1995!
We have analyzed the harmonic contents of the persistent current at high temperatures under the canonical
ensemble. Results show that the behavior of each harmonic is different from that under the grand-canonical
ensemble. The persistent current of the multiring system is also presented.
I. INTRODUCTION
Modern studies on persistent current in mesoscopic rings
have been renewed by Bu
¨
ttiker, Imry, and Landauer
1
in
1983. More detailed quantitative calculations have been
given by Cheung et al.
2–4
Experimental measurements have
been reported by various authors.
5–7
One of the main discov-
eries not predicted by former theories at that time was that
for an ensemble of 10
7
three-dimensional ~3D! rings the
persistent current varies with magnetic ﬂux with a dominant
period of h/2e (hc/2e in Gaussian units!. This contrasts to
that for a single ring whose dominant period is h/e (5 F
0
,
the magnetic ﬂux quantum!. This was subsequently
explained
8–12
by assuming that the number of electrons in
each ring is ﬁxed and independent of ﬂux. That is, canonical
ensemble should be used if the average properties of these
rings are to be calculated.
The variation of the persistent current in one- and higher-
dimensional perfect and disorder rings as a function of tem-
perature has been studied by Cheung et al.
2–4
All these stud-
ies adopted the grand-canonical ensemble, so the chemical
potential is ﬁxed and the electron occupation probability fol-
lows the Fermi-Dirac distribution. When T5 0 with the num-
ber of electrons ﬁxed, studies have been carried out by Bou-
chiat and Montambaux,
8
von Oppen and Riedel,
9
Schmid,
10
and Altshuler, Gefen, and Imry.
11
Their calculation logic is to
allow the chemical potential to vary with magnetic ﬂux such
that the number of electrons remains ﬁxed. This procedure is
perfectly correct at T5 0. At TÞ0, the persistent current un-
der the exact canonical ensemble has not been calculated so
far. The reason is that the canonical-ensemble occupation
probability is complicated and sensitive to the details of the
energy levels. Most of the time the canonical-ensemble oc-
cupation probability is not known.
At high temperatures, if one approximates the canonical-
ensemble occupation probability by the Fermi-Dirac distri-
bution with a suitably chosen chemical potential, one might
expect to get a reasonable answer for the persistent current.
The reason might be that in the high-temperature limit, ca-
nonical ensemble and grand-canonical ensemble give the
same occupation probability. In this paper we present exact
analysis on the persistent current on 1D perfect rings under
the canonical ensemble. Our results show that replacing the
canonical-ensemble occupation probability with the Fermi-
Dirac distribution cannot lead to correct results for the per-
sistent current. Our results send warnings to those using the
above approximation.
II. THE MODEL
The model we consider is 1D perfect rings with noninter-
acting electrons threaded by a magnetic ﬂux. The wave vec-
tor and energy of the eigenfunctions are given by
k
n
5
2
p
L
S
n1
F
F
0
D
,
E
n
5
\
2
2m
F
2
p
L
S
n1
F
F
0
D
G
2
, ~1!
where F is the magnetic ﬂux through the loop. The corre-
sponding current is given by
I
n
52
e\
mL
F
2
p
L
S
n1
F
F
0
D
G
, ~2!
where n50,61,62,.... The energy versus wave-vector
relation is described by a parabola. As the ﬂux increases,
levels on the left-hand side move down the parabola, move
across to the right-hand side, and then move up the parabola
on the right-hand side. If there are many electrons in the
system, the E versus k parabolic relation near the Fermi
energy can be approximated by two linear branches. Let us
rewrite the energy levels on the left branch and right branch
as
E
left
5 D
S
n2
F
F
0
D
,
~3!
E
right
5 D
S
n1
F
F
0
D
,
where n can take 2  to 1  and D is the energy spacing on
either branch. The current of each level on the left branch
and right branch is 1 I
0
and 2 I
0
, respectively. Electron lev-
els have twofold spin degeneracy. In our analysis we con-
sider g-fold degenerate levels.
We noticed that the above model is equivalent to a model
of small metallic particles studied by Denton, Mu
¨
hlschleger,
and Scalapino.
13
If the energy levels are uniformly spaced
and a magnetic ﬁeld is applied, the effect of the magnetic
PHYSICAL REVIEW B 15 JANUARY 1996-IVOLUME 53, NUMBER 3
53
0163-1829/96/53~3!/1006~4!/$06.00 1006 © 1996 The American Physical Society ﬁeld is to shift the spin-up levels downward and the spin- down levels upward linearly. So the energies of the spin-up and spin-down levels can be written as E up 5 nD2 h, ~4! E down 5 nD1 h , where n takes the value 2  to 1 . D is the energy spacing for spin-up levels ~or spin-down levels!, h is the external magnetic ﬁeld measured in suitable energy units. The mag- netic moment M of each level is 1 1 and 21 ~measured in suitable units! for spin-up and spin-down states, respectively. The energy levels of the two models are equivalent. Further- more, the magnetic moment of all the electrons in the metallic-particle model is almost equivalent to the persistent current in the 1D perfect-ring model. They are not exactly equivalent because in the metallic-particle model the spin-up energy levels are not linked to the spin-down energy levels, whereas in the 1D perfect-ring model the left branch is ac- tually connected to the right branch at the very bottom of the branches. The correct relation between them is ^ I & I 0 5 ^ M & 2 2gh D . ~5! When using this equation to ﬁnd the persistent current for 1D perfect rings, all the parameters in the metallic-particle model should be substituted by the corresponding parameters in the perfect-ring model ~i.e., h DF/F 0 ). In the following, we ﬁrst work out the partition function for the metallic-particle model under the canonical ensemble. From that we can calculate the average magnetic moment and subsequently we can deduce the persistent current. At low temperatures the persistent currents from canonical and grand-canonical ensembles are the same up to exponentially small corrections. This limit will not be discussed any further in this paper. Instead our objective is to investigate the high- temperature limit. Since the persistent current is periodic in the magnetic ﬂux with period F 0 , we express the persistent current as a Fourier series. We work out the Fourier coefﬁ- cients of the persistent current in the high-temperature limit. Consider the metallic-particle model with energy levels given by Eq. ~4!, where n goes from 2  to . Let b denote 1/k B T, where k B is the Boltzmann constant and T is the temperature. Following the calculation by Denton, Mu ¨ hlschleger, and Scalapino, 13 which is also described in an earlier paper by Chen and Cheung, 14 the partition function can be written as Z ~ h ! 5 1 2 p i R dz z z r ) n51  @ 11ze 2n b D2 b h # g @ 11ze 2n b D1 b h # g ) n 8 50  S 11 1 z e 2n 8 b D2 b h D g S 11 1 z e 2n 8 b D1 b h D g , ~6! where r is the number of holes of the ground-state uppermost ﬁlled level. The value of r is determined by the actual number of electrons in the system. Without loss of generality we take r to be from 0 to (2g2 1). Higher or lower value of r can be deduced because the energy-level ladders are translational invariant. Following the procedures and the mathematical identity mentioned in Denton, Mu ¨ hlschleger, and Scalapino, 13,15 one obtains Z ~ h ! 5 1 2 p Z B 2g q 2 g/2 E 2 p p d f e i ~ r2 g ! f S ( n 8 52  q ~ n11/2 ! 2 e i ~ n1 1/2 ! ~ f 1 i b h ! D g S ( n52  q ~ n11/2 ! 2 e i ~ n1 1/2 ! ~ f 2 i b h ! D g , ~7! where q5 e 2 b D/2 and Z B 5 ) m5 1  (12q 2m ) 21 . We calculate the sum over n by using the Poisson summation formula, then Z(h) can be written as Z ~ h ! 5 1 2 p Z B 2g e b Dg/4 S 2 p b D D g ( m j ,n j 52  E 2 p p d f e i ~ r2g ! f ~ 21 ! ( j51 g ~ m j 1n j ! ) j51 g ~ e ~ 21/2 b D ! ~ f 1 2m j p 2 i b h ! 2 e ~ 2 1/2 b D ! ~ f 1 2n j p 1 i b h ! 2 ! . ~8! We put in the condition 2 g< ((m j 1 n j ), g so that the integration limits could be extended from 2  to . After completing the square with respect to f in the exponent, the integration can be carried out readily. The result is Z ~ h ! 5 1 A 2g S 2 p b D D g2 1/2 Z B 2g e b Dr/22 b Dr 2 /4g1 g b h 2 /D ( 2 g<( j5 1 g ~ m j 1 n j ! , g exp S 2 2 p 2 b D ( j51 g ~ m j 2 1 n j 2 ! D 3 exp F p 2 g b D S ( j51 g ~ m j 1 n j ! D 2 G exp i2 p h D ( j51 g ~ m j 2 n j ! )exp S 2 ir p g ( j51 g ~ m j 1n j ! D . ~9! The second last term determines the order of the harmonics. The lth harmonic comes from the term where ((m j 2 n j )56l. After comparing their magnitudes and considering all possible combinations, the leading term for each harmonic in the high-temperature limit can be sorted out. In this limit Z(h) can be expressed as Z ~ h ! 5 1 A 2g Z B 2g S 2 p b D D g2 1/2 e b Dr/22 b Dr 2 /4g1 g b h 2 /D S 11 ( l5 1  4C k g 11 d k,g e ~ 2 p 2 / b Dg ! @ l 2 1 2k ~ g2 k ! # cos p rl g cos 2 p lh D D , ~10! 53 1007BRIEF REPORTS where p and k are deﬁned by l5 pg1k with p5 0,1,2, . . . and k5 1,2, ...,g. This is the ﬁnal expression for the canonical-ensemble partition function. Next we deduce the average magnetic moment and then the persistent current of the 1D perfect-ring model. III. PERSISTENT CURRENT IN THE RING It is possible to calculate the average magnetic moment from the occupation probability of the levels. A more direct approach is to ﬁnd the magnetic Gibbs function G * 52(1/ b )lnZ(h), then the average magnetic moment ~or magnetization up to some volume! can be found from the relation ^ M & 52( ] G * / ] h) T . After expanding the logarith- mic function, taking the derivative, and then considering all the combinations carefully, we ﬁnally obtained the average magnetic moment as ^ M & 5 2gh D 1 ( l51  ~ 21 ! p11 p d k,g 11 8 p l b D C k g 11 d k,g 3e ~ 2 p 2 / b Dg ! @ lg1k ~ g2k ! # cos p rl g sin 2 p lh D . ~11! As expected, the dominant term in the average magnetic mo- ment is linear in magnetic ﬁeld. Using the mapping between the metallic-particle model and the 1D perfect-ring model @i.e., Eq. ~5!#, the persistent current of 1D perfect rings is ^ I & gI 0 5 ( l51  ~ 21 ! p11 p d k,g 11 8 p l b Dg C k g 11 d k,g e ~ 2 p 2 / b Dg ! @ lg1k ~ g2k ! # 3cos p rl g sin 2 p lF F 0 . ~12! The current is expressed as a Fourier series, showing all the harmonics. Only the sine terms exist, reﬂecting the symme- try of the persistent current when the magnetic ﬂux is re- versed. For the same model using the grand-canonical en- semble ~keeping the same average number of electrons! would lead to the following result: 2 ^ I & gI 0 5 8 p b D ( l51  e 22l p 2 / b D cos ~ lk F L ! sin S 2l p F F 0 D . ~13! In both cases every harmonic decreases exponentially with temperature. The dominant term is the l5 1 term. We note that this term is proportional to exp @ 2 (2g21) p 2 / b Dg # in the canonical ensemble, but is proportional to exp(2 2 p 2 / b D) in the grand-canonical ensemble. The cur- rent amplitudes are very different even though the occupa- tion probabilities of the levels at high temperatures approach each other under the two ensembles. The two exponential factors agree with each other only in the high-degeneracy limit. We also noted that the amplitude of the harmonics is roughly exp(22l p 2 / b D) in the grand-canonical ensemble. The ratios of these exponential factors are the same for all pairs of adjacent harmonics, whereas in the canonical en- semble the amplitude of the harmonics is roughly exp$
2
p
2
@
lg1k(g 2k)]/
b
Dg
%
, and so the ratios of these ex-
ponential factors are not the same for different pairs of adja-
cent harmonics.
In experiments that measured persistent current,
5–7
a con-
stant magnetic ﬂux and a small time-varying ﬂux are applied.
In this way the amplitude for the ﬁrst few harmonics can be
deduced. The spin-degeneracy is usually lifted by the applied
magnetic ﬁeld. But if the magnetic ﬁeld does not act on the
ring ~only through the empty space enclosed by the ring!
then the levels would still have a spin degeneracy of 2. For
single-ring systems, our results show that if g5 2, the ampli-
tude of the ﬁrst two harmonics (l5 1 and 2! should be pro-
portional to exp(23
p
2
/2
b
D) and exp(22
p
2
/
b
D) in the
canonical ensemble, exp(22
p
2
/
b
D) and exp(2 4
p
2
/
b
D)
in the grand-canonical ensemble. If the amplitudes for the
ﬁrst two harmonics are measured as a function of tempera-
ture, it would be possible to deduce whether canonical or
grand-canonical ensemble is more appropriate for the par-
ticular experimental condition. We believe this could be a
sensitive test.
For systems with many rings, the electron number in each
ring may not be the same. One has an ensemble of rings with
different r, where r can vary from 0 to 2g2 1. Taking an
average over all possible values of r, most of the harmonics
shown in Eq. ~12! vanish. However, some higher-order terms
which are not shown in Eq. ~12! survive. We found that all
the odd harmonics of the average current are zero. All the
nonzero terms are described by l5 2l
8
, where
l
8
5 1,2,3,....Deﬁning p
8
and k
8
by l
8
5 p
8
g1 k
8
where
p
8
5 0,1,2, . . . and k
8
5 1,2, ...,g, the average persistent
current of multiring systems is
^^
I
&&
gI
0
5
(
l
8
51

8
p
l
8
b
Dg
~
C
k
8
g
!
2
11
d
k
8
,g
2p
8
11
~
2p
8
2
13p
8
!
d
k
8
,g
11
3e
~
22
p
2
/
b
Dg
!
@
gl
8
1k
8
~
g2k
8
!
#
sin
4l
8
p
F
F
0
. ~14!
Therefore the period would cross over from F
0
for single-
ring systems to F
0
/2 for multiring systems. This result
agrees with previous analysis
8–10
on g5 1 perfect and disor-
der rings at zero temperature. Analysis starting from Eq. ~9!
shows that this crossover of period is true over the whole
range of temperatures.
The realization of 1D perfect rings using semiconductors
may not be that far away. It would be most interesting to
have experiments on both single-ring and multiring systems
with the same type of rings. Right now the calculation is on
1D perfect rings. Whether similar features occur at higher-
dimensional rings with disorders is still open for investiga-
tion.
ACKNOWLEDGMENT
This work was supported by the Hong Kong UGC Re-
search Grants Council under Grant No. 904033.
1008 53BRIEF REPORTS

*
Permanent address: Centre for Fundamental Physics, University of
Science and Technology of China, Hefei, Anhui 230026, Peo-
ple’s Republic of China.
1
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¨
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2
H. F. Cheung, Y. Gefen, E. K. Riedel, and W. H. Shih, Phys. Rev.
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3
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4
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5
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6
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11
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12
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13
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53
1009BRIEF REPORTS
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Abstract: In this work, I present measurements of persistent currents in normal metal rings performed with cantilever torsional magnetometry. With this technique, the typical persistent current (the component that varies randomly from ring to ring) was measured with high sensitivity. I report measured magnitudes as low as 1 pA, over two orders of magnitude smaller than that observed in previous studies. These measurements extend the range of temperature and magnetic field over which the typical current has been observed. The wide magnetic field range allowed us to study the effect of magnetic field penetrating the ring. It also enabled the recording of many independent measurements of the current magnitude in a single sample. These independent measurements are necessary to characterize the persistent current magnitude because it is a random quantity. From these measurements of the persistent current, I also characterize the parametric dependence of the typical current on sample orientation and number of rings. In addition to presenting the experimental results, I thoroughly review the theory of the typical persistent current in the diffusive regime. I begin with the simplest model and build up to the case appropriate for the samples studied in our experiments. I also present in detail the experimental apparatus used to measure the persistent currents.

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### Cites background from "Persistent current of one-dimension..."

• ...We note that some authors have taken issue with the use of the grand canonical ensemble [48], while others have argued that the calculation in the grand canonical ensemble can be related to the one in the canonical ensemble by making the chemical potential flux dependent so that as the energy levels change with φ the number of occupied levels remains constant [40, 49, 50]....

[...]

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Abstract: We present theoretical studies of temperature dependent diamagnetic-paramagnetic transitions in thin quantum rings. Our studies show that the magnetic susceptibility of metal/semiconductor rings can exhibit multiple sign flips at intermediate and high temperatures depending on the number of conduction electrons in the ring ($N$) and whether or not spin effects are included. When the temperature is increased from absolute zero, the susceptibility begins to flip sign above a characteristic temperature that scales inversely with the number of electrons according to ${N}^{\ensuremath{-}1}$ or ${N}^{\ensuremath{-}1/2}$, depending on the presence of spin effects and the value of $N\phantom{\rule{0.16em}{0ex}}\mathrm{mod}\phantom{\rule{0.16em}{0ex}}4$. Analytical results are derived for the susceptibility in the low and high temperature limits, explicitly showing the spin effects on the ring Curie constant.

### Cites background from "Persistent current of one-dimension..."

• ...As has been wellestablished by now [22, 37–40], it is most appropriate to use the canonical ensemble when the number of particles on the ring is fixed, but it is often a good approximation to treat the ring using a grand canonical ensemble and Fermi-Dirac statistics....

[...]

• ...[40] M....

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##### References
More filters

Book
01 Jan 1902
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902 Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate All the formulas have been checked and many corrections made A complete bibliographical search has been conducted to present the references in modern form for ease of use A new foreword by Professor SJ Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text

8,955 citations

01 Jan 1935
Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity. A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text.

107 citations

##### Frequently Asked Questions (1)
###### Q1. What have the authors contributed in "Persistent current of one-dimensional perfect rings under the canonical ensemble" ?

The variation of the persistent current in oneand higherdimensional perfect and disorder rings as a function of temperature has been studied by Cheung et al. All these studies adopted the grand-canonical ensemble, so the chemical potential is fixed and the electron occupation probability follows the Fermi-Dirac distribution. In this paper the authors present exact analysis on the persistent current on 1D perfect rings under the canonical ensemble. Their calculation logic is to allow the chemical potential to vary with magnetic flux such that the number of electrons remains fixed. At high temperatures, if one approximates the canonicalensemble occupation probability by the Fermi-Dirac distribution with a suitably chosen chemical potential, one might expect to get a reasonable answer for the persistent current.