- Email: [email protected]
Excitation of hypersound in n-GaN films
ARTICLE IN PRESS
Microelectronics Journal 39 (2008) 740–743 www.elsevier.com/locate/mejo
Excitation of hypersound in n-GaN films F. Diaz, V. Grimalsky, M. Tecpoyotl, J. Escobedo, S. Koshevaya Autonomous University of Morelos, Research Center of Engineering and Applied Sciences, Avenue University 1001, Colony Chamilpa, Cuernavaca 62209, Morelos, Mexico Received 16 October 2007; accepted 15 December 2007 Available online 29 February 2008
Abstract The aim of this paper is the analysis of hypersound excitation in GaN films. Simulation of process is presented for a case of a thin film geometry on a non-piezoelectric substrate. The frequency range considered is from 50 GHz up to 200 GHz. The excitation is due to coupling with space charge waves (SCWs) in GaN film. The amplification of SCWs is related with negative differential conductivity in GaN films. Possible spatial increments are obtained. The amplified SCWs can excite hypersonic waves at the same frequency due to piezoeffect and deformation potential mechanisms. The first effect is stronger and causes an effective resonant excitation of hypersonic waves in the case of full mechanic contact of GaN film and non-piezoelectric substrate. r 2008 Elsevier Ltd. All rights reserved. Keywords: Excitation of hypersound; Space charge waves; Negative differential conductivity; Piezoeffect mechanism
1. Introduction There are traditional methods of ultrasound excitation, like comb and thin film transducers (at microwave frequencies f43 GHz [1]), but for excitation of hypersound (f430 GHz) actually these methods cannot work. A possible solution of this problem is the resonant coupling of acoustic waves (AWs) with the microwave electric field of space charge waves (SCWs) in materials possessing negative differential conductivity (NDC) (like GaAs) [2,3]. Namely, under propagation in the bias electric field higher than the critical value for observing NDC, the SCW is subject to amplification, and its microwave electric field can achieve high values. On the other hand, due to piezoeffect, deformation potential or electrostriction, this microwave electric field excites hypersonic AWs. In [2,3] it was demonstrated that this excitation has a resonant character with respect to frequency and the thickness of the GaAs film. But, the critical value of bias electric field in GaAs is Ec=3.5 kV/cm that limits the maximal values of the microwave electric Corresponding author. Tel./fax: +52 777 329 7084.
E-mail address: [email protected] (F. Diaz). 0026-2692/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.12.007
field of SCW and output values of amplitudes of excited AW. Also, the frequency range of amplification of SCW in GaAs films is fo50 GHz. To excite the powerful hypersonic AW at frequencies higher than 50 GHz, it is rather better to use new materials possessing NDC at higher frequencies, for example GaN [4–7] for f=100–300 GHz. The attracting properties of GaN are: (1) a high critical bias field Ec100 kV/cm; (2) extended frequency range for observing NDC fp500 GHz; and (3) high values of piezoelectric constants. In this article the excitation of hypersound in GaN films is researched. Simulation of process is preparing for case of this material for a thin film geometry on a solid substrate. The frequency range considered is f=30–200 GHz. Simulation of excitation of hypersonic AW is given for GaN films of a sub-micron thickness placed onto a non-piezoelectric substrate. The spatial increments of SCW due to NDC have been calculated. An amplification of SCW is possible for frequencies up to 200 or 300 GHz. The thin GaN film plays the role of a transducer of AW. The case of the full acoustic contact between GaN film and the substrate is considered. It has been demonstrated that the excitation of AW has a resonant character and the intensities of AW may reach the values of 1 W/cm2.
ARTICLE IN PRESS F. Diaz et al. / Microelectronics Journal 39 (2008) 740–743
2. SCWs amplification in thin GaN films It is necessary to analyze at first, the SCWs amplification in n-GaN film of a sub-micron thickness placed onto a semi-infinite dielectric substrate, see Fig. 1. The bias electric field is directed along OZ-axis, the SCW are excited by the input antenna I and propagate also in Z-direction. The dynamics of SCW is described by the equations of motion of electrons jointly with the Poisson equation for the electric field. In the frequency range fo200 GHz it is possible to use the simplest hydrodynamic diffusion-drift equation for the electron fluid: qn q ~ Dn rnÞ ¼ 0; þ r ð~ vðEÞn qt qz ~ E, ~ v ¼ mðjEjÞ rð0 ðxÞrjÞ ¼ eðn n0 Þ; ~ ¼ rj þ E 0 . E
(1)
Here n is the electron concentration, j is the electric potential of the alternative field, v is the electron velocity, n0 is the equilibrium electron concentration (equal to the donor one), Dn is the diffusion coefficient, m(E) is electron mobility, E0 is the bias electric field. The data for GaN are taken from [8]. A dependence of the diffusion coefficient Dn on the electric field is kB T e mðjEjÞ, jej 2 mðjEjÞv2 2 wðjEjÞ kB T e ¼ wðjEjÞ. 3 3 2
The dependence of electron velocity on electric field for wurtzite and zinc-blende GaN is obtained from [8]. In this article we use the zinc-blende structure material for the film construction. Amplification of SCW is analyzed below. We use at first, an approximation of a thin film where the electron concentration is assumed as two-dimensional. The linearized equations for the perturbation of electron concentra~ tion n˜(z,t) ¼ nn0 ðjnj5n 0 Þ and for the electric potential j are qn~ qn~ d~ v qE~ z q2 n~ þ v 0 þ n0 Dn 2 ¼ 0, qt qz dE qz x¼0 qz v0 ¼ vðE 0 Þ, q qj q2 j e ~ ðxÞ, ðxÞ þ ðxÞ 2 ¼ 2nld qx qx qz 0 qj . (3) E~ z ¼ qz n˜ and j are variable parts of electron concentration and electric potential, respectively; E0 is a bias electric field within the film, v0 ¼ m(E0)E0 is the constant part of electron drift velocity, m ¼ m(E0) ¼ v0/E0 and e is the electron charge. Consider the case of NDC: ðd~ v=dEÞo0. For linear SCW, we try to obtain a solution of Eq. (3) as: exp(i(otkz)). The dispersion equation k ¼ k(o) for SCW is
Dn ¼
o kv0 i (2)
At higher electric fields E4100 kV/cm, the value of diffusion coefficient is somewhat higher that at low fields (Dn(EE0) ¼ 25 cm2/s). The coordinate frame is aligned along the crystalline axes. The lower indexes 1 and 2 are related to the substrate and the film, respectively.
Fig. 1. The GaN film occupies the region 0oxo2l,x42l is vacuum, xo0 is a semi-infinite substrate. I is antenna input and II is output one.
741
2n0 ljej d~ v k iDn k2 ¼ 0, 0 ð1 þ 1 Þ dE
(4)
where k is the wave number, for spatial amplification of SCW (circular frequency o ¼ 2pf is real, k ¼ k0 +ik00 is complex), in some frequency range one can obtain k00 40. We use the following parameters of GaN film: the electron concentration is n0E1017 cm3, the bias electric field is E0 ¼ 150 kV/cm, the diffusion coefficient is Dn ¼ 50 cm2/s, the drift velocity is v0 ¼ 2.3 107 cm/s, the thickness of the film is 2l ¼ 0.1 mm. The dielectric permittivity of the substrate (like sapphire) is e1 ¼ 10. In the second approach, the finite thickness of the film has been taken into account. Our simulations have been demonstrated that there is no principal difference between the simplified and more complex approaches at frequencies fp200 GHz. The dependencies of spatial increment of SCW k00 on frequency f ¼ o/2p are given in Fig. 2. The curves 1–4 have been calculated when a finite thickness of the film has been considered, whereas the curve 10 has been calculated within an approximation of 2D electron gas. We can appreciate that an amplification of SCW in GaN film occurs in wide frequency range, and the maximal spatial increment is k00 ¼ 3 105 m1 at frequency f ¼ 150 GHz. When compared with a case of GaAs film, it is possible to observe an amplification of SCW in GaN films, at essentially frequencies higher than 100 GHz. In order to obtain an amplification of SCW of 20 dB, it is necessary to use the distance between input and output antennas of about 50 mm. An amplification of SCW in GaN films
ARTICLE IN PRESS F. Diaz et al. / Microelectronics Journal 39 (2008) 740–743
742
Fig. 2. Spatial increment of instability k00 (f) of SCW in GaN zinc-blende film. Curve 1 is for E0 ¼ 150 kV/cm, n0 ¼ 1017 cm3, curve 2 is for E0 ¼ 150 kV/cm, n0 ¼ 0.5 1017 cm3, curve 3 is for E0 ¼ 150 kV/cm, n0 ¼ 1.5 1017 cm3, curve 4 is for E0 ¼ 155 kV/cm, n0 ¼ 1 1017 cm3. For a comparison, the curve 10 is calculated under the same conditions of curve, 1 but using a 2D approximation (Eq. (3)).
should be realized in pulse regime (of a duration o1 ms), because of heating the GaN film. We do not consider here the frequency range f4200 GHz, because an applicability of hydrodynamic equations would be doubtful [6].
Fig. 3. Efficiency of excitation of AW due to piezoeffect (curve 1) and due to deformation potential (curve 2).
2ibE~ z 2 sin ðKl Þe2iKl , rs2 K o iGo 1=2 1þ 2 K¼ . s rs
U¼
(7)
The effective excitation of AW occurs when the following resonant condition takes place: p ð2m þ 1Þ; 2
3. Excitation of hypersonic AWs
Kl
The excitation of hypersonic AWs in the GaN film is possible at the same frequency due to piezoeffect. Because the sound velocity is two orders smaller than the velocity of SCW, the direction of propagation of excited AW is perpendicular to the surface of the film parallel to OX axis downwards. In the zinc-blende GaN (cubic symmetry), the nonzero piezoelectric module is b ¼ b3,12 ¼ 0.4 C/m2. Therefore, the alternative electric field E~ z of SCW can excite the shear AW (u ¼ u2) that propagates along the OX axis. The equation of elasticity theory for mechanical displacement u takes the form: q2 u q q q2 u 2 qu rs G r 2 ¼ þ . (5) qt qx qx qx qxqt
When the acoustic properties of the film and the substrate are different, the resonant condition (7) is valid, but the expression for the amplitude of the outgoing wave is more complicated. At the frequency f ¼ 100 GHz, the dissipation of shear AW is |K00 |500 cm1 [1], so the condition jK 00 jl3 103 51 is satisfied. For the amplitude of alternative electric field of SCW E~ z ¼ 10 kV=cm, the output intensity of AW Pa ¼ (1/2)rs3|KU|2 can reach the values Pa1 W/cm2 in the resonant case. The efficiency of excitation of AW is given in Fig. 3. A dependence of elastic deformation of AW at x ¼ 0 on a frequency is presented there for both, an excitation due to piezoeffect (curve 1) and due to deformation potential (curve 2). The value of the bias electric field is E0 ¼ 150 kV/cm, the electron concentration is n0 ¼ 1017 cm3. The longitudinal AW can be excited there due to deformation potential.
Here r is the density elastic medium, s is the shear acoustic velocity, G is viscosity. The expression for mechanical stress s ¼ s12 within the GaN film is s ¼ r2 s22
qu þ bE~ z . qx
(6)
The simulations have shown that within the film, 0oxo2l, the alternative electric field E~ z is uniform: qE~ z =qx 0. Therefore, only at the surfaces of the GaN film (x ¼ 0 and 2l, see Fig. 1) there exists an exciting force due to piezoeffect, and the film serves as a transducer for AW. The excitation has a resonance character. To simplify the formulas, consider a case when the elastic properties of the film and substrate are the same. The expression of the amplitude of AW outgoing to the substrate is
m ¼ 0; 1; 2 . . . .
(8)
4. Conclusion The excitation of AW has been considered here for the GaN film of the zinc blende structure. GaN films of the wurtzite structure are also piezoactive, and also can be used as transducers of AW. An excitation of AW can also occur due to deformation potential. The last interaction seems weaker than in the case due to piezoeffect. Our calculations have demonstrated that the output intensity is at least 10 times smaller, but the longitudinal AW (of a polarization u1) can be excited there.
ARTICLE IN PRESS F. Diaz et al. / Microelectronics Journal 39 (2008) 740–743
Acknowledgment This work was supported by CONACyT, Mexico (Project 48955).
References [1] D. Royer, E. Dieulessaint, Elastic Waves in Solids (2 Vols.), Springer, New York, 2000. [2] V. Grimalsky, E. Gutierrez-D, A. Garcia-B, S. Koshevaya, Resonant excitation of microwave acoustic modes in n-GaAs film, in: Proceedings of International Conference on Microelectronics ICM-2004, Paper No. 66, Tunis, Tunisia, December 2004, 4pp.
743
[3] V. Grimalsky, E. Gutierrez-D, A. Garcia-B, S. Koshevaya, Resonant excitation of microwave acoustic modes in n-GaAs, Microelectr. J. 37 (3) (2006) 395–403. [4] S.J. Pearton, J.C. Zolper, R.J. Shul, F. Ren, GaN: processing, defects, and devices, J. Appl. Phys. 86 (1) (1999) 1–79. [5] S.C. Jain, M. Willander, J. Narayan, R. Van Overstraeten, III-Nitrides: growth, characterization, and properties, J. Appl. Phys. 87 (3) (2000) 965–1006. [6] V. Gruzhinskis, P. Shiktorov, E. Starikov, J.H. Zhao, Comparative study of 200–300 GHz microwave power generation in GaN TEDs by the Monte Carlo technique, Semicond. Sci. Techn. 16 (8) (2001) 798–805. [7] M.E. Levinshtein, S.L. Rumyantsev, M.S. Shur, Properties of Advanced Semiconductor Materials: GaN, AlN, InN, WileyInterscience, New York, 2001. [8] Internet site: /http://www.ioffe.ru/SVA/NSM/Semicond/GaNS.
Microelectronics Journal 39 (2008) 740–743 www.elsevier.com/locate/mejo
Excitation of hypersound in n-GaN films F. Diaz, V. Grimalsky, M. Tecpoyotl, J. Escobedo, S. Koshevaya Autonomous University of Morelos, Research Center of Engineering and Applied Sciences, Avenue University 1001, Colony Chamilpa, Cuernavaca 62209, Morelos, Mexico Received 16 October 2007; accepted 15 December 2007 Available online 29 February 2008
Abstract The aim of this paper is the analysis of hypersound excitation in GaN films. Simulation of process is presented for a case of a thin film geometry on a non-piezoelectric substrate. The frequency range considered is from 50 GHz up to 200 GHz. The excitation is due to coupling with space charge waves (SCWs) in GaN film. The amplification of SCWs is related with negative differential conductivity in GaN films. Possible spatial increments are obtained. The amplified SCWs can excite hypersonic waves at the same frequency due to piezoeffect and deformation potential mechanisms. The first effect is stronger and causes an effective resonant excitation of hypersonic waves in the case of full mechanic contact of GaN film and non-piezoelectric substrate. r 2008 Elsevier Ltd. All rights reserved. Keywords: Excitation of hypersound; Space charge waves; Negative differential conductivity; Piezoeffect mechanism
1. Introduction There are traditional methods of ultrasound excitation, like comb and thin film transducers (at microwave frequencies f43 GHz [1]), but for excitation of hypersound (f430 GHz) actually these methods cannot work. A possible solution of this problem is the resonant coupling of acoustic waves (AWs) with the microwave electric field of space charge waves (SCWs) in materials possessing negative differential conductivity (NDC) (like GaAs) [2,3]. Namely, under propagation in the bias electric field higher than the critical value for observing NDC, the SCW is subject to amplification, and its microwave electric field can achieve high values. On the other hand, due to piezoeffect, deformation potential or electrostriction, this microwave electric field excites hypersonic AWs. In [2,3] it was demonstrated that this excitation has a resonant character with respect to frequency and the thickness of the GaAs film. But, the critical value of bias electric field in GaAs is Ec=3.5 kV/cm that limits the maximal values of the microwave electric Corresponding author. Tel./fax: +52 777 329 7084.
E-mail address: [email protected] (F. Diaz). 0026-2692/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2007.12.007
field of SCW and output values of amplitudes of excited AW. Also, the frequency range of amplification of SCW in GaAs films is fo50 GHz. To excite the powerful hypersonic AW at frequencies higher than 50 GHz, it is rather better to use new materials possessing NDC at higher frequencies, for example GaN [4–7] for f=100–300 GHz. The attracting properties of GaN are: (1) a high critical bias field Ec100 kV/cm; (2) extended frequency range for observing NDC fp500 GHz; and (3) high values of piezoelectric constants. In this article the excitation of hypersound in GaN films is researched. Simulation of process is preparing for case of this material for a thin film geometry on a solid substrate. The frequency range considered is f=30–200 GHz. Simulation of excitation of hypersonic AW is given for GaN films of a sub-micron thickness placed onto a non-piezoelectric substrate. The spatial increments of SCW due to NDC have been calculated. An amplification of SCW is possible for frequencies up to 200 or 300 GHz. The thin GaN film plays the role of a transducer of AW. The case of the full acoustic contact between GaN film and the substrate is considered. It has been demonstrated that the excitation of AW has a resonant character and the intensities of AW may reach the values of 1 W/cm2.
ARTICLE IN PRESS F. Diaz et al. / Microelectronics Journal 39 (2008) 740–743
2. SCWs amplification in thin GaN films It is necessary to analyze at first, the SCWs amplification in n-GaN film of a sub-micron thickness placed onto a semi-infinite dielectric substrate, see Fig. 1. The bias electric field is directed along OZ-axis, the SCW are excited by the input antenna I and propagate also in Z-direction. The dynamics of SCW is described by the equations of motion of electrons jointly with the Poisson equation for the electric field. In the frequency range fo200 GHz it is possible to use the simplest hydrodynamic diffusion-drift equation for the electron fluid: qn q ~ Dn rnÞ ¼ 0; þ r ð~ vðEÞn qt qz ~ E, ~ v ¼ mðjEjÞ rð0 ðxÞrjÞ ¼ eðn n0 Þ; ~ ¼ rj þ E 0 . E
(1)
Here n is the electron concentration, j is the electric potential of the alternative field, v is the electron velocity, n0 is the equilibrium electron concentration (equal to the donor one), Dn is the diffusion coefficient, m(E) is electron mobility, E0 is the bias electric field. The data for GaN are taken from [8]. A dependence of the diffusion coefficient Dn on the electric field is kB T e mðjEjÞ, jej 2 mðjEjÞv2 2 wðjEjÞ kB T e ¼ wðjEjÞ. 3 3 2
The dependence of electron velocity on electric field for wurtzite and zinc-blende GaN is obtained from [8]. In this article we use the zinc-blende structure material for the film construction. Amplification of SCW is analyzed below. We use at first, an approximation of a thin film where the electron concentration is assumed as two-dimensional. The linearized equations for the perturbation of electron concentra~ tion n˜(z,t) ¼ nn0 ðjnj5n 0 Þ and for the electric potential j are qn~ qn~ d~ v qE~ z q2 n~ þ v 0 þ n0 Dn 2 ¼ 0, qt qz dE qz x¼0 qz v0 ¼ vðE 0 Þ, q qj q2 j e ~ ðxÞ, ðxÞ þ ðxÞ 2 ¼ 2nld qx qx qz 0 qj . (3) E~ z ¼ qz n˜ and j are variable parts of electron concentration and electric potential, respectively; E0 is a bias electric field within the film, v0 ¼ m(E0)E0 is the constant part of electron drift velocity, m ¼ m(E0) ¼ v0/E0 and e is the electron charge. Consider the case of NDC: ðd~ v=dEÞo0. For linear SCW, we try to obtain a solution of Eq. (3) as: exp(i(otkz)). The dispersion equation k ¼ k(o) for SCW is
Dn ¼
o kv0 i (2)
At higher electric fields E4100 kV/cm, the value of diffusion coefficient is somewhat higher that at low fields (Dn(EE0) ¼ 25 cm2/s). The coordinate frame is aligned along the crystalline axes. The lower indexes 1 and 2 are related to the substrate and the film, respectively.
Fig. 1. The GaN film occupies the region 0oxo2l,x42l is vacuum, xo0 is a semi-infinite substrate. I is antenna input and II is output one.
741
2n0 ljej d~ v k iDn k2 ¼ 0, 0 ð1 þ 1 Þ dE
(4)
where k is the wave number, for spatial amplification of SCW (circular frequency o ¼ 2pf is real, k ¼ k0 +ik00 is complex), in some frequency range one can obtain k00 40. We use the following parameters of GaN film: the electron concentration is n0E1017 cm3, the bias electric field is E0 ¼ 150 kV/cm, the diffusion coefficient is Dn ¼ 50 cm2/s, the drift velocity is v0 ¼ 2.3 107 cm/s, the thickness of the film is 2l ¼ 0.1 mm. The dielectric permittivity of the substrate (like sapphire) is e1 ¼ 10. In the second approach, the finite thickness of the film has been taken into account. Our simulations have been demonstrated that there is no principal difference between the simplified and more complex approaches at frequencies fp200 GHz. The dependencies of spatial increment of SCW k00 on frequency f ¼ o/2p are given in Fig. 2. The curves 1–4 have been calculated when a finite thickness of the film has been considered, whereas the curve 10 has been calculated within an approximation of 2D electron gas. We can appreciate that an amplification of SCW in GaN film occurs in wide frequency range, and the maximal spatial increment is k00 ¼ 3 105 m1 at frequency f ¼ 150 GHz. When compared with a case of GaAs film, it is possible to observe an amplification of SCW in GaN films, at essentially frequencies higher than 100 GHz. In order to obtain an amplification of SCW of 20 dB, it is necessary to use the distance between input and output antennas of about 50 mm. An amplification of SCW in GaN films
ARTICLE IN PRESS F. Diaz et al. / Microelectronics Journal 39 (2008) 740–743
742
Fig. 2. Spatial increment of instability k00 (f) of SCW in GaN zinc-blende film. Curve 1 is for E0 ¼ 150 kV/cm, n0 ¼ 1017 cm3, curve 2 is for E0 ¼ 150 kV/cm, n0 ¼ 0.5 1017 cm3, curve 3 is for E0 ¼ 150 kV/cm, n0 ¼ 1.5 1017 cm3, curve 4 is for E0 ¼ 155 kV/cm, n0 ¼ 1 1017 cm3. For a comparison, the curve 10 is calculated under the same conditions of curve, 1 but using a 2D approximation (Eq. (3)).
should be realized in pulse regime (of a duration o1 ms), because of heating the GaN film. We do not consider here the frequency range f4200 GHz, because an applicability of hydrodynamic equations would be doubtful [6].
Fig. 3. Efficiency of excitation of AW due to piezoeffect (curve 1) and due to deformation potential (curve 2).
2ibE~ z 2 sin ðKl Þe2iKl , rs2 K o iGo 1=2 1þ 2 K¼ . s rs
U¼
(7)
The effective excitation of AW occurs when the following resonant condition takes place: p ð2m þ 1Þ; 2
3. Excitation of hypersonic AWs
Kl
The excitation of hypersonic AWs in the GaN film is possible at the same frequency due to piezoeffect. Because the sound velocity is two orders smaller than the velocity of SCW, the direction of propagation of excited AW is perpendicular to the surface of the film parallel to OX axis downwards. In the zinc-blende GaN (cubic symmetry), the nonzero piezoelectric module is b ¼ b3,12 ¼ 0.4 C/m2. Therefore, the alternative electric field E~ z of SCW can excite the shear AW (u ¼ u2) that propagates along the OX axis. The equation of elasticity theory for mechanical displacement u takes the form: q2 u q q q2 u 2 qu rs G r 2 ¼ þ . (5) qt qx qx qx qxqt
When the acoustic properties of the film and the substrate are different, the resonant condition (7) is valid, but the expression for the amplitude of the outgoing wave is more complicated. At the frequency f ¼ 100 GHz, the dissipation of shear AW is |K00 |500 cm1 [1], so the condition jK 00 jl3 103 51 is satisfied. For the amplitude of alternative electric field of SCW E~ z ¼ 10 kV=cm, the output intensity of AW Pa ¼ (1/2)rs3|KU|2 can reach the values Pa1 W/cm2 in the resonant case. The efficiency of excitation of AW is given in Fig. 3. A dependence of elastic deformation of AW at x ¼ 0 on a frequency is presented there for both, an excitation due to piezoeffect (curve 1) and due to deformation potential (curve 2). The value of the bias electric field is E0 ¼ 150 kV/cm, the electron concentration is n0 ¼ 1017 cm3. The longitudinal AW can be excited there due to deformation potential.
Here r is the density elastic medium, s is the shear acoustic velocity, G is viscosity. The expression for mechanical stress s ¼ s12 within the GaN film is s ¼ r2 s22
qu þ bE~ z . qx
(6)
The simulations have shown that within the film, 0oxo2l, the alternative electric field E~ z is uniform: qE~ z =qx 0. Therefore, only at the surfaces of the GaN film (x ¼ 0 and 2l, see Fig. 1) there exists an exciting force due to piezoeffect, and the film serves as a transducer for AW. The excitation has a resonance character. To simplify the formulas, consider a case when the elastic properties of the film and substrate are the same. The expression of the amplitude of AW outgoing to the substrate is
m ¼ 0; 1; 2 . . . .
(8)
4. Conclusion The excitation of AW has been considered here for the GaN film of the zinc blende structure. GaN films of the wurtzite structure are also piezoactive, and also can be used as transducers of AW. An excitation of AW can also occur due to deformation potential. The last interaction seems weaker than in the case due to piezoeffect. Our calculations have demonstrated that the output intensity is at least 10 times smaller, but the longitudinal AW (of a polarization u1) can be excited there.
ARTICLE IN PRESS F. Diaz et al. / Microelectronics Journal 39 (2008) 740–743
Acknowledgment This work was supported by CONACyT, Mexico (Project 48955).
References [1] D. Royer, E. Dieulessaint, Elastic Waves in Solids (2 Vols.), Springer, New York, 2000. [2] V. Grimalsky, E. Gutierrez-D, A. Garcia-B, S. Koshevaya, Resonant excitation of microwave acoustic modes in n-GaAs film, in: Proceedings of International Conference on Microelectronics ICM-2004, Paper No. 66, Tunis, Tunisia, December 2004, 4pp.
743
[3] V. Grimalsky, E. Gutierrez-D, A. Garcia-B, S. Koshevaya, Resonant excitation of microwave acoustic modes in n-GaAs, Microelectr. J. 37 (3) (2006) 395–403. [4] S.J. Pearton, J.C. Zolper, R.J. Shul, F. Ren, GaN: processing, defects, and devices, J. Appl. Phys. 86 (1) (1999) 1–79. [5] S.C. Jain, M. Willander, J. Narayan, R. Van Overstraeten, III-Nitrides: growth, characterization, and properties, J. Appl. Phys. 87 (3) (2000) 965–1006. [6] V. Gruzhinskis, P. Shiktorov, E. Starikov, J.H. Zhao, Comparative study of 200–300 GHz microwave power generation in GaN TEDs by the Monte Carlo technique, Semicond. Sci. Techn. 16 (8) (2001) 798–805. [7] M.E. Levinshtein, S.L. Rumyantsev, M.S. Shur, Properties of Advanced Semiconductor Materials: GaN, AlN, InN, WileyInterscience, New York, 2001. [8] Internet site: /http://www.ioffe.ru/SVA/NSM/Semicond/GaNS.