Oscillating magnetocaloric effect in quantum nanoribbons

Physica E 65 (2015) 44–50
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Physica E
journal homepage: www.elsevier.com/locate/physe
Oscillating magnetocaloric effect in quantum nanoribbons
Z.Z. Alisultanov a,b,c, R.P. Meilanov c,d, L.S. Paixão e,n, M.S. Reis e
a
Amirkhanov Institute of Physics Russian Academy of Sciences, Dagestan Science Centre, Makhachkala, Russia
Prokhorov General Physics Institute Russian Academy of Sciences, Moscow, Russia
Dagestan State University, Makhachkala, Russia
d
Institute of Geothermal Problems Russian Academy of Sciences, Dagestan Science Centre, Makhachkala, Russia
e
Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/n, 24210-346 Niterói, RJ, Brazil
b
c
H I G H L I G H T S
We studied magnetocaloric properties of quantum nanoribbon.
The entropy change of low-dimensional materials exhibits an oscillating behavior.
The model provides a relationship between the confinement potential and the nanoribbon width.
art ic l e i nf o
a b s t r a c t
Article history:
Received 10 June 2014
Received in revised form
23 July 2014
Accepted 20 August 2014
Available online 27 August 2014
We investigate the oscillating magnetocaloric effect on a diamagnetic nanoribbon, using the model of a
quasi-one-dimensional electron gas (Q1DEG) made with a parabolic confinement potential. We obtained
analytical expressions for the thermodynamic potential and for the entropy change. The entropy change
exhibits the same dependence on field and temperature observed for other diamagnetic systems. The
period of the field-oscillating pattern is 0.1 mT and the temperature of maximum entropy change is
0.1 K with an applied field of the order of 1 T. An interesting feature of the results is the dependence of
the oscillations with the strength of the confinement potential, as well as the possibility to provide a
relationship among this last with nanoribbon width. In the limit of null confinement potential our
expressions match those for the 2D diamagnetic system.
& 2014 Elsevier B.V. All rights reserved.
Keywords:
Magnetocaloric effect
Quasi 1D electron gas
Quantum wire
Magnetic oscillation
1. Introduction
Magnetocaloric effect (MCE) is a response of magnetic materials to a magnetic field change ΔB : Bi -Bf , which is related to an
entropy change in the magnetic subsystem. In adiabatic processes
(ΔS ¼ 0), a corresponding entropy change in the other subsystems
leads to a temperature change ΔT. In isothermal processes, the
entropy change is related to a heat exchange ΔQ ¼ T ΔS with a
thermal reservoir. Thus, the effect is characterized by the quantities ΔS and ΔT.
Nowadays, MCE is a hot topic of research, mainly due to its
application in magnetic refrigeration. The idea of using the MCE
for refrigeration purposes was first suggested by Debye [1] and
Giauque [2] in the late 1920s. Research of materials is particularly
focused in magnetically ordered materials, because MCE is stronger in the vicinity of a phase transition. The interested reader is
n
Corresponding author.
E-mail address: [email protected] (L.S. Paixão).
http://dx.doi.org/10.1016/j.physe.2014.08.012
1386-9477/& 2014 Elsevier B.V. All rights reserved.
referred to Refs. [3,4], which are recent reviews on magnetocaloric
effect and magnetocaloric materials applied to refrigeration.
However, magnetocaloric properties of diamagnetic materials
have been studied recently [5–8]. It is shown that both the entropy
change and the temperature change present oscillations when the
applied field is varied. These oscillations are caused by the crossing
of the Landau levels through the Fermi energy; a mechanism
analogous to the so-called de Haas–van Alphen effect. The oscillatory MCE was studied in 3D diamagnets [5,6] (a diamagnetic
material in bulk), in 2D diamagnets of non-relativistic behavior [7]
(a thin film of diamagnetic material), and in graphene [8]. Also, the
effect of the film thickness was reported [9]. The oscillating
magnetocaloric effect of diamagnetic materials is weaker than
that observed in ferromagnets, for instance. Thus, such materials
are not suitable for magnetic refrigeration applications. However,
due to the oscillations, diamagnetic materials could work as highly
sensitive magnetic field sensors [5,6,10].
Another system of reduced dimensionality is a nanoribbon,
which is the realization of a quasi-one-dimensional electron gas
(Q1DEG). Such model was first proposed by Sakaki [11] in 1980 to
describe a medium of high electron mobility, making it applicable
Z.Z. Alisultanov et al. / Physica E 65 (2015) 44–50
in high-speed electronic devices. Indeed, one-dimensional materials can be tailored with oxide interfaces [12] aiming at application
in electronics. The Q1DEG model can also be used to describe a
quantum wire working as an active laser medium [13]. If spin–
orbit coupling is included in the model, a quantum wire exhibits
exotic physics, such as Majorana fermions [14].
In the present work, we theoretically investigate the magnetocaloric effect in quantum nanoribbons, using a Q1DEG. In the next
section we briefly describe the model that will be used, discussing
the energy spectrum of the electron gas. In the following section
we use the energy spectrum to evaluate the grand canonical
potential from which we obtain the entropy. Then, we evaluate
and discuss the entropy change of the Q1DEG. By the end, the
appendix presents details on the evaluations.
45
oscillators are centered at
y0 ¼
p x ωc
;
mω
~2
ð8Þ
and the total system has a energy spectrum similar to the zero-field
case (Eq. (5)), and resumes as
p 2 ω2
1
ð9Þ
ϵ ¼ ϵn;px ¼ x 02 þ ℏω~ n þ :
2
2m ω
~
~ , that,
Note the gap between Landau levels changed from ℏω0 to ℏω
on its turn, depends on both, cyclotron frequency ωc and the strength
of the confinement potential ω0.
The centers of those harmonic oscillators on Eq. (7) must be
confined to the size of the nanoribbon and therefore the condition
0 r y0 r Ly must hold. As a consequence, px is bounded to
ω~ 2
¼ pm :
ωc
2. The model
0 r px r mLy
The present model considers a two dimensional electron gas
confined along one direction due to a lateral potential. This
situation mimics a quantum nanoribbon; and the size quantization
used for the present study is a parabolic approximation given by
On the other hand, the Born–von Karman boundary conditions
impose the quantization of the wave vector kx:
U¼
m 2 2
ω r :
2 0
ð1Þ
The present section then describes the energy spectra of the proposed
model for two cases: with and without applied magnetic field.
px ¼ ℏkx ¼ ℏ
The Hamiltonian of a 2D electron gas confined along the
y direction with a parabolic potential is
m
1 2
px þ p2y þ ω20 y2 :
H¼
ð2Þ
2m
2
The above Hamiltonian has a wave function of the form:
ip x
ψ ðx; yÞ ¼ χ ðyÞ exp x ;
ℏ
that leads to the following Schrödinger equation:
"
#
ℏ2 ∂ 2 m 2 2
p2
þ
ω
y
χ ðyÞ ¼ ϵ x χ ðyÞ;
0
2
2
2m ∂y
2m
in which the energy spectrum is
p2
1
ϵ ¼ ϵn;px ¼ x þ ℏω0 n þ :
2
2m
ð3Þ
ð4Þ
ð5Þ
ð11Þ
where l ¼ 0; 1; 2; …; and, consequently, from this information and
0 r y0 r Ly , it is possible to obtain the multiplicity of the Landau
levels, i.e., the maximum l value:
lmax ¼
2.1. Zero-field case
2π
l;
Lx
ð10Þ
~2
mLx Ly ω
:
2 π ℏ ωc
ð12Þ
3. Grand canonical potential
The grand canonical potential is given by the expression
Z 1
μϵ
dϵ;
Ω ¼ kB T
ρðϵÞ ln 1 þ exp
kB T
0
ð13Þ
where ρðϵÞ is the density of states, T is the temperature, μ is the
chemical potential and kB is the Boltzmann constant. The density
of states of a quantum nanoribbon is given by [15]
Z
1
L
Γ
ρðϵÞ ¼ x2 ∑
dp ;
ð14Þ
2π ℏ n ¼ 0 ðϵ ϵn;p Þ2 þ Γ 2 x
x
where Lx is nanoribbon size along the x-axis and Γ is the width of
Landau levels. Considering Γ ¼0, the density of states resumes as
Z
1
L
ρðϵÞ ¼ x ∑
δðϵ ϵn;px Þ dpx ;
ð15Þ
2π ℏ n ¼ 0
Note thus this system contains a plane wave along the x direction
and quantum harmonic oscillations depending on y. This last
represents the Landau levels and n is the Landau index.
and therefore
2.2. Magnetic field dependence
Due to the structure of the Poisson formula (see Eq. (A.2)), the
grand canonical potential has two contributions:
In the case of an applied magnetic field along the z direction
B ¼ ð0; 0; BÞ, i.e., perpendicular to the 2D electron gas, we can use the
gauge where A ¼ ð By; 0; 0Þ to rewrite the above Hamiltonian as
i m
1 h
H¼
ðpx eByÞ2 þ p2y þ ω20 y2 :
ð6Þ
2m
2
Ω ¼ Ω1 þ Ω2 ;
The wave function is again as the one in Eq. (3) and therefore the
Schrödinger equation reads as
"
#
ℏ2 ∂ 2 m 2
p2 ω2
~ ðy y0 Þ2 χ ðyÞ ¼ ϵ x 0 χ ðyÞ;
þ
ω
ð7Þ
2m ∂y2 2
2m ω
~2
~ 2 ¼ ω2c þ ω20 and ωc ¼ eB=m is the cyclotron frequency of the
where ω
system due to the applied magnetic field. In addition, these harmonic
Ω¼ Lx kB T 1
∑
2π ℏ n ¼ 0
Z
μ ϵn;px
dpx :
ln 1 þ exp
kB T
ð16Þ
ð17Þ
and below these two are described in further detail. The limits on
the px integral depend on the considered case, i.e., either with or
without applied magnetic field.
3.1. Zero-field case
Details on the evaluation of Eq. (16) are in Appendix A; and the
B¼0
B¼0
results for Ω1
and Ω2
are
pffiffiffiffiffiffiffi
pffiffiffi
2 2mL π
ΩB1 ¼ 0 2 x k2B T 2 μ þ const;
ð18Þ
3ℏ ω0 4
46
Z.Z. Alisultanov et al. / Physica E 65 (2015) 44–50
and L(x) is the Langevin function:
and
π
2π k μ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Lx kB T mℏω0 1 ð 1Þk cos ℏω0 4
B¼0
∑
Ω2 ¼
þ const:
2
3=2
4π ℏ
sinh 2πℏωkk0B T
k¼1 k
ð19Þ
The number of charge carriers is determined from
N¼
∂Ω
:
∂μ
ð20Þ
Using Eqs. (18) and (19), we evaluate the number of charge carriers
and then set T¼ 0 to obtain the Fermi level. Thus, neglecting an
oscillating contribution several orders of magnitude smaller than
the main term, we have
pffiffiffiffiffiffiffi
2 2mLx 3=2
ϵ :
ð21Þ
N
3π ℏ2 ω0 F
1
LðxÞ ¼ cothðxÞ :
x
If we set ω0 ¼ 0 in Eq. (29), i.e., if we remove the confining
potential, we recover the result found earlier [7,10] for the entropy
of a 2D electron gas.
From Eqs. (18) and (19) the zero-field magnetic entropy reads
as (see Appendix A)
pffiffiffiffiffiffiffi pffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
π 2mLx ϵF 2 Lx kB mℏω0
SB ¼ 0 k
T
þ
B
4π ℏ
3ℏ2 ω0
1 ð 1Þk
2π k ϵF π
T ðx0k Þ;
cos
ð32Þ
∑
3=2
ℏ ω0
4
k¼1 k
where
x0k ¼
3.2. Magnetic field dependence
Details on the evaluation of Eq. (16) are in Appendix B; and the
results for Ω1 and Ω2 are
Ω1 ¼ 2
~
Lx Ly kB T 2 π m ω
;
ωc
12ℏ2
ð22Þ
and
Ω2 ¼ Lx ϵF
I
Lx kB T
ð 1Þ
∑
;
∑ ð 1Þk k þ
k 2π ℏ k ¼ 1 k sinhðxx Þ
2π 2 ℏ k ¼ 1
n
1
1
k
I kþ
ð23Þ
where
xk ¼
2π 2 kkB T
;
~
ℏω
I nk ¼
Z
pm
sin
0
and
I kþ ¼
Z
ð24Þ
pm
cos
0
2π k p2x ω20
dpx ;
2
~ 2m ω
ℏω
~
ð25Þ
2π k
p2x ω20
dpx :
ϵ
F
~
ℏω
2m ω
~2
ð26Þ
The above results were obtained considering (i) the low
temperature regime and therefore μ ϵF and (ii) a large magnetic
field compared to parameters of the system (see Eq. (B.13)).
4. Oscillating magnetocaloric effect
This section is the aim of the present work. Thus, to obtain the
magnetic entropy change, we first need to evaluate the entropy
with and without magnetic field, from the grand canonical
potentials obtained before. The entropy is determined from
S¼ ∂Ω
;
∂T
ð27Þ
and the magnetic entropy change from
ΔS ¼ S B S B ¼ 0 :
ð28Þ
From Eqs. (22) and (23) the field-dependent magnetic entropy
reads as
~
Lx Ly kB T π mω
2
SB ¼
6ℏ ωc
2
þ
Lx kB 1 ð 1Þk þ
∑
I T ðxk Þ;
2π ℏ k ¼ 1 k k
ð29Þ
where
T ðxk Þ ¼
xk Lðxk Þ
;
sinhðxk Þ
ð30Þ
ð31Þ
2π 2 kkB T
:
ℏ ω0
ð33Þ
If we set ω0 ¼ 0 in Eq. (32), its first term diverges while the second
one vanishes. Although the entropy diverges, its value per area
must remain finite, given by [10]
0 ¼ 0
Sω
π mk2B T
B¼0
¼
:
Lx Ly
6ℏ2
ð34Þ
Comparing Eqs. (32) and (34) we find a relation between the
confining potential and the width of the ribbon:
rffiffiffiffiffiffiffiffi
1 8ϵF
:
ð35Þ
Ly ¼
ω0 m
The magnetic entropy change then reads as
L L πm
ω~
Lx kB 1 ð 1Þk
∑
ΔS ¼ x y 2 k2B T
1 þ
ωc
2π ℏ k ¼ 1 k
6ℏ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
ϵF π
mℏω0
0
T
ðx
cos
2
π
k
Þ
I kþ T ðxk Þ k :
1=2
ℏ ω0 4
2k
ð36Þ
Let us evaluate the entropy change in the limit of an infinite plane.
If we set ω0 ¼ 0 into Eq. (36), the first and last terms vanish. In the
remaining term, the function I kþ becomes (see Eqs. (26) and (B31))
2π k ϵF
;
ð37Þ
I kþ ¼ Ly mωc cos
ℏωc
consequently, we recover the previous result for the entropy
change per area [7]:
ΔSω0 ¼ 0 mωc kB 1 ð 1Þk
2π kϵF
∑
cos
T ðxk Þ:
¼
ð38Þ
Lx Ly
2π ℏ k ¼ 1 k
ℏωc
Fig. 1 presents the entropy change as a function of the inverse
magnetic field, as evaluated from Eq. (36). We see an oscillating
pattern, however the oscillations are asymmetric. The Fermi energy
used was ϵF ¼ 1 eV. The field difference between neighbor peaks is
0.1 mT.
The temperature dependence of the entropy change, presented
in Fig. 2, shows a pronounced maximum at low temperatures,
around 0.1 K. For higher temperatures the entropy change vanishes,
and this behavior is ruled by the function T ðxÞ. An interesting
characteristic of diamagnetic systems, which is not observed in
magnetically ordered materials is that the entropy change can be
positive or negative depending on the applied field.
The strength of the confining potential strongly affects the
entropy change, as shown in Fig. 3. The values of temperature and
field used were chosen to maximize the entropy change, however,
varying ω0 we observe oscillations. When ω0 ¼ 0 the entropy
change matches the value of the 2D system, which corresponds to
a maximum entropy change (negative because of the field).
Z.Z. Alisultanov et al. / Physica E 65 (2015) 44–50
Fig. 1. Field dependence of the entropy change. The dimensionless parameter
α ¼ 2ϵF =ℏωc is inversely proportional to the field.
47
determine the magnetocaloric properties of a quantum nanoribbon. The procedure employed was to evaluate the entropy from
the grand canonical potential. The entropy of the system depends
on the magnetic field, which acts on the magnetic moment of the
electrons, and also depends on temperature, which affects the
motion of the electrons. Here we report the dependence of the
entropy on the strength of the confinement potential, which
basically rules the width of the nanoribbon. Comparing the
entropy per area of the nanoribbon with the entropy of the 2D
electron gas we found a relation between the confining potential
ω0 and the effective width of the nanoribbon Ly (Eq. (35)).
The entropy change as a function of field and temperature
exhibits the same qualitative behavior observed for other diamagnetic systems. We observe an oscillating behavior with the inverse
of the field, with a field difference between neighboring peaks of
0.1 mT; and a pronounced maximum at low temperature, where
the entropy change is maximum around 0.1 K. The dependence of
the entropy change on the confinement potential is also oscillatory. In the limit of ω0 ¼ 0 the entropy change recovers the results
for the 2D electron gas.
Acknowledgments
Brazilian authors thank FAPERJ, CAPES, CNPq and PROPPi-UFF
for the financial support. Russian authors thank the grant from the
President of Dagestan.
Appendix A. Evaluation of zero-field
Ω1 and Ω2
As described in Eq. (16), the grand canonical potential is given by
Z 1
μ ϵn;px
L k T 1
dpx
ΩB ¼ 0 ¼ x B ∑
ln 1 þ exp
ðA:1Þ
2π ℏ n ¼ 0 1
kB T
Fig. 2. Temperature dependence of the entropy change.
where the energy spectrum is given by Eq. (5). The above equation
can be evaluated considering the Poisson formula:
Z 1
Z 1
1
1
f ðxÞ dxþ 2Re ∑
f ðxÞei2π kx dx
ðA:2Þ
∑ f ðnÞ ¼
1=2
n¼0
k¼1
1=2
and we obtain
ΩB ¼ 0 ¼ ΩB1 ¼ 0 þ ΩB2 ¼ 0
where
ΩB1 ¼ 0 ¼ ΩB2 ¼ 0 ¼ Lx kB T
2π ℏ
Z
ðA:3Þ
Z
1
1
dn
0
1
Lx kB T
Re ∑ ð 1Þk
2π ℏ
k¼1
1
Z
1
μ ϵn 1=2;px
dpx
ln 1 þ exp
kB T
ei2π kn dn
Z
1
1
0
ðA:4Þ
μ ϵn 1=2;px
ln 1 þexp
dpx
kB T
ðA:5Þ
B¼0
A.1. Solution to Ω1
Fig. 3. Oscillation of the entropy change as a function of the confining potential.
Integrating by parts over px, n and again over px, we obtain
Z 1
Lx
p4x dpx
ðA:6Þ
ΩB1 ¼ 0 ¼ 2
p2x
2
3 π m ℏ ω0 0
μ
þ1
exp 2m
kB T
5. Conclusions
or
In the present paper, we use a quasi-one-dimensional electron
gas with a parabolic confinement potential. With such model, we
ΩB1 ¼ 0 ¼ pffiffiffiffiffiffiffi Z
3=2
2 2mLx 1 ξ
dξ
2
3π ℏ ω0 0 expξkTμ þ1
B
ðA:7Þ
48
Z.Z. Alisultanov et al. / Physica E 65 (2015) 44–50
Thus, we need to calculate the integral
Z 1
f ðξÞ dξ
I¼
0
exp ξkB Tμ þ 1
we obtain
ðA:8Þ
where f ðξÞ ¼ ξ . Introducing the new variable z ¼ ðξ μÞ=kB T we
obtain
Z 1
f ðμ þ zkB TÞ dz
I ¼ kB T
ez þ 1
μ=kB T
!
Z 1
Z μ=kB T
f ðμ zkB TÞ dz
f ðμ þ zkB TÞ dz
þ
¼ kB T
ðA:9Þ
ez þ1
ez þ 1
0
0
ΩB2 ¼ 0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi
Z 1
1 ð 1Þk e iπ =4
Lx kB T mℏω0
μz
Re ∑
ei2π kz=ℏω0 ln 1 þ exp
dz
2
1=2
kB T
2π ℏ ω0
0
k
k¼1
ðA:21Þ
3=2
Taking into account that
1
1
¼ 1 z
e z þ1
e þ1
ðA:10Þ
we obtain
Z μ
Z
I¼
f ðξÞ dξ þ kB T
1
0
0
ðf ðμ þ zkB TÞ f ðμ zkB TÞÞ
dz
ez þ 1
ðA:11Þ
where we put μ=kB T ¼ 1. Expanding the numerator in the
integrand in a Taylor series, we have
Z μ
Z 1
z dz
2
0
I¼
f ðξÞ dξ þ 2kB T 2 f ðμÞ
ez þ 1
0
0
Z 1 3
4
k T4
z dz
þ⋯
ðA:12Þ
þ B f ‴ðμÞ
ez þ1
3
0
Taking into account that
Z 1 2n 1
z
dz 22n 1 1 2n
¼
π Bn
z þ1
e
2n
0
ðA:13Þ
where Bn is the Bernoulli numbers:
B1 ¼ 16 ;
1
B2 ¼ 30
;
1
B3 ¼ 42
Then
Z μ
4
π2 2 0
7π 4 kB T 4
f ‴ðμÞ þ ⋯
f ðξÞ dξ þ kB T 2 f ðμÞ þ
I¼
6
360
0
Neglecting terms of higher order, we obtain
Z μ
π2 2 0
f ðξÞ dξ þ kB T 2 f ðμÞ
I
6
0
Then
ΩB1 ¼ 0 and
SB1 ¼ 0
!
pffiffiffiffiffiffiffi
5=2
2 2mLx 2ϵF
π 2 2 2 pffiffiffiffiffi
þ
k
T
ϵ
F
5
4 B
3π ℏ2 ω0
ðA:14Þ
ðA:23Þ
ðA:24Þ
where x0k ¼ 2π 2 kkB T=ℏω0 .
Appendix B. Evaluation of field dependent
Ω1 and Ω2
As described in Eq. (16), the grand canonical potential is given by
Z pm μ ϵn;px
L k T 1
dpx
Ω¼ x B ∑
ln 1 þ exp
2π ℏ n ¼ 0 0
kB T
where the energy spectra are given by Eq. (9). Again, the above
equation can be evaluated considered the Poisson formula (see
Eq. (A.2)).
Thus, as mentioned before, the above structure leads to consider
Ω ¼ Ω1 þ Ω2 ;
Ω1 ¼ ðB:1Þ
Lx kB T
2π ℏ
Z
1
1=2
0
μ ϵn;px
dpx dn;
ln 1 þ exp
kB T
ðB:2Þ
Integrating
Ω1 ¼ Ωn ~
Lx ω
4π
ðB:3Þ
Ω1
B.1. Solution to
ðA:20Þ
pm
Z 1 Z pm
1
Lx kB T
Re ∑
πℏ
k ¼ 1 1=2 0
μ ϵn;px
ei2π kn dpx dn;
ln 1 þexp
kB T
ðA:18Þ
Introducing the new variable z ¼ ℏω0 n þ p2x =2m we have (see
Ref. [16])
Z 1
1
L k T
ΩB2 ¼ 0 ¼ x 2B Re ∑ ð 1Þk
ei2π kz=ℏω0
2 π ℏ ω0 k ¼ 1
0
Z 1
2
μz
dz
ln 1 þ exp
dpx e iπ kpx =mℏω0
ðA:19Þ
kB T
1
Z
Ω2 ¼ ðA:17Þ
A.2. Solution to Ω
1
For the SB2 ¼ 0 , consequently, we have
pffiffiffiffiffiffiffiffiffiffiffiffiffi
Lx kB mℏω0 1 ð 1Þk
2π kϵF π
∑
T ðx0k Þ
SB2 ¼ 0 ¼
cos
3=2
4π ℏ
ℏω0
4
k¼1 k
and
ðA:16Þ
B¼0
2
Integrating over px
Z 1
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
dpx e iπ kpx =mℏω0 ¼ e iπ =4 mℏω0 =k
we obtain finally
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
L mℏω μ 1 ð 1Þk L k T mℏω0
ΩB2 ¼ 0 x pffiffiffi 2 0 ∑ 3=2 þ x B
4π ℏ
4 2π ℏ k ¼ 1 k
π
2π kϵF
1 ð 1Þk cos
ℏω0 4
∑
2
3=2
sinh 2πℏωkk0B T
k¼1 k
ðA:22Þ
where
ðA:15Þ
pffiffiffiffiffiffiffi pffiffiffiffiffi
π 2mLx ϵF 2
kB T
3ℏ2 ω0
Integrating by parts and taking into account that
Z 1 iα t
e dt
iπ
;
¼
t
1
þ
e
sinhð
παÞ
1
Ω1 by parts it turns to be
Z
pm
0
Z
1
1=2
1
z dz dpx ;
1 þev
ðB:4Þ
where
v¼
ϵz;px μ
kB T
Ωn ¼
Lx kB T
4π ℏ
¼
Z
p2 ω20
ω~ 2
x
~ ðz þ 1=2Þ þ 2m
ℏω
kB T
pm
0
ln½1 þ e v0 dpx
μ
;
ðB:5Þ
ðB:6Þ
and
v0 ¼
p2x ω20
2m ω
~2
μ
:
kB T
ðB:7Þ
Z.Z. Alisultanov et al. / Physica E 65 (2015) 44–50
A simple trick to go further on this evaluation is to consider the
temperature derivative
Z pm Z 1
~
∂Ω1
∂Ωn Lx ω
v
¼
þ
z dz dpx
ðB:8Þ
2
∂T
∂T
4π T 0
1=2 4cosh v
Considering Eq. (B.5), it is possible to write
(
)
Z 1
2 Z
∂Ω1
∂Ωn
Lx kB T pm
v2
¼
þ
dp
dv
x
2
∂T
∂T
π ℏ2 ω~ 0
v0 cosh ðvÞ
(
)
Z 1
2 ω2 Z pm
kB Lx
px 0
v
~
~
ω
μ
ℏ
ω
=2
dp
dv
þ
x
2
2m ω
~
~2
2π ℏ2 ω
v0 4 cosh ðvÞ
0
ðB:9Þ
From now on, we will consider (i) the case of low temperatures,
i.e.,
μ c kB T
ðB:10Þ
and therefore μ ϵF and (ii)
μc
p2m ω20
2m ω
~2
ðB:11Þ
and therefore
ϵF c
mL2y ω
~ 2 ω20
ðB:12Þ
ω2c
2
In other words, this last assumption implies that the magnetic
field must be
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ℏL ω2
m
ðB:13Þ
μB B c y 0
2
2ϵF mL2y ω2
0
Considering these approximations, it is possible to write v0 - 1
and therefore the integrals on Eq. (B.9) turn to be
Z 1
v
dv ¼ 0
ðB:14Þ
2
1 cosh ðvÞ
Z
1
v2
1
cosh ðvÞ
2
dv ¼
π2
ðB:15Þ
6
In addition, a minor calculation on Eq. (B.6) leads to
∂ Ωn
-0
∂T
ðB:16Þ
Finally, Eq. (B.9) reads as
π m ω~
ωc
6ℏ
Ω
2
∂ 1 Lx Ly kB T
¼
2
∂T
Ω1 ¼ 2
~
L x L y kB T 2 π m ω
ωc
12ℏ2
þ const
ðB:18Þ
Lx Ly kB T 2 π m
where
Z pm
2π k p2x ω20
dpx
exp i
Ik ¼
~ 2m ω
ℏω
~2
0
B.2. Solution to
þ const
ðB:19Þ
Ω2
To evaluate Ω2 on Eq. (B.3) we first consider
1
p 2 ω2
ζ ¼ ℏω~ n þ þ x 02
2
2m ω
~
ðB:23Þ
Note that ω0 ¼ 0 leads to I k ¼ mωc Ly . Furthermore
Z 1 ϵF ζ
2π kζ
exp i
dζ
ln 1 þ exp
~
kB T
ℏω
0
~
ℏω
ϵF
ln 1 þ exp
¼
i2π k
kB T
Z 1
~
ℏω
ζ ϵF 1
2π kζ
þ
dζ
1 þ exp
exp i
~
i2π kkB T 0
kB T
ℏω
ðB:24Þ
and considering the limits of these evaluations, we consider
ϵF
ϵF
ðB:25Þ
ln 1 þ exp
kB T
kB T
and
Z 1
ζ ϵF 1
2π kζ
1 þ exp
exp i
dζ
~
kB T
ℏω
0
Z 1
1
2π kðtkB T þ ϵF Þ
¼ kB T
1 þ et
exp i
dt
~
ℏω
μ=kB T
Z 1
1
2π kϵF
2π kkB Tt
dt
1 þ et
exp i
kB T exp i
~
~
ℏω
ℏω
1
Taking into account that
Z 1 iα t
e dt
iπ
¼
t
sinhðπαÞ
1 1þe
ðB:26Þ
ðB:27Þ
we then obtain this contribution to the grand canonical potential:
Iþ
L x ϵF 1
I n Lx kB T 1 ð 1Þk
k
∑
∑ ð 1Þk k þ
k 2π ℏ k ¼ 1 k sinh 2π 2 kkB T
2π 2 ℏ k ¼ 1
ðB:28Þ
~
ℏω
where
Z pm
2π k p2x ω20
sin
dpx
I nk ¼
2
~ 2m ω
ℏω
~
0
and
I kþ ¼
2
12ℏ2
ðB:21Þ
The values ζ μ ϵF are significant in the oscillating part. Considering therefore the limits imposed on Eqs. (B.10) and (B.12), it is
possible to equate to zero the lower limit in the first integral. Then
Z 1
1
L k T
Ω2 ¼ x 2B Re ∑ ð 1Þk Ik
π ℏ ω~ k ¼ 1
0
ϵF ζ
2π kζ
exp i
dζ ;
ln 1 þ exp
ðB:22Þ
~
kB T
ℏω
ðB:17Þ
Note when ω0 ¼ 0 the above contribution to the grand canonic
potential turns to be
Ω1ω0 ¼ 0 ¼ ~
ϵF ζ
2π k
p2 ω2 ℏω
exp i
dpx dζ ;
ζ x 02 ln 1 þ exp
~
ℏω
kB T
2m ω
2
~
Ω2 ¼ that implies to
49
Z
pm
cos
0
2π k
p2x ω20
ϵ
dpx
F
~
ℏω
2m ω
~2
For the pm -0 limit the above integral resumes as
ω~ 2
2π kϵF
I kþ ¼ mLy
cos
~
ωc
ℏω
ðB:29Þ
ðB:30Þ
ðB:31Þ
References
ðB:20Þ
and then, after a minor calculation, that contribution to the grand
canonical potential can be rewritten as
Z 1
Z pm
1
L k T
Ω2 ¼ x 2B Re ∑
π ℏ ω~ k ¼ 1 ðp2x =2mÞω20 =ω~ 2 0
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