Thermal conductance in a quantum waveguide modulated with quantum dots

Thermal conductance in a quantum waveguide modulated with quantum dots

ARTICLE IN PRESS Physica E 40 (2008) 2862–2868 www.elsevier.com/locate/physe Thermal conductance in a quantum waveguide modulated with quantum dots ...

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ARTICLE IN PRESS

Physica E 40 (2008) 2862–2868 www.elsevier.com/locate/physe

Thermal conductance in a quantum waveguide modulated with quantum dots Ling-Jiang Yao, Lingling Wang, Xiao-Fang Peng, B.S. Zou, Ke-Qiu Chen Department of Applied Physics, Hunan University, Changsha 410082, China Received 29 October 2007; received in revised form 11 January 2008; accepted 11 January 2008 Available online 14 February 2008

Abstract We investigate the thermal conductance in a quantum waveguide modulated with quantum dots at low temperatures. It is found that the thermal conductance sensitively depends on the geometrical parameters of the structure and boundary conditions. When the stressfree boundary conditions are applied in the structure, the universal quantum of thermal conductance can be found regardless of the geometry details in the limit T ! 0. For an uniform quantum waveguide, a thermal conductance plateau can be observed at very low temperatures; while for the quantum waveguide modulated with quantum dots, the plateau disappears, instead a decrease of the thermal conductance can be observed as the temperature goes up in the low temperature region, and its magnitude can be adjusted by the radius of the quantum dot. Moreover, it is found that the quantum waveguide with two coupling quantum dots exhibits oscillatory decaying thermal conductance behavior with the distance between two quantum dots. However, when the hard-wall boundary conditions are applied, the thermal conductance displays different behaviors. r 2008 Elsevier B.V. All rights reserved. PACS: 63.22.þm; 73.23.Ad; 44.10.þi Keywords: Phonons in low-dimensional structures; Ballistic transport; Thermal conduction

1. Introduction In the past decade, the thermal transport associated with a set of discrete acoustic phonon modes in semiconductor nanostructures has been paid much attention [1–19]. Theoretically, some groups have predicated that the thermal conductance for ballistic phonon transport in the low temperature limit T ! 0 should be quantized, and the value is p2 k2B T=3h for each acoustic phonon mode, regardless of the geometry details [20,21]. For an ideal quantum wire, a quantum plateau can be observed in a very low temperature region where only zero mode is excited. Experimentally, Schwab et al. [22] verified the theoretical prediction of the universal quantum of thermal conductance. However, the quantum plateau was not observed in the experiment in the low temperature region, instead a decrease of thermal conductance appears as the Corresponding author.

E-mail address: [email protected] (K.-Q. Chen). 1386-9477/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.01.008

temperature goes up in the temperature region. It is suggested that the decrease of the thermal conductance results from the attached scattering to the transport phonons due to the presence of discontinuity in main quantum wire such as the imperfect acoustic coupling between the wire and the reservoirs [23,24], the scattering due to surface roughness [25,26], the scattering by the presence of the defects in the structure [10,27], or the scattering by stub structures [28–31]. It should be noted that in the limit T ! 0 the thermal conductance is always kept to be a universal unit, p2 k2B T=3h for each independent acoustic mode. In the present work, based on the scalar model of elasticity, we calculate the thermal conductance in ballistic quantum waveguide modulated with quantum dots. Note that the scalar model of elasticity is a rather good approximation in calculations of ballistic acoustic phonon transmission coefficients and thermal conductance at low temperatures, and has also been applied to study the thermal transport mechanism in various kinds of

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nanostructures [4,7,10,24,28,30–32]. The present work primarily focuses on the influences of geometry and boundary conditions on the thermal conductance. Calculated results show some interesting physical effects: (i) the behavior of the thermal conductance versus temperature is qualitatively different for the different types of boundary conditions. (ii) The thermal conductance is of different dependence on the radius of quantum dot in both stressfree and hard-wall boundary conditions. (iii) The coupling effects between two quantum dots play an important role in thermal transport of the quantum waveguide modulated with two quantum dots, and is different for different types of boundary conditions. This paper is organized as follows. In Section 2, we present a brief description of the model and the formulae used in calculations. In Section 3, we numerically investigate the thermal conductance. Finally, a summary is given in Section 4. 2. Model and formalism We consider the model structures shown in Fig. 1. In such structures, the mode mixing effects that may occur at boundaries or interfaces can be ignored at very low temperatures, then there exist three types of acoustic modes in the structures namely longitudinally polarized P mode, vertically polarized SV mode, and horizontally polarized SH mode [28,33]. Their polarization directions are along x, y and z direction, respectively. For simplicity, we assume here that the structures have same thickness in each region, and the thickness is small with respect to the other dimensions and also to the wavelength of the elastic waves, and thus a two-dimensional calculation is adequate. In such case, the SH mode is decoupled from the other two

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phonon modes. So in the present paper, for simplicity, we only discuss the thermal conductance of SH mode. Here, we also assume that the temperatures in regions I and III (region V for Fig. 1(b)) are T 1 and T 2 , respectively; and the temperature difference dT ððdT  T 1  T 2 Þ40Þ is small that we can adopt the mean temperature T ðT  ðT 1 þ T 2 Þ=2Þ as the temperature in our calculation. For the structure we consider here, the expression of thermal conductance K can be written as [4,20] K¼

_2 X 1 kB T 2 m 2p

Z

1

tm ðoÞ om

o2 eb_o do. ðeb_o  1Þ2

(1)

Here tm ðoÞ is the transmission coefficient from mode m of region I at frequency o across all the interfaces into the region III (region V for Fig. 1(b)), om is the cutoff frequency of the mth mode, b ¼ 1=ðkB TÞ, kB is the Boltzman constant, T is temperature, and _ is Planck’s constant. A central issue in calculation of the thermal conductance is to calculate the transmission coefficient, tm ðoÞ. Here, we employ the scalar model of elasticity to calculate the transmission coefficient of the acoustic phonon. In the elastic approximation, the elastic equation of motion for SH wave is q2 C  v2SH r2 C ¼ 0, qt2

(2)

where v2SH ¼ c44 =r is the sound velocity of SH mode related to the mass density r and elastic stiffness constant c44 . To obtain the solution of Eq. (2), we first subdivide the quantum dot structure into a number of subregions along the transmitted direction so that each subregion is of an uniform width. Then the solution to Eq. (2) in each region

Fig. 1. Structure of quantum waveguides modulated with a quantum dot (a) and two coupling quantum dots (b).

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x can be expressed as Cx ðx; yÞ ¼

Nx X

x

x

½Axm eikm x þ Bxm eikm x fxm ðyÞ,

while hard boundary conditions are employed at the edges, we have sffiffiffiffiffi 2 mp x y ðma0Þ. (6) Fm ðyÞ ¼ sin dx dx

(3)

m¼0

where kxn can be expressed in terms of incident phonon frequency o, the SH wave velocity vxSH , and the transverse dimension d x of region x as

Then by using the scattering matrix method [24,34–36], we can calculate the transmission coefficient tm . Note that the cutoff frequency om in Eq. (1) can be obtained by x2 x2 2 2 the2 energy conservation condition: o ¼ k v þ o m m ¼ 2 2 2 kxm vx þ m2 p2 vx =d x . The sum over m includes all propagating and evanescent modes (imaginary k). However, in the real calculations, we take all the propagating modes and several lowest evanescent modes into account to meet the desired precision. In the following calculations, we employ the values of elastic stiffness constants and the mass densities of GaAs referred to Ref. [37].

2

2

o ¼

2 2 kxm vxSH

þ

m2 p2 vxSH d 2x

,

(4)

and fxm ðyÞ represents transverse wave function of acoustic mode m in each subregion x and its expression is dependent on the boundary conditions. When stress-free boundary conditions are employed at the edges, we can obtain the expression of 8 sffiffiffiffiffi > 2 2mp > > cos y ðma0Þ; > > dx < dx Fxm ðyÞ ¼ sffiffiffiffiffi (5) > 1 > > > ðm ¼ 0Þ; > : dx

3. Numerical results and discussion In Fig. 2, we first show the dependence of the thermal conductances divided by temperature reduced by the zerotemperature universal p2 k2B =3h on the reduced temperature kB T=_D (D ¼ omþ1  om ¼ pvSH =d) for the structure given

1.2 K/T (π2kB2/3h)

K/T (π2kB2/3h)

3

2

1 0.5 0.05 0.2

0.4

0.6

0.8

1 0.8 0.6 0.4 0.05

1

T (2πkB/hΔ)

0.6

K/T (π2kB2/3h)

K/T (π2kB2/3h)

1

0.5

0.8

0.4 0.2 0 0.05

0.5 T (2πkB/hΔ)

0.5 T (2πkB/hΔ)

1

0.4

0.2

0 0.05

0.5

1

T (2πkB/hΔ)

Fig. 2. Thermal conductance divided by temperature K=T reduced by the zero-temperature universal value p2 k2B =3h as a function of the reduced temperature 2pkB T=hD (D ¼ omþ1  om ¼ pvSH =d) under the stress-free boundary conditions for different radius R of the quantum dot in the structure shown in Fig. 1(a): (a) is for the total K=T, and (b)–(d) corresponds to K=T of modes 0, 1, and 2, respectively. Note that the total K=T should include the contributions of all the propagation modes. By our calculations, however, only the first six modes can make their contributions to the total thermal conductance for the explored temperature scope. The solid, dashed, dotted, and dash-dotted curves correspond to R ¼ 5, 7.5, 10, and 15 nm, respectively. Here, we choose d ¼ 10 nm.

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that from the figure that with the increase of the radius R of the quantum dot, the reduced thermal conductance K=T of zero mode is always decreased. However, this does not always happen for the modes with higher index m. From Figs. 2(c) and (d), we find that the value K=T with larger radius is even bigger than that with small radius for a certain temperature range. From Fig. 3, however, it can be found that when the hard-wall boundary conditions are applied in the lead, the thermal conductance displays different behavior. From Fig. 3(a), it is observed that when the temperature is lower than the threshold temperature the total K=T is always zero. When the temperature is larger than the threshold temperature, the total K=T is always increased monotonically with increasing temperature. In such case, the universal quantum of thermal conductance cannot be observed. These results originate from the fact that under hard-wall boundary conditions ballistic transport for zero mode is impossible, and the threshold temperature of the first mode (m ¼ 1) is larger than zero. In comparison with zero mode or m ¼ 1 mode under stress-free boundary conditions, this mode has different contribution to the thermal conductance. This is because their transversal wave functions are different. From Fig. 3, we also find that the thermal conductance of the first mode is always decreased, while for the modes with higher index m, the value K=T with larger radius is not always bigger than that with small radius. This is similar to the case under stress-free

in Fig. 1(a): Fig. 2(a) corresponds to the total reduced thermal conductance, and Figs. 2(b)–(d) correspond to the reduced thermal conductance of modes 0, 1 and 2, respectively. Here, the stress-free boundary conditions are applied in the structure. From Fig. 2(a), we can see clearly that the thermal conductance reaches the universal quantum of thermal conductance at zero temperature regardless of the geometry details. It is known that at o ! 0, the wavelength of acoustic wave is much bigger than the feature size of the structure, the attached scattering by the quantum dot is very small. As a result, the phonon transmission approaches unity, and the thermal conductance approaches the ideal universal value, p2 k2B T=3h, for each acoustic mode. When the structure is a perfect quantum wire, a perfect quantized plateau appears. However, while doR=2, the plateau disappears, instead a decrease of the thermal conductance can be observed as the temperature goes up in low temperature region where only zero mode is excited, and its magnitude can be adjusted by the magnitude of the radius of the quantum dot. This results from the scattering effect of discontinuities on the zero mode, and the scattering effect is enhanced with the increase of the radius R of the quantum dot. When the temperature continues to go up, the value of K=T increases monotonously. This is because the higher transverse modes m ðm40Þ are excited and contribute to the thermal conductance at higher temperatures. These results agree with the experimental results qualitatively [22]. It is also found

0.8

1.5

K/T (π2kB2/3h)

K/T (π2kB2/3h)

2

1 0.5 0

0

0.2

0.4

0.6

0.8

0.4

0

1

0

0.2

T (2πkB/hΔ)

0.4

0.6

0.8

1

T (2πkB/hΔ)

0.5

0.3

0.4 K/T (π2kB2/3h)

K/T (π2kB2/3h)

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0.3 0.2 0.1 0

0

0.2

0.4

0.6

T (2πkB/hΔ)

0.8

1

0.2

0.1

0

0

0.2

0.4

0.6

0.8

1

T (2πkB/hΔ)

Fig. 3. Thermal conductance divided by temperature K=T reduced by the zero-temperature universal value p2 k2B =3h as a function of the reduced temperature 2pkB T=hD under the hard-wall boundary conditions for different radius R in the structure shown in Fig. 1(a): (a) is for the total K=T, and (b)–(d) corresponds to K=T of modes 1, 2, and 3, respectively. The structural parameters and explanations for curves are the same as Fig. 2.

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mode–mode coupling. And the higher the reduced temperature, the deeper the valley. This is because at the higher temperature, the more modes are excited, and so the stronger mode–mode coupling leads to the strong scattering to the transport phonons. This phenomenon is different from the case of stress-free boundary conditions being applied in the structure. These results show that the thermal conductance can be controlled by adjusting the radius, which may be useful for the design of thermal quantum devices. We now turn to study the influence of two coupling quantum dots on the thermal conductance in the structure shown in Fig. 1(b). Fig. 5 describes the reduced total thermal conductance versus the distance L between two quantum dots for the different temperatures. From Fig. 5(a), we can see clearly that when the reduced temperature is 0.06, the reduced thermal conductance (solid curve) is decreased monotonically with the increase of L, and the change is gradually becomes smaller for larger L. When the reduced temperature is further increased, more modes than one are excited, the reduced thermal conductance presented oscillatory decaying behaviors with the increase of L. These results can be understood. It is known that the transversal wave function to describe the acoustic phonon mode in the structure is

boundary conditions. By comparing Fig. 2 with Fig. 3, it can be found that the stress-free boundary condition is more favorable than the hard-wall boundary conditions for the thermal transport of the acoustic phonon. To further explore the influence of the radius of the quantum dot on the thermal conductance, in Fig. 4, we describe the reduced total thermal conductance versus the radius for both stress-free (Fig. 4(a)) and hard-wall (Fig. 4(b)) boundary conditions: the solid, dashed, dotted, and dot-dashed curves in Fig. 4(a) correspond to the reduced temperature 2pkB T=hD ¼ 0:06, 0.17, 0.4, and 0.8, while the solid and dashed curves in Fig. 4(b) correspond to 2pkB T=hD ¼ 0:4 and 0.8, respectively. Here, we choose d ¼ 10 nm. From Fig. 4(a), it can be found that when the reduced temperature is 0.06, only zero mode can be excited in the system, the thermal conductance is monotonically decreased with the increase of the radius R. At higher temperatures where more acoustic phonon modes can be excited, the thermal conductance displays a slight oscillatory decaying behaviors with the radius R. This results from the coupling between the incident mode in the lead and the mode in scattering region (quantum dot). However, from Fig. 4(b), we can find that the thermal conductance is of a deep valley at the certain radius, which shows that the strong reflection occurs due to strong

2.5

1.2 2

K/T (π2kB2/3h)

1 1.5

0.8

0.6

1

0.4 0.5 0.2

0

5

10

20 R (nm)

30

40

0

5

10

20 R (nm)

30

40

Fig. 4. The total thermal conductance divided by temperature K=T reduced by the zero-temperature universal value p2 k2B =3h as a function of the radius R in the structure shown in Fig. 1(a). (a) is for the stress-free boundary conditions: the solid, dashed, dotted, and dot-dashed curves correspond to the reduced temperature 2pkB T=hD ¼ 0:06, 0.17, 0.4, and 0.8, respectively. When 2pkB T=hD ¼ 0:06, only the zero mode can be excited, and the total K=T is factually that of zero mode. When 2pkB T=hD ¼ 0:17, mode 1 is excited, when 2pkB T=hD ¼ 0:4, modes 2 and 3 are all excited, and when 2pkB T=hD ¼ 0:8, all six modes are excited. (b) is for the hard-wall boundary conditions: the solid and dashed curves correspond to 2pkB T=hD ¼ 0:4 and 0.8, respectively. Here, we choose d ¼ 10 nm.

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0.6 1.3 0.5 1.2 0.4

1.1

1

0.3

0.9 0.2 0.8 0.1

0.7

0.6

0

5

10

15

20

0

0

5

10

15

20

L (nm)

L (nm)

Fig. 5. The total thermal conductance divided by temperature K=T reduced by the zero-temperature universal value p2 k2B =3h as a function of the distance L between two quantum dots in the structure shown in Fig. 1(b). (a) is for the stress-free boundary conditions: the solid, dashed, and dotted curves correspond to 2pkB T=hD ¼ 0:06, 0.17, and 0.8, respectively. (b) is for the hard-wall boundary conditions: the solid and dashed curves correspond to 2pkB T=hD ¼ 0:4 and 0.8. Here, we choose d ¼ 10 nm and R ¼ 7:5 nm.

pffiffiffiffiffiffiffiffiffiffi expressed as Fxm ðyÞ ¼ 2=d x cosð2mp=d x Þy ðma0Þ. For pffiffiffiffiffiffiffiffiffiffi zero mode, the expression is 1=d x ðm ¼ 0Þ. When the reduced temperature is 0.06, only zero mode is excited in the structure. In such case, no mode–mode coupling occurs in the structure, only the structural scattering leads to the decrease of the thermal conductance monotonically with the change of the distance L. However, when the modes with the index m being larger than 0 are excited, mode–mode coupling will occur due to the presence of the phase factor cosð2mp=d x Þy containing the transversal parameter y. This will induce a nonlinear behavior of the thermal conductance. Also considering the fact that the coupling effect between two quantum dots will become weaker for larger L, the reduced thermal conductance displays a complex oscillatory decaying behaviors with the change of L. The thermal conductance at the reduced temperature 0.17 being smaller than that at 0.06 results from the combination of the stronger scattering of the quantum dots to the zero mode at higher temperature and the small contribution to the thermal conductance from the first mode just excited. When the hard-wall boundary conditions are applied in the structure, it is also found from Fig. 5(b) that the quantum structure exhibits oscillatory decaying thermal conductance with the distance L at low

temperature, even only the first mode being excited. This is because under hard-wall boundary conditions the wave function of the first mode contains the phase factor, and so mode–mode coupling also leads to a nonlinear thermal conductance behavior with the structural parameters. These results show that we can adjust the thermal conductance by changing the distance L. 4. Summary In conclusion, we have presented a numerical study of the thermal conductance in a ballistic quantum waveguide modulated with quantum dots under both stress-free and hard-wall boundary conditions at low temperatures. The results show that the thermal conductance is dependent on the boundary conditions, the radius R of the quantum dot, the distance L between the two coupling quantum dots, and the temperature. When the stress-free boundary conditions are applied in the structure, zero mode can be excited, and the universal quantum of thermal conductance can be observed. When the temperature is low enough where only zero mode can be excited, the reduced thermal conductance displays monotonic behavior with the structural parameters. However, the reduced thermal

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conductance of the first mode under the hard-wall boundary condition exhibits an oscillatory decaying behavior due to the fact that the wave function of the mode contains phase factor dependent on the transversal parameters. With the increase of the temperature, more modes are excited in the structure, the reduced thermal conductance displays a complex oscillatory decaying behavior with the radius of the quantum dot or the distance L. These results show that the thermal conductance can be controlled to a certain degree by adjusting the radius of the quantum dot or the distance L between the two coupling quantum dots, which may be useful for application in devices. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 10674044, 90606001), by the Hunan Provincial Natural Science Foundation of China (No. 06JJ20004), and by the Ministry of Science and Technology of China (No. 2006CB605105). References [1] T.S. Tighe, J.M. Worlock, M.L. Roukes, Appl. Phys. Lett. 70 (1997) 2687. [2] G. Chen, Phys. Rev. B 57 (1998) 14958. [3] J. Zou, A. Balandin, J. Appl. Phys. 89 (2001) 2932. [4] M.C. Cross, R. Lifshitz, Phys. Rev. B 64 (2001) 085324. [5] Q.-F. Sun, P. Yang, H. Guo, Phys. Rev. Lett. 89 (2002) 175901. [6] D. Li, Y.Y. Wu, P. Kim, L. Shi, P. Yang, A. Majumdar, Appl. Phys. Lett. 83 (2003) 2934. [7] W.-X. Li, K.-Q. Chen, W. Duan, J. Wu, B.-L. Gu, Appl. Phys. Lett. 85 (2004) 822. [8] B. Li, L. Wang, G. Casati, Phys. Rev. Lett. 93 (2004) 184301. [9] Y. Tanaka, F. Yoshida, S. Tamura, Phys. Rev. B 71 (2005) 205308.

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