- Email: [email protected]
Transmission of longitudinal phonons through a mass-spring nanoring
Author’s Accepted Manuscript Transmission of longitudinal phonons through a mass-spring nanoring Hassan Rabani, Mohammad Mardaani
www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(14)00425-1 http://dx.doi.org/10.1016/j.physe.2014.11.020 PHYSE11788
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 22 June 2014 Revised date: 19 November 2014 Accepted date: 25 November 2014 Cite this article as: Hassan Rabani and Mohammad Mardaani, Transmission of longitudinal phonons through a mass-spring nanoring, Physica E: Lowdimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2014.11.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Transmission of longitudinal phonons through a mass-spring nanoring Hassan Rabania,b,∗, Mohammad Mardaani a,b a Department
of Physics, Faculty of Sciences, Shahrekord University P. O. Box 115, Shahrekord, Iran Research Center, Shahrekord University, 8818634141, Shahrekord, Iran
b Nanotechnology
Abstract In this paper, we study the coherent phonon transport through a cyclic mass-spring structure which is embedded between two longitudinal phononic leads within the harmonic approximation. We assume only the in-plane vibrations for the atoms of the structure and also the nearest neighbor interaction between them. By starting from the system potential energy and then using the Green’s function method, we construct a formalism to compute the phononic transmission coefficient and density of states/modes of the system. The numerical calculations are performed for a hexagonal mass-spring ring in the presence and absence of a massive impurity. The results reveal that, the variation of value of the masses or spring constants in the ring leads to appearance of the Fano resonance in the transmission spectrum. This phenomenon occurs at a special phonon frequency independent of the impurity position in the structure. Keywords: Phonon transmission, Nanoring, Harmonic approximation, Green’s function
1. Introduction The experimental challenges have made that the study of thermal properties of nano-devices has been investigated less than the electrical ones by the scientists [1, 2]. However, the recent high-rapidly developments in nanofabrication and measurement technologies, enables us today to study experimentally the thermal properties of nanostructures [3, 4]. For instance, thermal rectifiers [5], transistors [6], switches [7, 8], and memories [9] have been experimentally demonstrated. Also the size and geometry dependent [4, 10], electron-, magnon- and phonon-phonon scattering and related phenomena [11–13] in heat transport, have theoretically been illustrated in the recent publications. Therefore, this field attracted the attention of researchers on this research area in both experimental and theoretical viewpoints. On the other hand, in the field of micro and nano-electronics, the influence of heat dissipation and transport on electronic responses of the system is an important issue. The thermal properties of a structure are originated mainly from the electrons and atomic vibrations. Indeed, the consideration of these contributions even separately helps us to understand the underlying physics and mechanisms of the thermal phenomena. Since, a significant amount of thermal conductance of molecular structures, originates from phononic contribution, therefore the calculation of phonon transmission coefficient becomes necessary. By means of this quantity the computation of other thermodynamic quantities like Seebeck coefficient [14], Raman response coefficient [15], specific heat [16], and etc is possible. In this paper, we study the problem of phonon transport through a cyclic molecule including one or none impurity embedded between two simple phononic leads. We replace the molecule bonds by springs and allocate them to the corresponding spring constants in order to construct a simple mass-spring model. We also suppose the atomic displacements are in the bond directions and we employ the Green’s function technique within the harmonic approximation. In this manner, we insert the effect of phononic leads as self-energies in the phonon Green’s function. Then, the phononic transmission and density of modes are evaluated by using the system Green’s function. Depending on the position of impurity and output lead contact as well as bond structure of the molecule, there are some constructive ∗ Corresponding
author Email addresses: [email protected] (Hassan Rabani), [email protected] (Mohammad Mardaani)
Preprint submitted to Physica E
November 26, 2014
3
2
...
-1
0
4
1
6
7
8
...
5
Figure 1: A benzene-like ring connected to two simple mass-spring chains. The corresponding springs of the single and double bonds are represented by solid and bold-solid wavy lines, respectively. The numbers show the labels of the atoms in the hexagonal ring and leads.
and destructive interferences which lead to interesting phenomena such as Fano and anti-resonances in the system phononic spectra. This paper is organized as follows. In Sec. 2, we present the harmonic model, the system geometry and the Green’s function model in order to formulate the phonon transmission coefficient and density of modes for the present system. These quantities are calculated numerically for some different configurations and situations in Sec. 3. We also discuss and compare the numerical results in this section. We conclude the paper in Sec. 4. 2. Model and Formalism We consider a mass-spring benzene-like molecule embedded between two similar semi-infinite phononic leads within the harmonic approximation. We label atoms in the center ring according to Fig. 1. We indicate all forceconstants, in the left (L) and right (R) leads by C s . We assume the Kekul´e-like structure for the ring including two different bands with different spring constants. So, the corresponding spring constant for single and double bonds are represented by C s and C d , respectively. In the harmonic approximation, the potential energy of the system reads U=
i< j
Ui j =
2 1 Ci j nˆ i j · (ri − r j ) , 2 i< j
(1)
where U i j is the potential energy between i-th and j-th masses, C i j can takes the values C s or Cd , nˆ i j is the unit vector in the i − j bond direction; and finally r i( j) is the in-plane vector displacement from equilibrium position of i( j)-th atoms. For a extended hexagonal mass-spring ring depicted in Fig. 1, the explicit forms of U i j are 1 U01 = C s x21 , 2 1 1 U12 = C s (x1 − x2 ) + 2 2 1 U23 = Cd (x2 − x3 )2 , 2 1 1 U34 = C s (x3 − x4 ) − 2 2 1 1 U45 = Cd (x4 − x5 ) + 2 2 1 U56 = C s (x5 − x6 )2 , 2 1 1 U61 = Cd (x3 − x4 ) − 2 2 1 2 U47 = C s x4 , 2
(2a) √
2 3 (y1 − y2 ) , 2
(2b) (2c)
√
2 3 (y3 − y4 ) , 2 √ 2 3 (y4 − y5 ) , 2
(2d) (2e) (2f)
√
2 3 (y6 − y1 ) , 2
(2g) (2h)
The inverse Green’s function matrix of the ring in the presence of phononic leads is G−1 = (mω2 − ΣL δi,1 − ΣR δi,2n f −1 )I − A, 2
(3)
where ω is frequency of the incident phonon from leads, m is the mass of each atom, Σ L(R) is the self-energy of the ring due to the existence of left (right) for which the explicit form will be given later. Here, I is the unit matrix with dimension 2N × 2N and δ i, j refers the Kronecker delta. We assume that one lead is attached to the ring from first atom and another is connected via the atom number n f of the ring. Moreover, A is the force-constant matrix with the following elements ∂2 U Ai, j = , (4) ∂αi ∂α j where αi = x(i+1)/2 when i is odd and α i = yi/2 when i is even. Here we suppose the atoms are vibrating only in the plane of the ring with coordinates (x, y). The left (right) self energy is given by [15] ΣL(R) = C s exp(ıθL(R) ),
(5)
where according to dispersion relation of a mass-spring simple chain, θ L(R) is cos θL(R) = 1 − ω2 /2ω20
in which ω0 = written as
√ C s /m. At the end, the total phonon density of states and transmission coefficient, respectively, are DOS(ω) = −
N 2ω ImG i,i (ω), πN i=1
(6)
and T (ω) = 4 Im ΣL Im ΣR |G1,2n f −1 |2 .
(7)
We mention here that if one of the masses in the ring is replaced by a mass m + δm, then mω 2 should be changed to (1 + η)mω2 at the corresponding element of G −1 in Eq. (3) where η = δm/m. The above formalism enables us to evaluate the phonon transmission coefficient through an atomic ring for some different situations. We will examine numerically this scenario for a benzene-like mass-spring ring in the following cases: (i) the corresponding springs of the single and double bands can be different, (ii) the position of connection atom of the ring with output lead can be changed (iii) one of the masses in the ring will be massive with respect to others. 3. Numerical results for a hexagonal ring In this section, as a test case, we study the phonon transport properties of a hexagonal mass-spring ring which is enclosed by two simple phononic leads (Fig. 1). In our calculations, we take all the masses in the leads the same and preferably similar to masses of the ring. We also assume all the spring constants in the leads and contacts are identical and equal to C s = 1. We first investigate the case in which all the masses in the ring are the same but the spring constants can take alternatively different values. Figures 2(a) and (b), respectively, show the phonon DOS and transmission coefficient of a 6-atom ring embedded between two simple phononic leads as functions of dimensionless phonon frequency for different values δC. Here, δC = C d − C s is the difference between spring constants of sequential bands in the ring when the masses alternatively connected by different springs together. According to Fig. 1, the number of atoms in the ring which are connected to the leads are chosen as 1 and 4. From these plots, we see that by increasing δC the peaks of DOS and T curves shift to the right edge of band frequency. For δC 0, a Fano resonance appears in the transmission spectrum. In this case, the upper and lower paths for the phonon waves in the ring are different, which causes the phononic constructive and destructive interferences which consequently leads to the Fano resonance phenomenon. The Fig. 2(b) shows that by increasing of δC, the position of Fano resonance in the band frequency moves to the lower frequencies. Moreover, at the middle frequencies, the transmission coefficient tends to zero for large values of δC. Now, we assume that all the spring constants in the ring are specified and one mass can be taken different of the others (massive impurity). The phonon transmission coefficient of an extended hexagonal mass-spring structure including one massive impurity is plotted in Fig. 3(a) for some different values of η. Here η, which is defined below the Eq. (7), is the relative ratio of difference of impurity mass and one of the other masses. We first fixed the position 3
Figure 2: Phonon (a) DOS and (b) transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains for some different values of δC = Cd − C s .
of impurity at the atom with the number two of the ring and we put δC = 0.25. Also, the leads are connected to the ring via the atoms 1 and 4. It can be seen that by increasing η, the area under the curves increases. This means the phonon transmission coefficient arises at the most frequencies. By variation of impurity mass there is a small change in the position of Fano resonance at the frequency band. By comparison of Fig. 2(b) and Fig. 3(a), one can find that the corresponding frequency of Fano phenomenon is independent of impurity mass and only depends on δ C . Now, we study the effect of position variation of this heavy mass on the transmission coefficient for a fixed value of η = 0.2 at Fig. 3(b). The positions of resonance peaks depend on the position of the heavy mass. The reason is due to the changing of the system eigenfrequencies in different impurity position. Again the Fano resonance position at the band frequency, approximately has not be changed which shows this phenomenon is independent of the position and mass of the impurity. Finally, we assume the output lead can be connected to the ring via a mass with number n f which can take different values of 2, .., 6. Figures. 4(a) and (b) display the logarithm of transmission coefficient of a mass-spring nanoring for δC = 0 and δC 0 cases, respectively. According to Fig. 4(a), except for n f = 4 that is the symmetric case, a Fano phenomenon is occurred in the transmission spectrum. Indeed, for the cases n f 4, the phonon paths in the lower and upper arms of the ring is different which causes the more constructive and destructive interferences phononic wave functions. For an isolated benzene, there are several separated normal modes while for a simple mass-spring chain, there is a continuous range of allowed frequency [0, 2ω 0]. This means that for a benzene which is connected to two simple phononic chain, the phonon states of leads appear in the phononic DOS of benzene molecule (tunneling effect). We mention here that the phononic DOS of an infinite benzene chain (poly(p-phenylene) polymer), is exactly equal to zero at gap region. In fact the gap region of this structure lies in the range of that 0 < ω < 0.6ω 0 which corresponds to tunneling region of the studied structure in the paper. The phonon transmission in the gap region (ω 0.6ω 0 ), which has the tunneling behavior, for the symmetric case (n f = 4) is more than the other cases. As it can be seen 4
Figure 3: Phonon transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains. In (a) the mass which is located at j = 2 is δm = (m η) heavier than other masses and the transmission is plotted for some different values of η. In (b) the transmission is drown for some position of the massive mass ( j) for η = 0.2. In these plots, we choose the difference between the corresponding spring constants of double and single bands equal to δC = 0.25.
in Fig. 4(b), the behavior of transmission curve strongly depends on output lead position. Specially, the positions of anti-resonances vary by the variation of n f . Also, for the case of n f = 4, with respect to other cases, the least fluctuation is observed in the curve. 4. Concluding Remarks We have proposed a theoretical model based on Green’s function technique in order to describe the phonon transport properties of a mass-spring nanoring which is embedded between two longitudinal phononic leads. We have used the harmonic approximation and have supposed the in-plane motions for the vibrating masses of the system. As an illustrative example, we have calculated the phonon density of states/modes and transmission coefficient for some different situations of a six-atom nanoring. When a mass is heavier than the other masses in the ring, a Fano resonance will appear in the transmission spectrum. In this case, the position of this phenomenon in the band frequency is merely independent of the massive atom position. It turns out that for a nanoring which its masses are alternatively connected by two different springs, there is a Fano resonance in the transmission spectrum. When the difference of spring constants arises, this resonance becomes clearer and its position shifts to higher frequencies. At the low frequencies, the considered system can be taken as a uniform mass-spring chain including a heavy mass which implies the phonon transmission coefficient tends to zero. Acknowledgments We acknowledge the Iranian Nanotechnology Initiative for its partial financial support. This work has also been supported by Shahrekord University through a research fund.
5
Figure 4: Phonon transmission coefficient of a hexagonal ring embedded between two simple phononic chains as a function of phonon frequency for different values of nf . Here n f represents the atom number of the ring which is connected to output lead. In (a) δC is zero and in (b) δC is 0.25.
References [1] W. Lee, K. Kim, W. Jeong, L. A. Zotti, F. Pauly, J. C. Cuevas and P. Reddy, Nature 498, (2013) 209. [2] A. N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications, Springer-Verlag, NewYork, 2003. [3] M. N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M. T. Bulsara, A. J. Schmidt, A. J. Minnich, S. Chen, M. S. Dresselhaus, Z. Ren, E. A. Fitzgerald and G. Chen, Science 338 (2012) 936. [4] M. Tsutsui, T. Morikawa, A. Arima and M. Taniguchi, Scientific Reports 3 (2013) 3326. [5] C. W. Chang, D. Okawa, A. Majumdar and A. Zettl, Science 314 (2006) 1121. [6] B. Li, L. Wang and Giulio Casati, Applied Physics Letters 88 (2006) 143501. [7] A. Goker, E. Gedik, J. Phys.: Condens. Matter 25 (2013) 365301. [8] C. Dames, Nature Nanotechnology 7 (2012) 82. [9] L. Wang and B. Li, Phys. Rev. Lett., 101 2008 267203. [10] M. Mardaani, H. Rabani, M. Keshavarz, Physica E 44 (2012) 1342. [11] M. Mardaani and H. Rabani, physica status solidi (b)251 (2014) 1001. [12] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara, G. E. W. Bauer, S. Maekawa and E. Saitoh, J. Appl. Phys. 111 (2012) 103903. [13] N. Mingo, Physical Review B 74 (2006) 125402. [14] Thermoelectrics Handbook. Macro to Nano, edited by D.M. Rowe (CRC Press, Boca Raton, FL, 2006). [15] M. Mardaani and H. Rabani, Solid State Communications 151 (2011) 311. [16] R. Prasher, T. Tong and A. Majumdar, Nano Lett. 8 (2008) 99.
6
Figure Captions Figure 1: A benzene-like ring connected to two simple mass-spring chains. The corresponding springs of the single and double bonds are represented by solid and bold-solid wavy lines, respectively. The numbers show the labels of the atoms in the hexagonal ring and leads. Figure 2: Phonon (a) DOS and (b) transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains for some different values of δC. Figure 3: Phonon transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains. In (a) the mass which is located at j = 2 is δm = (m η) heavier than other masses and the transmission is plotted for some different values of η. In (b) the transmission is drown for some position of the massive mass ( j) for η = 0.2. In these plots, we choose the difference between the corresponding spring constants of double and single bands equal to δC = 0.25. Figure 4: Phonon transmission coefficient of a hexagonal ring embedded between two simple phononic chains as a function of phonon frequency for different values of n f . Here n f represents the atom number of the ring which is connected to output lead. In (a) δC is zero and in (b) δC is 0.25.
7
www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(14)00425-1 http://dx.doi.org/10.1016/j.physe.2014.11.020 PHYSE11788
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 22 June 2014 Revised date: 19 November 2014 Accepted date: 25 November 2014 Cite this article as: Hassan Rabani and Mohammad Mardaani, Transmission of longitudinal phonons through a mass-spring nanoring, Physica E: Lowdimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2014.11.020 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Transmission of longitudinal phonons through a mass-spring nanoring Hassan Rabania,b,∗, Mohammad Mardaani a,b a Department
of Physics, Faculty of Sciences, Shahrekord University P. O. Box 115, Shahrekord, Iran Research Center, Shahrekord University, 8818634141, Shahrekord, Iran
b Nanotechnology
Abstract In this paper, we study the coherent phonon transport through a cyclic mass-spring structure which is embedded between two longitudinal phononic leads within the harmonic approximation. We assume only the in-plane vibrations for the atoms of the structure and also the nearest neighbor interaction between them. By starting from the system potential energy and then using the Green’s function method, we construct a formalism to compute the phononic transmission coefficient and density of states/modes of the system. The numerical calculations are performed for a hexagonal mass-spring ring in the presence and absence of a massive impurity. The results reveal that, the variation of value of the masses or spring constants in the ring leads to appearance of the Fano resonance in the transmission spectrum. This phenomenon occurs at a special phonon frequency independent of the impurity position in the structure. Keywords: Phonon transmission, Nanoring, Harmonic approximation, Green’s function
1. Introduction The experimental challenges have made that the study of thermal properties of nano-devices has been investigated less than the electrical ones by the scientists [1, 2]. However, the recent high-rapidly developments in nanofabrication and measurement technologies, enables us today to study experimentally the thermal properties of nanostructures [3, 4]. For instance, thermal rectifiers [5], transistors [6], switches [7, 8], and memories [9] have been experimentally demonstrated. Also the size and geometry dependent [4, 10], electron-, magnon- and phonon-phonon scattering and related phenomena [11–13] in heat transport, have theoretically been illustrated in the recent publications. Therefore, this field attracted the attention of researchers on this research area in both experimental and theoretical viewpoints. On the other hand, in the field of micro and nano-electronics, the influence of heat dissipation and transport on electronic responses of the system is an important issue. The thermal properties of a structure are originated mainly from the electrons and atomic vibrations. Indeed, the consideration of these contributions even separately helps us to understand the underlying physics and mechanisms of the thermal phenomena. Since, a significant amount of thermal conductance of molecular structures, originates from phononic contribution, therefore the calculation of phonon transmission coefficient becomes necessary. By means of this quantity the computation of other thermodynamic quantities like Seebeck coefficient [14], Raman response coefficient [15], specific heat [16], and etc is possible. In this paper, we study the problem of phonon transport through a cyclic molecule including one or none impurity embedded between two simple phononic leads. We replace the molecule bonds by springs and allocate them to the corresponding spring constants in order to construct a simple mass-spring model. We also suppose the atomic displacements are in the bond directions and we employ the Green’s function technique within the harmonic approximation. In this manner, we insert the effect of phononic leads as self-energies in the phonon Green’s function. Then, the phononic transmission and density of modes are evaluated by using the system Green’s function. Depending on the position of impurity and output lead contact as well as bond structure of the molecule, there are some constructive ∗ Corresponding
author Email addresses: [email protected] (Hassan Rabani), [email protected] (Mohammad Mardaani)
Preprint submitted to Physica E
November 26, 2014
3
2
...
-1
0
4
1
6
7
8
...
5
Figure 1: A benzene-like ring connected to two simple mass-spring chains. The corresponding springs of the single and double bonds are represented by solid and bold-solid wavy lines, respectively. The numbers show the labels of the atoms in the hexagonal ring and leads.
and destructive interferences which lead to interesting phenomena such as Fano and anti-resonances in the system phononic spectra. This paper is organized as follows. In Sec. 2, we present the harmonic model, the system geometry and the Green’s function model in order to formulate the phonon transmission coefficient and density of modes for the present system. These quantities are calculated numerically for some different configurations and situations in Sec. 3. We also discuss and compare the numerical results in this section. We conclude the paper in Sec. 4. 2. Model and Formalism We consider a mass-spring benzene-like molecule embedded between two similar semi-infinite phononic leads within the harmonic approximation. We label atoms in the center ring according to Fig. 1. We indicate all forceconstants, in the left (L) and right (R) leads by C s . We assume the Kekul´e-like structure for the ring including two different bands with different spring constants. So, the corresponding spring constant for single and double bonds are represented by C s and C d , respectively. In the harmonic approximation, the potential energy of the system reads U=
i< j
Ui j =
2 1 Ci j nˆ i j · (ri − r j ) , 2 i< j
(1)
where U i j is the potential energy between i-th and j-th masses, C i j can takes the values C s or Cd , nˆ i j is the unit vector in the i − j bond direction; and finally r i( j) is the in-plane vector displacement from equilibrium position of i( j)-th atoms. For a extended hexagonal mass-spring ring depicted in Fig. 1, the explicit forms of U i j are 1 U01 = C s x21 , 2 1 1 U12 = C s (x1 − x2 ) + 2 2 1 U23 = Cd (x2 − x3 )2 , 2 1 1 U34 = C s (x3 − x4 ) − 2 2 1 1 U45 = Cd (x4 − x5 ) + 2 2 1 U56 = C s (x5 − x6 )2 , 2 1 1 U61 = Cd (x3 − x4 ) − 2 2 1 2 U47 = C s x4 , 2
(2a) √
2 3 (y1 − y2 ) , 2
(2b) (2c)
√
2 3 (y3 − y4 ) , 2 √ 2 3 (y4 − y5 ) , 2
(2d) (2e) (2f)
√
2 3 (y6 − y1 ) , 2
(2g) (2h)
The inverse Green’s function matrix of the ring in the presence of phononic leads is G−1 = (mω2 − ΣL δi,1 − ΣR δi,2n f −1 )I − A, 2
(3)
where ω is frequency of the incident phonon from leads, m is the mass of each atom, Σ L(R) is the self-energy of the ring due to the existence of left (right) for which the explicit form will be given later. Here, I is the unit matrix with dimension 2N × 2N and δ i, j refers the Kronecker delta. We assume that one lead is attached to the ring from first atom and another is connected via the atom number n f of the ring. Moreover, A is the force-constant matrix with the following elements ∂2 U Ai, j = , (4) ∂αi ∂α j where αi = x(i+1)/2 when i is odd and α i = yi/2 when i is even. Here we suppose the atoms are vibrating only in the plane of the ring with coordinates (x, y). The left (right) self energy is given by [15] ΣL(R) = C s exp(ıθL(R) ),
(5)
where according to dispersion relation of a mass-spring simple chain, θ L(R) is cos θL(R) = 1 − ω2 /2ω20
in which ω0 = written as
√ C s /m. At the end, the total phonon density of states and transmission coefficient, respectively, are DOS(ω) = −
N 2ω ImG i,i (ω), πN i=1
(6)
and T (ω) = 4 Im ΣL Im ΣR |G1,2n f −1 |2 .
(7)
We mention here that if one of the masses in the ring is replaced by a mass m + δm, then mω 2 should be changed to (1 + η)mω2 at the corresponding element of G −1 in Eq. (3) where η = δm/m. The above formalism enables us to evaluate the phonon transmission coefficient through an atomic ring for some different situations. We will examine numerically this scenario for a benzene-like mass-spring ring in the following cases: (i) the corresponding springs of the single and double bands can be different, (ii) the position of connection atom of the ring with output lead can be changed (iii) one of the masses in the ring will be massive with respect to others. 3. Numerical results for a hexagonal ring In this section, as a test case, we study the phonon transport properties of a hexagonal mass-spring ring which is enclosed by two simple phononic leads (Fig. 1). In our calculations, we take all the masses in the leads the same and preferably similar to masses of the ring. We also assume all the spring constants in the leads and contacts are identical and equal to C s = 1. We first investigate the case in which all the masses in the ring are the same but the spring constants can take alternatively different values. Figures 2(a) and (b), respectively, show the phonon DOS and transmission coefficient of a 6-atom ring embedded between two simple phononic leads as functions of dimensionless phonon frequency for different values δC. Here, δC = C d − C s is the difference between spring constants of sequential bands in the ring when the masses alternatively connected by different springs together. According to Fig. 1, the number of atoms in the ring which are connected to the leads are chosen as 1 and 4. From these plots, we see that by increasing δC the peaks of DOS and T curves shift to the right edge of band frequency. For δC 0, a Fano resonance appears in the transmission spectrum. In this case, the upper and lower paths for the phonon waves in the ring are different, which causes the phononic constructive and destructive interferences which consequently leads to the Fano resonance phenomenon. The Fig. 2(b) shows that by increasing of δC, the position of Fano resonance in the band frequency moves to the lower frequencies. Moreover, at the middle frequencies, the transmission coefficient tends to zero for large values of δC. Now, we assume that all the spring constants in the ring are specified and one mass can be taken different of the others (massive impurity). The phonon transmission coefficient of an extended hexagonal mass-spring structure including one massive impurity is plotted in Fig. 3(a) for some different values of η. Here η, which is defined below the Eq. (7), is the relative ratio of difference of impurity mass and one of the other masses. We first fixed the position 3
Figure 2: Phonon (a) DOS and (b) transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains for some different values of δC = Cd − C s .
of impurity at the atom with the number two of the ring and we put δC = 0.25. Also, the leads are connected to the ring via the atoms 1 and 4. It can be seen that by increasing η, the area under the curves increases. This means the phonon transmission coefficient arises at the most frequencies. By variation of impurity mass there is a small change in the position of Fano resonance at the frequency band. By comparison of Fig. 2(b) and Fig. 3(a), one can find that the corresponding frequency of Fano phenomenon is independent of impurity mass and only depends on δ C . Now, we study the effect of position variation of this heavy mass on the transmission coefficient for a fixed value of η = 0.2 at Fig. 3(b). The positions of resonance peaks depend on the position of the heavy mass. The reason is due to the changing of the system eigenfrequencies in different impurity position. Again the Fano resonance position at the band frequency, approximately has not be changed which shows this phenomenon is independent of the position and mass of the impurity. Finally, we assume the output lead can be connected to the ring via a mass with number n f which can take different values of 2, .., 6. Figures. 4(a) and (b) display the logarithm of transmission coefficient of a mass-spring nanoring for δC = 0 and δC 0 cases, respectively. According to Fig. 4(a), except for n f = 4 that is the symmetric case, a Fano phenomenon is occurred in the transmission spectrum. Indeed, for the cases n f 4, the phonon paths in the lower and upper arms of the ring is different which causes the more constructive and destructive interferences phononic wave functions. For an isolated benzene, there are several separated normal modes while for a simple mass-spring chain, there is a continuous range of allowed frequency [0, 2ω 0]. This means that for a benzene which is connected to two simple phononic chain, the phonon states of leads appear in the phononic DOS of benzene molecule (tunneling effect). We mention here that the phononic DOS of an infinite benzene chain (poly(p-phenylene) polymer), is exactly equal to zero at gap region. In fact the gap region of this structure lies in the range of that 0 < ω < 0.6ω 0 which corresponds to tunneling region of the studied structure in the paper. The phonon transmission in the gap region (ω 0.6ω 0 ), which has the tunneling behavior, for the symmetric case (n f = 4) is more than the other cases. As it can be seen 4
Figure 3: Phonon transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains. In (a) the mass which is located at j = 2 is δm = (m η) heavier than other masses and the transmission is plotted for some different values of η. In (b) the transmission is drown for some position of the massive mass ( j) for η = 0.2. In these plots, we choose the difference between the corresponding spring constants of double and single bands equal to δC = 0.25.
in Fig. 4(b), the behavior of transmission curve strongly depends on output lead position. Specially, the positions of anti-resonances vary by the variation of n f . Also, for the case of n f = 4, with respect to other cases, the least fluctuation is observed in the curve. 4. Concluding Remarks We have proposed a theoretical model based on Green’s function technique in order to describe the phonon transport properties of a mass-spring nanoring which is embedded between two longitudinal phononic leads. We have used the harmonic approximation and have supposed the in-plane motions for the vibrating masses of the system. As an illustrative example, we have calculated the phonon density of states/modes and transmission coefficient for some different situations of a six-atom nanoring. When a mass is heavier than the other masses in the ring, a Fano resonance will appear in the transmission spectrum. In this case, the position of this phenomenon in the band frequency is merely independent of the massive atom position. It turns out that for a nanoring which its masses are alternatively connected by two different springs, there is a Fano resonance in the transmission spectrum. When the difference of spring constants arises, this resonance becomes clearer and its position shifts to higher frequencies. At the low frequencies, the considered system can be taken as a uniform mass-spring chain including a heavy mass which implies the phonon transmission coefficient tends to zero. Acknowledgments We acknowledge the Iranian Nanotechnology Initiative for its partial financial support. This work has also been supported by Shahrekord University through a research fund.
5
Figure 4: Phonon transmission coefficient of a hexagonal ring embedded between two simple phononic chains as a function of phonon frequency for different values of nf . Here n f represents the atom number of the ring which is connected to output lead. In (a) δC is zero and in (b) δC is 0.25.
References [1] W. Lee, K. Kim, W. Jeong, L. A. Zotti, F. Pauly, J. C. Cuevas and P. Reddy, Nature 498, (2013) 209. [2] A. N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications, Springer-Verlag, NewYork, 2003. [3] M. N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M. T. Bulsara, A. J. Schmidt, A. J. Minnich, S. Chen, M. S. Dresselhaus, Z. Ren, E. A. Fitzgerald and G. Chen, Science 338 (2012) 936. [4] M. Tsutsui, T. Morikawa, A. Arima and M. Taniguchi, Scientific Reports 3 (2013) 3326. [5] C. W. Chang, D. Okawa, A. Majumdar and A. Zettl, Science 314 (2006) 1121. [6] B. Li, L. Wang and Giulio Casati, Applied Physics Letters 88 (2006) 143501. [7] A. Goker, E. Gedik, J. Phys.: Condens. Matter 25 (2013) 365301. [8] C. Dames, Nature Nanotechnology 7 (2012) 82. [9] L. Wang and B. Li, Phys. Rev. Lett., 101 2008 267203. [10] M. Mardaani, H. Rabani, M. Keshavarz, Physica E 44 (2012) 1342. [11] M. Mardaani and H. Rabani, physica status solidi (b)251 (2014) 1001. [12] K. Uchida, T. Ota, H. Adachi, J. Xiao, T. Nonaka, Y. Kajiwara, G. E. W. Bauer, S. Maekawa and E. Saitoh, J. Appl. Phys. 111 (2012) 103903. [13] N. Mingo, Physical Review B 74 (2006) 125402. [14] Thermoelectrics Handbook. Macro to Nano, edited by D.M. Rowe (CRC Press, Boca Raton, FL, 2006). [15] M. Mardaani and H. Rabani, Solid State Communications 151 (2011) 311. [16] R. Prasher, T. Tong and A. Majumdar, Nano Lett. 8 (2008) 99.
6
Figure Captions Figure 1: A benzene-like ring connected to two simple mass-spring chains. The corresponding springs of the single and double bonds are represented by solid and bold-solid wavy lines, respectively. The numbers show the labels of the atoms in the hexagonal ring and leads. Figure 2: Phonon (a) DOS and (b) transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains for some different values of δC. Figure 3: Phonon transmission vs. phonon frequency through a hexagonal mass-spring ring embedded between two simple phononic chains. In (a) the mass which is located at j = 2 is δm = (m η) heavier than other masses and the transmission is plotted for some different values of η. In (b) the transmission is drown for some position of the massive mass ( j) for η = 0.2. In these plots, we choose the difference between the corresponding spring constants of double and single bands equal to δC = 0.25. Figure 4: Phonon transmission coefficient of a hexagonal ring embedded between two simple phononic chains as a function of phonon frequency for different values of n f . Here n f represents the atom number of the ring which is connected to output lead. In (a) δC is zero and in (b) δC is 0.25.
7