Thermoelectric amplification of phonons in graphene
Amplification of acoustic in-plane phonons due to an external temperature gradient (∇T) in single-layer graphene (SLG) was studied theoretically.
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Thermoelectric amplification of phonons in graphene / K.A. Dompreh, N.G. Mensah, S.Y. Mensah, S.K. Fosuhene // Физика низких температур. — 2016. — Т. 42, № 6. — С. 596-599. — Бібліогр.: 29 назв. — англ. |
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| author | Dompreh, K.A. Mensah, N.G. Mensah, S.Y. Fosuhene, S.K. |
| author_facet | Dompreh, K.A. Mensah, N.G. Mensah, S.Y. Fosuhene, S.K. |
| citation_txt | Thermoelectric amplification of phonons in graphene / K.A. Dompreh, N.G. Mensah, S.Y. Mensah, S.K. Fosuhene // Физика низких температур. — 2016. — Т. 42, № 6. — С. 596-599. — Бібліогр.: 29 назв. — англ. |
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| description | Amplification of acoustic in-plane phonons due to an external temperature gradient (∇T) in single-layer graphene (SLG) was studied theoretically.
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 6, pp. 596–599
Thermoelectric amplification of phonons in graphene
K.A. Dompreh1, N.G. Mensah2, S.Y. Mensah1, and S.K. Fosuhene3
1Department of Physics, College of Agriculture and Natural Sciences, University of Cape Coast, Ghana
2Department of Mathematics, College of Agriculture and Natural Sciences, University of Cape Coast, Ghana
3Ghana Space Science and Technology Institute, Ghana Atomic Energy Commission, Ghana
E-mail: kwadwo.dompreh@ucc.edu.gh
Received October 28, 2015, revised February 2, 2016, published online April 25, 2016
Amplification of acoustic in-plane phonons due to an external temperature gradient ( T∇ ) in single-layer graphene
(SLG) was studied theoretically. The threshold temperature gradient 0( )gT∇ and the threshold voltage 0( )g
TV in SLG
were evaluated. For T = 77 K, the calculated value for 0( ) =gT∇ 746.8 K/cm and 0( ) = 6.6 mV.g
TV The calculation
was done in the hypersound regime. Further, the dependence of the normalized amplification ( 0/Γ Γ ) on the fre-
quency qω and /T T∇ were evaluated numerically and presented graphically. The calculated threshold tempera-
ture gradient 0( )gT∇ for SLG was higher than that obtained for homogeneous semiconductors (n-InSb)
hom
0( )T∇ ≈ 103 K/cm, superlattices 0( ) =SLT∇ 384 K/cm, and cylindrical quantum wire 0( )cqwT∇ ≈ 102 K/cm.
This makes SLG a much better material for thermoelectric phonon amplification.
PACS: 73.22.Pr Electronic structure of graphene;
79.10.–n Thermoelectronic phenomena;
63.22.Rc Phonons in graphene.
Keywords: thermoelectric, graphene, acoustic phonon, amplification.
Introduction
The successful exfoliation of SLG sheets has attracted
lots of research due to its unusual material properties such
as the high carrier mobilities, unusual transport phenomena
characteristic for two-dimensional Dirac fermions [1], the
anomalous integer quantum Hall effect and Shubnikov–de
Haas oscillations that exhibit a phase shift of π due to Ber-
ry’s phase [2]. SLG is characterized by vibrations of two
types of phonons: 1) in-plane vibrations with linear and
longitudinal acoustic branches (LA and TA), and 2) out-of-
plane vibrations known as flexural phonons (ZA and ZO)
[3]. In SLG, these two forms of phonon differ in their cou-
pling to charge carriers: while the coupling is conventional
for in-plane phonons; reflection symmetry demands out-of-
plane displacement for flexurals phonons. Moreover, scat-
tering of Dirac fermions by flexural phonons requires ab-
sorption (or emission) of two phonons which is quadratic
in nature whilst the in-plane phonons have linear disper-
sion. Due to the flexibility of SLG, its flexural mode (also
called the ZA mode, bending mode, or out-of-plane trans-
verse acoustic mode) can not be ignored. SLG is 2D mate-
rial with an in-plane symmetry thus only in-plane phonons
modes can couple linearly to electrons [4]. The amplifica-
tion (absorption) of acoustic phonons in graphene [5–7]
and other low-dimensional materials such as superlattices
[8–11], carbon nanotubes (CNT) [12] and cylindrical quan-
tum wires (CQW) [12] has attracted lots of attention re-
cently. For SLG, Nunes and Fonseca [7] studied amplifica-
tion of acoustic phonons and determined the drift velocity
DV at which amplification occurs but Dompreh et al. [14]
further showed that even at = 0DV , absorption of acoustic
phonons can occur. Acoustoelectric effect (AE) involves
the transfer of momentum from phonons to conducting
charge carriers which leads to the generation of dc current
in the sample. This has been studied both theoretically
[14,15] and experimentally [16] in graphene. The interac-
tion between electrons and phonons in the presence of an
external temperature gradient ( T∇ ) can lead to thermo-
electric effect [17–20] and thermoelectric amplification of
phonons. Thermoelectric amplification of phonons has
been studied in bulk [21,22] and low-dimensional materi-
als such as cylindrical quantum wire (CQW) [23] and
superlattices [24]. This phenomena was predicted by
Gulyeav (1967) [21] but was thoroughly developed by
Sharma and Singh (1974) [25] from a hydrodynamic ap-
proach << 1ql (q is the acoustic wave number, l is the
electron mean free path). Epstein further explained this
effect for sound in the opposite limiting case, >> 1ql and
showed that amplification is also possible in an electrically
open-circuited sample (i.e., in the absence of an electric
© K.A. Dompreh, N.G. Mensah, S.Y. Mensah, and S.K. Fosuhene, 2016
Thermoelectric amplification of phonons in graphene
current) [26]. In n-InSb, Epstein calculated a threshold
temperature gradient of ≈ 103 K/cm at 77 K. However, in
superlattices, Mensah and Kangah (1991) [27] calculated
the threshold temperature gradient necessary for amplifica-
tion to occur to be hom
0 0( ) = 2.6( )SLT T∇ ∇ (where 0( )SLT∇ is
the threshold temperature gradient of superlattice) but in the
lowest energy mini-band ( = 0.1 eV∆ ) obtained
0( ) = 384 K/cmSLT∇ . In all these materials, the threshold
temperature gradient for the amplification was found to
depend on the scattering mechanism and sound frequency
where the relaxation time is independent of energy [24].
Graphene differs significantly from the other low-
dimensional materials. It has the highest value for thermal
conductivity at room temperature (≈ 3000–5000 W/mK)
[28]. This extremely high thermal conductivity opens up a
variety of applications. The most interesting property of
graphene is its linear energy dispersion ( ) = | |Fk V kε ±
(the Fermi velocity 810 cm/sFV ≈ ) at the Fermi level with
low-energy excitation. Conductivity in SLG is restricted by
scattering of impurities but in the absence of extrinsic scat-
tering sources, in-plane phonons constitute an intrinsic
source of scattering of electrons to produce measurable
temperature gradient ( T∇ ) [25]. In the works of Mariani et
al. [3], the threshold temperature below which flexural
phonons or above which in-plane phonons dominate was
calculated to be T = 70 K. To date, there is no study of
thermoelectric amplification of acoustic phonons in SLG.
In this paper, thermoelectric amplification of acoustic pho-
nons is theoretically studied in SLG with degenerate ener-
gy dispersion. Here the threshold temperature gradient
0( )gT∇ above which amplification occur is calculated in
the regime >> 1ql . The paper is organized as follows: In
the theory section, the equation underlying the thermoelec-
tric amplification of acoustic phonon in graphene is pre-
sented. In the numerical analysis section, the final equation
is analyzed and presented in a graphical form. Lastly, the
conclusion is presented in Sec. 4.
Theory
The kinetic equation for the acoustic phonon population
( )N tq in the graphene sheet is given by
2
,
,
( ) 2= | | {[ ( ) 1] (1 )s v
N t
g g C N t f f
t ′ ′
′
∂ π
δ + − ×
∂ ∑q
q k k q k k
k k
( ) ( ) (1 ) ( )}.N t f f′ ′ ′× δ ε − ε + ω − − δ ε − ε − ωk k q q k k k k q
(1)
Here, we have ignored other processes rather than phonon
emission or absorption by electrons. The = = 2s vg g ac-
count for the spin and valley degeneracy, respectively,
( )N tq represent the number of phonons with a wave vector
q at time t . The factor 1N +q accounts for the presence of
Nq phonons in the system when the additional phonon is
emitted. The (1 )f f−k k represent the probability that the
initial k state is occupied and the final electron state ′k is
empty whilst the factor (1 )N f f′ −q k k is that of the boson
and fermions statistics. From [29], the matrix element | |qC
in Eq. (1) is given as
2
2
1 10
02
acoustic phonons,
2
| |=
2
( )( ) optical phonons,
s
q
q
V
C
k k
q
− −
∞
Λ
ρ
π ρω
−
where Λ is the deformation potential constant, ρ is the
crystal density, sV is the sound velocity, 0ω is the frequen-
cy of an optical phonon, 1k−∞ and 1
0k− are, respectively, the
low frequency and optical permeabilities of the crystal. In
Eq. (1), the summation over k and k ′ can be transformed
into integrals by the prescription
2
2 2
4
,
,
(2 )k k
A d kd k
′
′→
π
∑ ∫
where A is the area of the sample. Assuming that
( ) >> 1qN t yields
= ,
N
N
t
∂
Γ
∂
q
q q (2)
where
2 22
3
0 0 0 0
| |= {[ ( ) ( )]
(2 )
q
F s
A q kdk k dk d d f k f k
V V
∞ ∞ π πΛ ′ ′ ′Γ ϕ θ − ×
π ρ ∫ ∫ ∫ ∫
1( ( ))}q
F
k k
V
′×δ − − ω
(3)
with
1= ( )q
F
k k
V
′ − ω
,
ρ is the density of the graphene sheet, ( )f k is the distribu-
tion function and sV is the velocity of sound. Here the
acoustic wave will be considered as phonons of frequency
( qω ) in the short-wave region. The linear approximation of
the distribution function ( )f k is given as
0 1( ) = ( ( )) ( ( ))f k f k f kε + ε , (4)
1( )f k is derived from the Boltzmann transport equation as
0
1
( )
( ( )) = [( ( ) ) ] ( )
f pTf k k v k
T
∂∇
ε τ ε − ξ
∂ε
. (5)
Here ( ) = ( )/v k k k∂ε ∂ is the electron velocity, ξ is the
chemical potential, τ is the relaxation time and T∇ is the
temperature gradient. The unperturbed electron distribution
function is given by the shifted Fermi–Dirac function,
1
0 ( ) = {exp ( ( ) ) 1} ,Ff k k −βε −βε + (6)
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 6 597
K.A. Dompreh, N.G. Mensah, S.Y. Mensah, and S.K. Fosuhene
where = 1/ Bk Tβ ( Bk is the Boltzmann constant, T is the
absolute temperature and Fε is the Fermi energy). At low
temperatures ( << 1Bk T ), =Fε ξ, the Fermi–Dirac equilib-
rium distribution function become
0 ( ( )) = exp ( ( ( ) )).f k kε −β ε − ξ (7)
Inserting Eqs. (5), (6) and (7) into Eq. (4) and setting = 0ξ
for intrinsic graphene, gives
2
2
3
0
| |= ( ){exp ( ( ))
(2 )
q
F
FF s
kA q k V k
VV V
∞ ωΛ
Γ − −β −
π ρ ∫
( ) exp( ) exp( ( ))q
F F F F
F
TV V k V k V k
T V
ω∇
−β τ −β − −β − −
( ( )) exp ( ( ))}q q
F F F
F F
TV V k V k
V T V
ω ω∇
−β τ − −β −
. (8)
Using standard integrals in Eq. (8) and after cumbersome
calculations yield the final equation as
0= {(2 )(1 exp ( ))q qΓ Γ −β ω − −β ω −
[6(1 exp ( )) (2 exp ( ))] }F q q q q
TV
T
∇
−τ + β ω −β ω +β ω β ω ,
(9)
where
2
0 3 3 4
2 | |=
2 F s
A q
V V
Λ
Γ
πβ ρ
. (10)
In Eq. (9), the temperature-dependent parameter is the
diffusion coefficient. Increasing temperature increases the
particle kinetic energy and the diffusion component of the
system. The threshold temperature gradient 0( )gT∇ nec-
essary for diffusion to occur is calculated by letting
= 0Γ as a consequence of the laws of conservation
which yields
0( ) =gT∇
(2 )(1 exp ( ))
[6(1 exp ( )) (2 exp ( ))]
q q
F q q q q
T
V
−β ω − −β ω
=
τ + β ω −β ω +β ω β ω
.
(11)
The 0( )gT∇ is dependent on the temperature (T ), the
frequency ( qω ) and the relaxation time (τ) as well as the
acoustic wavenumber (q). The threshold voltage 0( )g
TV
in graphene is the minimum gate-to-source voltage dif-
ferential that is needed to create a conducting path be-
tween the source and drain terminals. From Eq. (11), the
source-to-drain voltage 0( )gV is deduced as
_____________________________________________________
0 0
[6(1 exp ( )) (2 exp ( ))]
( ) = ( )
(2 )(1 exp ( ))
F q q q qg g
T
q q
k V
V T
e
βτ + β ω −β ω +β ω β ω
∇
−β ω − −β ω
. (12)
________________________________________________
Numerical analysis
To understand the complex expressions in Eqs. (9), (11)
and (12), numerical method was adopted with the follow-
ing parameters used: = 9eVΛ , =sV 2.1⋅103 m/s, =τ
= 5⋅10–10 s, =qω 1.5⋅1012 s–1 and 7 1= 10 mq − . At T = 77 K,
the calculated values for the threshold temperature gradient is
0( ) =gT∇ 746.8 K/cm and threshold voltage 0( ) = 6.6 mV.gV
Using these values, the Eq. (9) is analyzed numerically and
presented graphically (see Figs. 1, 2). Figure 1(a) shows a
normalized graph of 0/Γ Γ on qω for varying T∇ . For
=T∇ 550 K/cm, it was observed that the amplitude of the
absorption graph reduces and the absorption switches
to amplification at = 10 THzqω . Interestingly, when
0> ( )gT T∇ ∇ , that is =T∇ 750 K/cm and 950 K/cm, the
graph switches completely to amplification ( 0/ < 0Γ Γ ) (see
Fig. 1(a)). This agrees with the theory of thermoelectric
amplification of phonons.
Figure 1(b) shows the graph of 0/Γ Γ on various T∇ for
varying qω . The graphs increases to a maximum then de-
creased with the peak shifting to the right for higher values
of qω . This means the temperature gradient ( T∇ ) attains a
maximum at 0/ > 0Γ Γ then decreases. A 3D graph is pre-
sented to further elucidate the graphs obtained (see Fig. 2).
Figure 2 shows the dependence of 0/Γ Γ on T∇ and qω .
Along the T∇ axis, the graph increases to a maximum then
decreases. On the qω axis, the graph attains a maximum at
lower T∇ and a minimum at higher .T∇
Conclusion
Thermoelectric amplification of in-plane phonons in
graphenes is studied. We observed that absorption switches
over to amplification at values greater than the threshold
values. The threshold value calculated at T = 77 K in
graphene is 0( ) =gT∇ 746.8 K/cm and threshold voltage
0( ) = 6.6 mVg
TV . The threshold value is far higher than that
calculated in homogeneous semiconductor using n-InSb
( =sV 2.3⋅105 cm/s at 77 K) and was found to be hom
0( )T∇ ≈
≈ 103 K/cm [21], superlattice 0( )SLT∇ ≈ 384 K/cm for
mini-band width = 0.1 eV∆ at 77 K [24], and finally for cy-
lindrical quantum wires (CQW) to be 0( )CWQT∇ ≈ 102 K/cm
at liquid nitrogen temperature of 77 K [23]. These makes
graphene a much better material for thermoelectric phonon
amplification.
598 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 6
Thermoelectric amplification of phonons in graphene
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Fig. 1. The normalized graph of 0/Γ Γ on qω for varying T∇ (a)
and on T∇ for varying qω (b).
Fig. 2. (Color online) The dependence of 0/Γ Γ on qω and T∇ .
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 6 599
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Introduction
Theory
Numerical analysis
Conclusion
|
| id | nasplib_isofts_kiev_ua-123456789-129136 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T16:37:08Z |
| publishDate | 2016 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Dompreh, K.A. Mensah, N.G. Mensah, S.Y. Fosuhene, S.K. 2018-01-16T15:35:11Z 2018-01-16T15:35:11Z 2016 Thermoelectric amplification of phonons in graphene / K.A. Dompreh, N.G. Mensah, S.Y. Mensah, S.K. Fosuhene // Физика низких температур. — 2016. — Т. 42, № 6. — С. 596-599. — Бібліогр.: 29 назв. — англ. 0132-6414 PACS: 73.22.Pr, 79.10.–n, 3.22.Rc https://nasplib.isofts.kiev.ua/handle/123456789/129136 Amplification of acoustic in-plane phonons due to an external temperature gradient (∇T) in single-layer graphene (SLG) was studied theoretically. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низкоразмерные и неупорядоченные системы Thermoelectric amplification of phonons in graphene Article published earlier |
| spellingShingle | Thermoelectric amplification of phonons in graphene Dompreh, K.A. Mensah, N.G. Mensah, S.Y. Fosuhene, S.K. Низкоразмерные и неупорядоченные системы |
| title | Thermoelectric amplification of phonons in graphene |
| title_full | Thermoelectric amplification of phonons in graphene |
| title_fullStr | Thermoelectric amplification of phonons in graphene |
| title_full_unstemmed | Thermoelectric amplification of phonons in graphene |
| title_short | Thermoelectric amplification of phonons in graphene |
| title_sort | thermoelectric amplification of phonons in graphene |
| topic | Низкоразмерные и неупорядоченные системы |
| topic_facet | Низкоразмерные и неупорядоченные системы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/129136 |
| work_keys_str_mv | AT domprehka thermoelectricamplificationofphononsingraphene AT mensahng thermoelectricamplificationofphononsingraphene AT mensahsy thermoelectricamplificationofphononsingraphene AT fosuhenesk thermoelectricamplificationofphononsingraphene |