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Influence of the electromagnetic wave on the transversal conductivity of the graphene superlattice
Superlattices and Microstructures 60 (2013) 524–532
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Influence of the electromagnetic wave on the transversal conductivity of the graphene superlattice S.V. Kryuchkov a,b,⇑, E.I. Kukhar’ a, O.S. Nikitina a a Volgograd State Socio-Pedagogical University, Physical Laboratory of Low-Dimensional Systems, 27, V.I. Lenin Avenue, 400005 Volgograd, Russia1 b Volgograd State Technical University, 28, V.I. Lenin Avenue, 400005 Volgograd, Russia
a r t i c l e
i n f o
Article history: Received 27 May 2013 Accepted 29 May 2013 Available online 7 June 2013 Keywords: Graphene Superlattice Cnoidal wave
a b s t r a c t The transversal conductivity of the graphene superlattice under electromagnetic radiation and constant electric field applied along the superlattice axis was calculated. The cases of sinusoidal and cnoidal electromagnetic waves are considered. Conductivity dependences on the electromagnetic wave amplitude and on the longitudinal electric field intensity were investigated. Such dependences were shown to have the character of oscillations. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the transport properties of graphene-based structures were intensively studied experimentally and theoretically [1–12]. Such investigations are of fundamental and practical interest [13–18]. The remarkable electronic and optical properties of graphene structures are related with its unusual electron spectrum. In [19,20] the quantum theory of the magneto-optical conductivity was developed. The nonlinear electromagnetic (EM) response of the carbon systems to the excitations of different shapes was investigated in [21] where the frequency multiplication effect was calculated. A quasi-classical kinetic theory of the nonlinear EM response of graphene was developed in
⇑ Corresponding author at: Volgograd State Socio-Pedagogical University, Physical Laboratory of Low-Dimensional Systems, 27, V.I. Lenin Avenue, 400005 Volgograd, Russia. E-mail addresses: [email protected] (S.V. Kryuchkov), [email protected] (E.I. Kukhar’), [email protected] (O.S. Nikitina). 1 http://edu.vspu.ru/physlablds 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.05.036
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[22–24]. Also in [22–24] the possible applications of the predicted effects for generation of terahertz radiation were discussed. The so-called mutual rectification of the EM waves in graphene was studied in [25–27]. New opportunities for building of the optoelectronic devices can be opened by the predicted nonlinear optical properties of the graphene-based structures. The electron spectrum of the graphene with the additional periodic potential was studied in [28–36]. The band structure of graphene superlattice (GSL) with periodically arranged rows of vacancies and lines of adsorbed hydrogen atoms was calculated in [28,29] correspondingly. In [32] a new method to obtain the superlattice (SL) potential in the graphene was proposed where the last was suggested to be deposited on the periodic substrate h-BN/SiO2. Dispersion low of GSL was studied theoretically in [32–36]. Intensive investigations of the electronic and optical features of carbon structures (in particularly – graphene structures) with additional SL potential are of interest due to the so-called Bloch oscillator problem [37–39]. Moreover SL is the suitable medium for the formation of the nonlinear and solitary EM waves [40–44]. For instance to form the cnoidal waves and the solitons in the semiconductor SL a relatively small electric fields (103 V/cm) in compared with bulk semiconductors [40–42] are required. That is why the structures with SL are of fundamental and practical interest [45–48]. The optical properties of GSL were investigated in [34,35,49–51], where the constant electric current induced in GSL by the EM waves of different polarization was calculated. In [35,50] one of the waves was of cnoidal form. The negative differential conductivity and absolute negative conductivity of the GSL axis was shown to be possible under the simultaneous action of EM wave polarized elliptically and constant electric field applied along the GSL axis. In [37] the alternate electric field influence on the longitudinal current–voltage characteristics of the semiconductor SL was studied. The GSL current–voltage characteristic along the GSL axis in the presence of the alternate electric field was calculated in [34]. In experiment, however, it is very difficult to orient the electric field vector along the SL axis strictly. There always exists a small transverse component of the applied electric field intensity. The problem of non-coincidence of the electric field intensity with certain crystallographic axis was formulated in [52]. In [53] the dependence of the spectrum of Bloch oscillations on the orientation of an applied electric field was discussed. Thus, it was of interest to calculate the transversal conductivity of the GSL in the presence of a weak transverse electric field in addition to a strong constant electric field applied along the GSL axis and EM radiation. The transverse field component can either be related to uncertainly of the field orientation along the GSL axis or it can be specially applied so as to obey certain conditions. Such conditions lead to the constant component of the electric current appearing in the GSL plane transversally to the GSL axis. The dependence of the transversal conductivity on the longitudinal electric field intensity is shown below to have a resonant character. 2. Transversal conductivity of GSL in s-approximation GSL is considered to be obtained by a sheet of graphene deposited on a banded substrate formed by periodically alternating layers of SiO2 and SiC. The layers are arranged so that the hexagonal crystal lattice of SiC was under the hexagonal lattice of graphene. Due to this, in the areas of graphene plane located above the layers of SiC an energy gap 2D = 0.26 eV arises [2,34]. Let the graphene is located on the plane xz. The electron spectrum of GSL has the approximate view [34,35]:
eðpÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e21 pd 1 cos z e20 þ p2x t2F þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h e20 þ p2x t2F
ð1Þ
where e0 = 0.059 eV, e1 = 0.029 eV, d = 2 106 cm is the SL period, tF = 108 cm/s is the Fermi surface velocity, Oz is the SL axis. The vector potential A of the EM wave is considered to be applied along the SL axis (Fig. 1). The constant electric field vector is E ¼ ðEx ; 0; Ez Þ. The mean free path of electrons is assumed to be much shorter than the wavelength. This allows us to neglect the coordinate dependence of the EM fields and the distribution function. Under these conditions the electric current
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Fig. 1. Schematic of the process.
density arising through the axis Ox in the constant relaxation time s approximation is calculated with the following formula:
jx ðtÞ ¼
e
Z
s
t
tt 0
dt 0 e
s
1
X e f0 ðpÞV x ðp þ ðAðtÞ Aðt 0 ÞÞ eEðt t0 ÞÞ c p
ð2Þ
where f0 ðpÞ is equilibrium state function. The electron velocity along the axis Ox is:
tF qx c2 tF ð1 cos qz Þ ffiþ VðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 1 þ q2x ð1 þ q2x Þ
ð3Þ
Here we introduce a new notations: qx ¼ px tF =e0 , qz ¼ pz d= h, c ¼ e1 =e0 . After substitution (3) into (2) we write 0
jx ðtÞ ¼
þ
Z
X qx bðt t0 Þ B f0 ðpÞ@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 p 1 þ ðqx bðt t 0 ÞÞ 1 c2 ðqx bðt t0 ÞÞ cosðqz xB ðt t0 Þ þ a a0 ÞA etF
t
dt0 e
s
tt0
s
2 3=2
ð1 þ ðqx bðt t 0 ÞÞ Þ
ð4Þ
where xB ¼ eEz d= h is frequency of Bloch oscillations, aðtÞ ¼ edAðtÞ=c h, a0 ¼ aðt0 Þ, b ¼ etF Ex =e0 . Further the transversal field is considered to be weak: bs 1. Taking into account the evenness of the equilibrium distribution function, we find that, in the first approximation with respect to the parameter Ex , formula (4) takes the form:
jx ðtÞ ¼
e2 t2F Ex
e0 s
Z
t
Z
t
dt0 e
tt0
s
ðt t0 Þ
1
p
0 tt
dt 0 e
X
s
2 c2 e2 t2F Ex 2 1 2qx þ 1 c 3=2 2 2 1 þ q e0 s ð1 þ qx Þ x f0 ðpÞ
ðt t 0 Þ cosðxB ðt t0 Þ a þ a0 Þ
1
X ð1 2q2x Þ cos qz f0 ðpÞ 5=2 ð1 þ q2x Þ p
ð5Þ
If the next condition is performed:
exp
e l 0 1 h
ð6Þ
where l is the chemical potential, then the electron gas is nondegenerate [54]. In this case the equilibrium state function f0 ðpÞ takes the form of the Boltzmann state function. For the gap modification of graphene the inequality (6) leads to the next condition:
h
n0 h
2 2 F
e0
t
ð7Þ
Here n0 is the surface concentration of free electrons of the GSL. After calculation of the sums by momentums, we obtain
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jx ðtÞ ¼
e2 t2F Ex F 1 ðc; e0 =hÞ e0 s F 0 ðc; e0 =hÞ
Z
t
tt 0
dt 0 e
s
ðt t 0 Þ þ
1
c2 e2 t2F Ex F 2 ðc; e0 =hÞ e0 s F 0 ðc; e0 =hÞ
Z
t
dt 0 e
tt0
s
ðt t 0 Þ
1
cosðxB ðt t 0 Þ a þ a0 Þ
ð8Þ
where
F 0 ðc; e0 =hÞ ¼
Z
1
0
¼
Z
1
0
¼
Z
e0 c2 e0 1 c2 þ z2 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi dz; F 1 ðc; e0 =hÞ exp h 1 þ z2 h 1 þ z2 2 2 2 1 c þ ð1 þ 2c Þz e0 c2 e0 1 c2 þ z2 p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi exp dz; F 2 ðc; e0 =hÞ I 0 5=2 h h 1 þ z2 1 þ z2 ð1 þ z2 Þ 1 2z2 e0 c2 e0 1 c2 þ z2 p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi exp dz I 1 5=2 h h 1 þ z2 1 þ z2 ð1 þ z2 Þ
I0
1
0
In ðxÞ is the modified Bessel function, h is the electron gas temperature in energy units. At low temperatures:
h e0
ð9Þ
we can write instead of (8):
Z jx ðtÞ ¼ r? Ex 1 c2 þ c2
1
xex cosðxB sx aðtÞ þ aðt sxÞÞdx
ð10Þ
0
where r? ¼ n0 e2 t2F s=e0 is the GSL conductivity in the direction perpendicular to the GSL axis in the absence of electric fields applied along the GSL axis. For the following parameters values [1,2]: n0 = 1010 cm2, D = 0.13 eV, tF = 108 cm/s, inequalities (7) and (9) allow for formula (16) to be used if the temperature is within the range: 10 K h 103 K. Boiling point of liquid nitrogen (h 77.4 K) is suitable. Further we consider two cases for the orientation of the EM wave polarization plane. In the case when the polarization plane is perpendicular to the GSL plane the EM wave is considered to be of sinusoidal form. If the EM wave polarization plane coincides with the GSL plane then it has the form of cnoidal wave [35,44,50]. 3. Transversal conductivity of GSL under the sinusoidal EM wave At first we consider the case of sinusoidal EM wave. The vector potential of the EM wave is
cE0 AðtÞ ¼ 0; 0; sin xt
ð11Þ
x
where E0 and x are the amplitude and the frequency of the EM wave correspondingly. After the substitution (11) into (10) and averaging of the current density over the wave period we have 2
hjx i ¼ r? Ex 1 c þ c
2
1 X
x x
xÞ2 s2 2 xÞ2 s2 Þ
1ð Bþn J 2n ðaÞ n¼1 ð1 þ ð B þ n
! ð12Þ
The transversal conductivity of GSL is
rxx ¼ r? 1 c2 þ c2 J20 ðaÞ
1 x2B s2 2 2 2 Þ B
ð1 þ x
s
1 X J 2n ðaÞ n¼1
s 1 ðxB nxÞ2 s2 þ 2 2 ð1 þ ðxB þ nxÞ2 s2 Þ ð1 þ ðxB nxÞ2 s2 Þ 1 ðxB þ nxÞ
2 2
þ c2
!! ð13Þ
Here a ¼ eE0 d= hx, J n ðxÞ is the Bessel function of whole order. Graphs of dependences of the GSL transverse conductivity on the longitudinal electric field intensity and on the EM wave amplitude are shown in Figs. 2 and 3 correspondingly. The dependence of rxx on the longitudinal electric field
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Fig. 2. Dependence of the GSL transverse conductivity on the longitudinal electric field intensity. xs = 10.
Fig. 3. Dependence of the GSL transverse conductivity on the EM wave amplitude. xs = 10.
intensity is seen from Fig. 2 to have the view of the alternating resonances. The dependence of rxx on the wave amplitude is seen from Fig. 3 to have the character of oscillations. Such effect of the electric fields applied along the GSL axis on the transverse conductivity is due to the nonadditivity of the electronic spectrum of GSL (1). In the absence of the longitudinal component of the constant electric field ðEz ¼ 0Þ the transversal conductivity of the GSL is equal
rxx ¼ r? 1 c2 ð1 J20 ðaÞÞ þ 2c2
1 X 1 n2 x2 s2 J 2n ðaÞ 2 ð1 þ n2 x2 s2 Þ n¼1
! ð14Þ
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In the absence of the EM wave ða ¼ 0Þ we obtain for GSL transversal conductivity: 2
rxx ¼ r? 1 c
ð3 þ x2B s2 Þx2B s2 ð1 þ x2B s2 Þ
! ð15Þ
2
4. Transversal conductivity of GSL under the cnoidal EM wave Now we consider the case when the EM wave propagates in direction perpendicular to the GSL axis so that polarization plane of this wave coincides with the GSL plane. In this situation the EM wave is of cnoidal form [35,44,50]. To calculate the transversal conductivity of the GSL exposed to the cnoidal EM wave we substitute the following Fourier series expansion in the formula (10): 1 X
eiaðtÞ ¼
cn ðjÞeinXðjÞt
ð16Þ
n¼1
where
2EðjÞ np2 1 pX0 j ; if 0 < j 6 1XðjÞ ¼ ; 1; cn ¼ 2 n=2 nþ1 n=2 KðjÞ Kð j1 Þ K ðjÞ qj þ ð1Þ qj 2j2 Eðj1 Þ 4np2 j2 1 eE0 d ; cn ¼ 2 ; if j > 1; j ¼ ; c0 ¼ 1 2j2 þ 3n 1 Kðj Þ 2hX0 K ðj1 Þ qn 1=j q1=j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! xpl u u2 pKð 1 j2 Þ 2ede1 pn0 F 3 ðc; e0 =hÞ ; X0 ¼ F ðc; e0 =hÞ qj ¼ exp 1 c2 ; xpl ¼ h KðjÞ e0 F 0 ðc; e0 =hÞ 3 c Z 1 1 e0 c2 e0 1 c2 þ z2 pffiffiffiffiffiffiffiffiffiffiffiffiffi I1 pffiffiffiffiffiffiffiffiffiffiffiffiffi exp pffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ 2 2 h h 1þz 1þz 1 þ z2 0
XðjÞ ¼
pX0
2KðjÞ
; c0 ¼
E0 is the amplitude of cnoidal wave, u is the wave velocity, c is the wave velocity in the absence of electrons, KðxÞ b EðxÞ are complete elliptic integrals of the first and second kind correspondingly. After substituting (16) into (10), calculation of integrals, and averaging over the wave period, we obtain the next formula for the GSL transversal conductivity at low temperatures (9):
0
rxx
0 11 2 2 2 2 1 2 2 2 2 X 1 x s c ð1 ð x þ n X Þ s Þ c ð1 ð x n X Þ s Þ B B CC B B 2 B ¼ r? @1 c2 þ c2 c20 þ n @ n þc 2 AA 2 2 2 2 2 2 2 1 þ xB s ð1 þ ðxB þ nXÞ s Þ n¼1 1 þ ðxB nXÞ s2 ð17Þ
In the case 0 < j 6 1 we obtain
0 11 2 2 2 2 1 X 1x s 1 ðxB nXÞ s CC B 1 ðxB þ nXÞ s B ¼ r? @1 c2 þ c2 c20 þ c2 c2n @ 2 þ 2 AA 2 2 2 ð1 þ x2B s2 Þ n¼1 1 þ ðxB þ nXÞ s 1 þ ðxB nXÞ2 s2 0
rxx
2 2 B
ð18Þ Graphs of dependences of the GSL transverse conductivity on the longitudinal electric field intensity plotted with the formula (18) are shown in Fig. 4. In the absence of the longitudinal component of the constant electric field ðEZ ¼ 0Þ we obtain the next formula for the transversal conductivity of the GSL:
rxx ¼ r? 1 c2 ð1 c20 Þ þ 2c2
1 X 1 n2 X2 s2 c2n 2 ð1 þ n2 X2 s2 Þ n¼1
!
In the absence of the Em wave ðj ¼ 0Þ formula (18) gives (15). In the case j > 1 we have
ð19Þ
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Fig. 4. Dependence of the GSL transverse conductivity on the longitudinal electric field intensity. O0s = 5, 0 6 j < 1.
rxx ¼ r? 1 c2 þ c2 c20
1 X 1 ðxB þ nXÞ2 s2 1 ðxB nXÞ2 s2 4n þ c2 c2n þ q 2 1=j 2 2 2 2 2 2 ð1 þ xB s Þ n¼1 ð1 þ ðxB þ nXÞ s Þ ð1 þ ðxB nXÞ2 s2 Þ
1 x2B s2
!!
ð20Þ Graphs of dependences of the GSL transverse conductivity on the longitudinal electric field intensity and on the EM wave amplitude plotted with the formula (20) are shown in Fig. 5. Dependences of the GSL transverse conductivity on the cnoidal EM wave amplitude plotted with the formulas (18) and (20) are shown in Fig. 6. In the absence of the longitudinal component of the constant electric field ðEz ¼ 0Þ the transversal conductivity is
Fig. 5. Dependence of the GSL transverse conductivity on the longitudinal electric field intensity. O0s = 5, j > 1.
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531
Fig. 6. Dependence of the GSL transverse conductivity on the EM wave amplitude. O0s = 5.
2
rxx ¼ r? 1 c ð1
c20 Þ
1 X 1 n2 X2 s2 þc c2n ð1 þ q4n 1=j Þ 2 ð1 þ n2 X2 s2 Þ n¼1
!
2
ð21Þ
5. Discussion The dependence of rxx on the longitudinal electric field intensity is seen from Figs. 2, 4 and 5 to have the view of the alternating resonances. In the case of sinusoidal wave conductivity peaks occur when the next condition is performed:
xB ¼ nx where n is the whole number. In the case of cnoidal wave conductivity peaks occur if
xB ¼ nXðjÞ
ð22Þ
As can be seen from Figs. 4 and 5 and from the condition (22) in contrast to sinusoidal EM waves the positions of conductivity resonances can be regulated by changing of the amplitude of cnoidal EM wave. In the case j > 1 the conductivity rxx dependence on the longitudinal component of electric field vector can be seen from Fig. 5 to be not symmetric, unlike the case of 0 < j 6 1 (Fig. 4). This fact is due to the constant component of electric field of cnoidal wave in the case when j > 1. The dependence of rxx on the wave amplitude is seen from Figs. 3 and 6 to have the character of oscillations. The possibility of the transversal conductivity resonances and oscillations is explained by the non-additivity of the GSL spectrum (1). In the bulk SL with additive spectrum such effects are impossible. Acknowledgments The work was supported by the RFBR Grant No. 13-02-97033 r_povolzhye_a, and by the project of state jobs to the research work of the Ministry of Education and Science of the Russian Federation in 2013 No. 2.8298.2013.
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Influence of the electromagnetic wave on the transversal conductivity of the graphene superlattice S.V. Kryuchkov a,b,⇑, E.I. Kukhar’ a, O.S. Nikitina a a Volgograd State Socio-Pedagogical University, Physical Laboratory of Low-Dimensional Systems, 27, V.I. Lenin Avenue, 400005 Volgograd, Russia1 b Volgograd State Technical University, 28, V.I. Lenin Avenue, 400005 Volgograd, Russia
a r t i c l e
i n f o
Article history: Received 27 May 2013 Accepted 29 May 2013 Available online 7 June 2013 Keywords: Graphene Superlattice Cnoidal wave
a b s t r a c t The transversal conductivity of the graphene superlattice under electromagnetic radiation and constant electric field applied along the superlattice axis was calculated. The cases of sinusoidal and cnoidal electromagnetic waves are considered. Conductivity dependences on the electromagnetic wave amplitude and on the longitudinal electric field intensity were investigated. Such dependences were shown to have the character of oscillations. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the transport properties of graphene-based structures were intensively studied experimentally and theoretically [1–12]. Such investigations are of fundamental and practical interest [13–18]. The remarkable electronic and optical properties of graphene structures are related with its unusual electron spectrum. In [19,20] the quantum theory of the magneto-optical conductivity was developed. The nonlinear electromagnetic (EM) response of the carbon systems to the excitations of different shapes was investigated in [21] where the frequency multiplication effect was calculated. A quasi-classical kinetic theory of the nonlinear EM response of graphene was developed in
⇑ Corresponding author at: Volgograd State Socio-Pedagogical University, Physical Laboratory of Low-Dimensional Systems, 27, V.I. Lenin Avenue, 400005 Volgograd, Russia. E-mail addresses: [email protected] (S.V. Kryuchkov), [email protected] (E.I. Kukhar’), [email protected] (O.S. Nikitina). 1 http://edu.vspu.ru/physlablds 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.05.036
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[22–24]. Also in [22–24] the possible applications of the predicted effects for generation of terahertz radiation were discussed. The so-called mutual rectification of the EM waves in graphene was studied in [25–27]. New opportunities for building of the optoelectronic devices can be opened by the predicted nonlinear optical properties of the graphene-based structures. The electron spectrum of the graphene with the additional periodic potential was studied in [28–36]. The band structure of graphene superlattice (GSL) with periodically arranged rows of vacancies and lines of adsorbed hydrogen atoms was calculated in [28,29] correspondingly. In [32] a new method to obtain the superlattice (SL) potential in the graphene was proposed where the last was suggested to be deposited on the periodic substrate h-BN/SiO2. Dispersion low of GSL was studied theoretically in [32–36]. Intensive investigations of the electronic and optical features of carbon structures (in particularly – graphene structures) with additional SL potential are of interest due to the so-called Bloch oscillator problem [37–39]. Moreover SL is the suitable medium for the formation of the nonlinear and solitary EM waves [40–44]. For instance to form the cnoidal waves and the solitons in the semiconductor SL a relatively small electric fields (103 V/cm) in compared with bulk semiconductors [40–42] are required. That is why the structures with SL are of fundamental and practical interest [45–48]. The optical properties of GSL were investigated in [34,35,49–51], where the constant electric current induced in GSL by the EM waves of different polarization was calculated. In [35,50] one of the waves was of cnoidal form. The negative differential conductivity and absolute negative conductivity of the GSL axis was shown to be possible under the simultaneous action of EM wave polarized elliptically and constant electric field applied along the GSL axis. In [37] the alternate electric field influence on the longitudinal current–voltage characteristics of the semiconductor SL was studied. The GSL current–voltage characteristic along the GSL axis in the presence of the alternate electric field was calculated in [34]. In experiment, however, it is very difficult to orient the electric field vector along the SL axis strictly. There always exists a small transverse component of the applied electric field intensity. The problem of non-coincidence of the electric field intensity with certain crystallographic axis was formulated in [52]. In [53] the dependence of the spectrum of Bloch oscillations on the orientation of an applied electric field was discussed. Thus, it was of interest to calculate the transversal conductivity of the GSL in the presence of a weak transverse electric field in addition to a strong constant electric field applied along the GSL axis and EM radiation. The transverse field component can either be related to uncertainly of the field orientation along the GSL axis or it can be specially applied so as to obey certain conditions. Such conditions lead to the constant component of the electric current appearing in the GSL plane transversally to the GSL axis. The dependence of the transversal conductivity on the longitudinal electric field intensity is shown below to have a resonant character. 2. Transversal conductivity of GSL in s-approximation GSL is considered to be obtained by a sheet of graphene deposited on a banded substrate formed by periodically alternating layers of SiO2 and SiC. The layers are arranged so that the hexagonal crystal lattice of SiC was under the hexagonal lattice of graphene. Due to this, in the areas of graphene plane located above the layers of SiC an energy gap 2D = 0.26 eV arises [2,34]. Let the graphene is located on the plane xz. The electron spectrum of GSL has the approximate view [34,35]:
eðpÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e21 pd 1 cos z e20 þ p2x t2F þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h e20 þ p2x t2F
ð1Þ
where e0 = 0.059 eV, e1 = 0.029 eV, d = 2 106 cm is the SL period, tF = 108 cm/s is the Fermi surface velocity, Oz is the SL axis. The vector potential A of the EM wave is considered to be applied along the SL axis (Fig. 1). The constant electric field vector is E ¼ ðEx ; 0; Ez Þ. The mean free path of electrons is assumed to be much shorter than the wavelength. This allows us to neglect the coordinate dependence of the EM fields and the distribution function. Under these conditions the electric current
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Fig. 1. Schematic of the process.
density arising through the axis Ox in the constant relaxation time s approximation is calculated with the following formula:
jx ðtÞ ¼
e
Z
s
t
tt 0
dt 0 e
s
1
X e f0 ðpÞV x ðp þ ðAðtÞ Aðt 0 ÞÞ eEðt t0 ÞÞ c p
ð2Þ
where f0 ðpÞ is equilibrium state function. The electron velocity along the axis Ox is:
tF qx c2 tF ð1 cos qz Þ ffiþ VðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 3=2 1 þ q2x ð1 þ q2x Þ
ð3Þ
Here we introduce a new notations: qx ¼ px tF =e0 , qz ¼ pz d= h, c ¼ e1 =e0 . After substitution (3) into (2) we write 0
jx ðtÞ ¼
þ
Z
X qx bðt t0 Þ B f0 ðpÞ@qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 p 1 þ ðqx bðt t 0 ÞÞ 1 c2 ðqx bðt t0 ÞÞ cosðqz xB ðt t0 Þ þ a a0 ÞA etF
t
dt0 e
s
tt0
s
2 3=2
ð1 þ ðqx bðt t 0 ÞÞ Þ
ð4Þ
where xB ¼ eEz d= h is frequency of Bloch oscillations, aðtÞ ¼ edAðtÞ=c h, a0 ¼ aðt0 Þ, b ¼ etF Ex =e0 . Further the transversal field is considered to be weak: bs 1. Taking into account the evenness of the equilibrium distribution function, we find that, in the first approximation with respect to the parameter Ex , formula (4) takes the form:
jx ðtÞ ¼
e2 t2F Ex
e0 s
Z
t
Z
t
dt0 e
tt0
s
ðt t0 Þ
1
p
0 tt
dt 0 e
X
s
2 c2 e2 t2F Ex 2 1 2qx þ 1 c 3=2 2 2 1 þ q e0 s ð1 þ qx Þ x f0 ðpÞ
ðt t 0 Þ cosðxB ðt t0 Þ a þ a0 Þ
1
X ð1 2q2x Þ cos qz f0 ðpÞ 5=2 ð1 þ q2x Þ p
ð5Þ
If the next condition is performed:
exp
e l 0 1 h
ð6Þ
where l is the chemical potential, then the electron gas is nondegenerate [54]. In this case the equilibrium state function f0 ðpÞ takes the form of the Boltzmann state function. For the gap modification of graphene the inequality (6) leads to the next condition:
h
n0 h
2 2 F
e0
t
ð7Þ
Here n0 is the surface concentration of free electrons of the GSL. After calculation of the sums by momentums, we obtain
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jx ðtÞ ¼
e2 t2F Ex F 1 ðc; e0 =hÞ e0 s F 0 ðc; e0 =hÞ
Z
t
tt 0
dt 0 e
s
ðt t 0 Þ þ
1
c2 e2 t2F Ex F 2 ðc; e0 =hÞ e0 s F 0 ðc; e0 =hÞ
Z
t
dt 0 e
tt0
s
ðt t 0 Þ
1
cosðxB ðt t 0 Þ a þ a0 Þ
ð8Þ
where
F 0 ðc; e0 =hÞ ¼
Z
1
0
¼
Z
1
0
¼
Z
e0 c2 e0 1 c2 þ z2 pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi dz; F 1 ðc; e0 =hÞ exp h 1 þ z2 h 1 þ z2 2 2 2 1 c þ ð1 þ 2c Þz e0 c2 e0 1 c2 þ z2 p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi exp dz; F 2 ðc; e0 =hÞ I 0 5=2 h h 1 þ z2 1 þ z2 ð1 þ z2 Þ 1 2z2 e0 c2 e0 1 c2 þ z2 p ffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffi exp dz I 1 5=2 h h 1 þ z2 1 þ z2 ð1 þ z2 Þ
I0
1
0
In ðxÞ is the modified Bessel function, h is the electron gas temperature in energy units. At low temperatures:
h e0
ð9Þ
we can write instead of (8):
Z jx ðtÞ ¼ r? Ex 1 c2 þ c2
1
xex cosðxB sx aðtÞ þ aðt sxÞÞdx
ð10Þ
0
where r? ¼ n0 e2 t2F s=e0 is the GSL conductivity in the direction perpendicular to the GSL axis in the absence of electric fields applied along the GSL axis. For the following parameters values [1,2]: n0 = 1010 cm2, D = 0.13 eV, tF = 108 cm/s, inequalities (7) and (9) allow for formula (16) to be used if the temperature is within the range: 10 K h 103 K. Boiling point of liquid nitrogen (h 77.4 K) is suitable. Further we consider two cases for the orientation of the EM wave polarization plane. In the case when the polarization plane is perpendicular to the GSL plane the EM wave is considered to be of sinusoidal form. If the EM wave polarization plane coincides with the GSL plane then it has the form of cnoidal wave [35,44,50]. 3. Transversal conductivity of GSL under the sinusoidal EM wave At first we consider the case of sinusoidal EM wave. The vector potential of the EM wave is
cE0 AðtÞ ¼ 0; 0; sin xt
ð11Þ
x
where E0 and x are the amplitude and the frequency of the EM wave correspondingly. After the substitution (11) into (10) and averaging of the current density over the wave period we have 2
hjx i ¼ r? Ex 1 c þ c
2
1 X
x x
xÞ2 s2 2 xÞ2 s2 Þ
1ð Bþn J 2n ðaÞ n¼1 ð1 þ ð B þ n
! ð12Þ
The transversal conductivity of GSL is
rxx ¼ r? 1 c2 þ c2 J20 ðaÞ
1 x2B s2 2 2 2 Þ B
ð1 þ x
s
1 X J 2n ðaÞ n¼1
s 1 ðxB nxÞ2 s2 þ 2 2 ð1 þ ðxB þ nxÞ2 s2 Þ ð1 þ ðxB nxÞ2 s2 Þ 1 ðxB þ nxÞ
2 2
þ c2
!! ð13Þ
Here a ¼ eE0 d= hx, J n ðxÞ is the Bessel function of whole order. Graphs of dependences of the GSL transverse conductivity on the longitudinal electric field intensity and on the EM wave amplitude are shown in Figs. 2 and 3 correspondingly. The dependence of rxx on the longitudinal electric field
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Fig. 2. Dependence of the GSL transverse conductivity on the longitudinal electric field intensity. xs = 10.
Fig. 3. Dependence of the GSL transverse conductivity on the EM wave amplitude. xs = 10.
intensity is seen from Fig. 2 to have the view of the alternating resonances. The dependence of rxx on the wave amplitude is seen from Fig. 3 to have the character of oscillations. Such effect of the electric fields applied along the GSL axis on the transverse conductivity is due to the nonadditivity of the electronic spectrum of GSL (1). In the absence of the longitudinal component of the constant electric field ðEz ¼ 0Þ the transversal conductivity of the GSL is equal
rxx ¼ r? 1 c2 ð1 J20 ðaÞÞ þ 2c2
1 X 1 n2 x2 s2 J 2n ðaÞ 2 ð1 þ n2 x2 s2 Þ n¼1
! ð14Þ
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529
In the absence of the EM wave ða ¼ 0Þ we obtain for GSL transversal conductivity: 2
rxx ¼ r? 1 c
ð3 þ x2B s2 Þx2B s2 ð1 þ x2B s2 Þ
! ð15Þ
2
4. Transversal conductivity of GSL under the cnoidal EM wave Now we consider the case when the EM wave propagates in direction perpendicular to the GSL axis so that polarization plane of this wave coincides with the GSL plane. In this situation the EM wave is of cnoidal form [35,44,50]. To calculate the transversal conductivity of the GSL exposed to the cnoidal EM wave we substitute the following Fourier series expansion in the formula (10): 1 X
eiaðtÞ ¼
cn ðjÞeinXðjÞt
ð16Þ
n¼1
where
2EðjÞ np2 1 pX0 j ; if 0 < j 6 1XðjÞ ¼ ; 1; cn ¼ 2 n=2 nþ1 n=2 KðjÞ Kð j1 Þ K ðjÞ qj þ ð1Þ qj 2j2 Eðj1 Þ 4np2 j2 1 eE0 d ; cn ¼ 2 ; if j > 1; j ¼ ; c0 ¼ 1 2j2 þ 3n 1 Kðj Þ 2hX0 K ðj1 Þ qn 1=j q1=j sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! xpl u u2 pKð 1 j2 Þ 2ede1 pn0 F 3 ðc; e0 =hÞ ; X0 ¼ F ðc; e0 =hÞ qj ¼ exp 1 c2 ; xpl ¼ h KðjÞ e0 F 0 ðc; e0 =hÞ 3 c Z 1 1 e0 c2 e0 1 c2 þ z2 pffiffiffiffiffiffiffiffiffiffiffiffiffi I1 pffiffiffiffiffiffiffiffiffiffiffiffiffi exp pffiffiffiffiffiffiffiffiffiffiffiffiffi dz ¼ 2 2 h h 1þz 1þz 1 þ z2 0
XðjÞ ¼
pX0
2KðjÞ
; c0 ¼
E0 is the amplitude of cnoidal wave, u is the wave velocity, c is the wave velocity in the absence of electrons, KðxÞ b EðxÞ are complete elliptic integrals of the first and second kind correspondingly. After substituting (16) into (10), calculation of integrals, and averaging over the wave period, we obtain the next formula for the GSL transversal conductivity at low temperatures (9):
0
rxx
0 11 2 2 2 2 1 2 2 2 2 X 1 x s c ð1 ð x þ n X Þ s Þ c ð1 ð x n X Þ s Þ B B CC B B 2 B ¼ r? @1 c2 þ c2 c20 þ n @ n þc 2 AA 2 2 2 2 2 2 2 1 þ xB s ð1 þ ðxB þ nXÞ s Þ n¼1 1 þ ðxB nXÞ s2 ð17Þ
In the case 0 < j 6 1 we obtain
0 11 2 2 2 2 1 X 1x s 1 ðxB nXÞ s CC B 1 ðxB þ nXÞ s B ¼ r? @1 c2 þ c2 c20 þ c2 c2n @ 2 þ 2 AA 2 2 2 ð1 þ x2B s2 Þ n¼1 1 þ ðxB þ nXÞ s 1 þ ðxB nXÞ2 s2 0
rxx
2 2 B
ð18Þ Graphs of dependences of the GSL transverse conductivity on the longitudinal electric field intensity plotted with the formula (18) are shown in Fig. 4. In the absence of the longitudinal component of the constant electric field ðEZ ¼ 0Þ we obtain the next formula for the transversal conductivity of the GSL:
rxx ¼ r? 1 c2 ð1 c20 Þ þ 2c2
1 X 1 n2 X2 s2 c2n 2 ð1 þ n2 X2 s2 Þ n¼1
!
In the absence of the Em wave ðj ¼ 0Þ formula (18) gives (15). In the case j > 1 we have
ð19Þ
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Fig. 4. Dependence of the GSL transverse conductivity on the longitudinal electric field intensity. O0s = 5, 0 6 j < 1.
rxx ¼ r? 1 c2 þ c2 c20
1 X 1 ðxB þ nXÞ2 s2 1 ðxB nXÞ2 s2 4n þ c2 c2n þ q 2 1=j 2 2 2 2 2 2 ð1 þ xB s Þ n¼1 ð1 þ ðxB þ nXÞ s Þ ð1 þ ðxB nXÞ2 s2 Þ
1 x2B s2
!!
ð20Þ Graphs of dependences of the GSL transverse conductivity on the longitudinal electric field intensity and on the EM wave amplitude plotted with the formula (20) are shown in Fig. 5. Dependences of the GSL transverse conductivity on the cnoidal EM wave amplitude plotted with the formulas (18) and (20) are shown in Fig. 6. In the absence of the longitudinal component of the constant electric field ðEz ¼ 0Þ the transversal conductivity is
Fig. 5. Dependence of the GSL transverse conductivity on the longitudinal electric field intensity. O0s = 5, j > 1.
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531
Fig. 6. Dependence of the GSL transverse conductivity on the EM wave amplitude. O0s = 5.
2
rxx ¼ r? 1 c ð1
c20 Þ
1 X 1 n2 X2 s2 þc c2n ð1 þ q4n 1=j Þ 2 ð1 þ n2 X2 s2 Þ n¼1
!
2
ð21Þ
5. Discussion The dependence of rxx on the longitudinal electric field intensity is seen from Figs. 2, 4 and 5 to have the view of the alternating resonances. In the case of sinusoidal wave conductivity peaks occur when the next condition is performed:
xB ¼ nx where n is the whole number. In the case of cnoidal wave conductivity peaks occur if
xB ¼ nXðjÞ
ð22Þ
As can be seen from Figs. 4 and 5 and from the condition (22) in contrast to sinusoidal EM waves the positions of conductivity resonances can be regulated by changing of the amplitude of cnoidal EM wave. In the case j > 1 the conductivity rxx dependence on the longitudinal component of electric field vector can be seen from Fig. 5 to be not symmetric, unlike the case of 0 < j 6 1 (Fig. 4). This fact is due to the constant component of electric field of cnoidal wave in the case when j > 1. The dependence of rxx on the wave amplitude is seen from Figs. 3 and 6 to have the character of oscillations. The possibility of the transversal conductivity resonances and oscillations is explained by the non-additivity of the GSL spectrum (1). In the bulk SL with additive spectrum such effects are impossible. Acknowledgments The work was supported by the RFBR Grant No. 13-02-97033 r_povolzhye_a, and by the project of state jobs to the research work of the Ministry of Education and Science of the Russian Federation in 2013 No. 2.8298.2013.
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