- Email: [email protected]
Direct generation of longitudinal acoustic waves by electric field in semiconductor heterostructures
Physb1=21574=Gracy=Venkatachala=BG
Physica B 284}288 (2000) 1153}1154
Direct generation of longitudinal acoustic waves by electric "eld in semiconductor heterostructures Valery I. Khizhnyi Institute of Radiophysics and Electronics of NASU, 12 Acad.Proskura st., 310085, Kharkov, Ukraine
Abstract The mechanism of sound generation by an external variable electric "eld in semiconductor heterostructure is considered. We take into account a spatially non-uniform electric charged area in the whole heterostructure. Provided that Coulomb forces only are responsible for the sound generation, the phenomenological `piezo-likea constant is derived. For the case of low temperatures, the temperature dependence of an electric "eld-sound conversion signal is considered. 2000 Elsevier Science B.V. All rights reserved. Keywords: Longitudinal acoustic waves; Semiconductor heterostructure; Sound generation
1. Introduction The modulation } doping concept [1] has enabled not only essentially to increase mobility of charged carriers in semiconductor heterostructures (HS), but also has allowed to put forward an idea [2] about an opportunity of observation of linear generation of sound by an external electromagnetic "eld. In experiments [3] it was shown that the high-frequency electric "eld applied in the growth direction of Si/SiGe HS generates longitudinal acoustic waves. In this paper we consider sound generation caused by spatially separated charged regions in HS. By solving the sound wave equation and the Poisson equation we have derived the amplitude of sound generated as a function of HS parameters and temperature.
2. Sound amplitude Let us consider the model of HS with the single quantum well and a Schottky barrier (SB) at the surface of the HS. The space charge region of the SB should not extend to the quantum well under small AC bias voltage [4]. Let the width of SB region be ¸ and the width of the E-mail address: [email protected] (V.I. Khizhnyi)
charged depletion region at the side of the quantum well be ¸ . These regions are separated by a neutral region with free carriers (at some temperature ¹O0) and with the width of ¸. The lengths of all these layers are determined by band parameters. The valence band of such HS is presented in Ref. [4]. Let us consider the case when uq;1, where u is the angular frequency of the sound wave and q\ is the relaxation frequency of carriers in region ¸, and the sound wavelength j<¸ , ¸ , ¸. (1) The phenomenon of sound excitation by an external electric "eld E""x (x is the crystal growth axis) can be described (without attenuation) by the wave equation: *u 1 #ku"! f . *x c
(2)
Here u(x, t)"u(x) exp(iut) is the displacement of the lattice, k"u/< is the sound wave vector, <"(c/o ) , c is the longitudinal elastic modulus, o is the density of the material, f is the density of volume forces acting on the lattice. The f can be determined as f (x)"o(x)E (x), (3) where o(x) is the charge density distribution along the x direction and E (x) is the variable electric "eld distribution caused by an external applied AC bias voltage. By
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 5 6 2 - 4
Phyb=21574=GV=VVC=BG
1154
V.I. Khizhnyi / Physica B 284}288 (2000) 1153}1154
neglecting the secondary e!ect of acoustic-wave transformation into electric "eld and assuming that the HS is loaded by a semi-in"nite acoustic-bu!er rod with the same acoustic impedance, one can obtain the solution of Eq. (2) for the acoustic-wave amplitude u"u(x):
ie " u"! e IV cos(kx )E (x )o(x ) dx , (4) kc where e is the electron charge, D is the sum of all `activea region lengths. The "eld distribution E (x) is de"ned as the "rst approximation that the small variable potential <(x) in the structure causes small charge variation do(x);o(x) in ¸ layer only. By using the boundary conditions of potential and electric "eld at the interfaces x"¸ and x"¸ #¸, the electrical "eld distribution could be determined by solving the Poisson equation do(x) d<(x) "! , dx e
ieN (E ¸ #E ¸ ), u+! kc
(6)
where N is the doping concentration, and E "< (¸ #¸ )\, (7) " E "¸\(E ¸ #< )e\* *" , (8) " where < is the external AC potential at the surface and ¸ is the screening length for Maxwell}Boltzmann statis" tics
E n +(n N ) exp ! , k ¹;E , 2k ¹ n "2(mk ¹/2p ) .
Acknowledgements This work was partially supported by the STCU Project N346. The author thanks V. Litvinov for aquaintance with results of an unpublished work.
(5)
where e is the dielectric constant. Thus, from Eqs. (2)}(4) and according to approximation (1), the "nal result for the sound wave amplitude could be shown to be
ek ¹ ¸ " , " en
Here k is the Boltzmann constant, E is the ionization energy of acceptors and m is the e!ective hole mass. As visible from Eq. (8) the temperature dependence of sound generation e$ciency is determined by the parameter ¸ . " For the conditions of experiment [3] the formulae give the deep fall of sound amplitude at ¹ less than 20 K. This temperature is de"ned by the acceptors ionization energy E . Thus, the above-described mechanism of sound exci tation can be useful for study of dynamical electrical properties of HS by acoustic methods.
(9)
References [1] L. Esaki, R. Tsu, IBM Res. Rep. RC (1969) 2418. [2] J.J. Quinn, U. Strom, L.L. Chang, Solid State Commun. 45 (1983) 111. [3] V.I. Khizhnyi, O.A. Mironov, E.H.C. Parker et al., Appl. Phys. Lett. 69 (1996) 960. [4] J.B. Wang, F. Lu, S.K. Zhang et al., Phys. Rev. 54 (1996) 7979.
Physica B 284}288 (2000) 1153}1154
Direct generation of longitudinal acoustic waves by electric "eld in semiconductor heterostructures Valery I. Khizhnyi Institute of Radiophysics and Electronics of NASU, 12 Acad.Proskura st., 310085, Kharkov, Ukraine
Abstract The mechanism of sound generation by an external variable electric "eld in semiconductor heterostructure is considered. We take into account a spatially non-uniform electric charged area in the whole heterostructure. Provided that Coulomb forces only are responsible for the sound generation, the phenomenological `piezo-likea constant is derived. For the case of low temperatures, the temperature dependence of an electric "eld-sound conversion signal is considered. 2000 Elsevier Science B.V. All rights reserved. Keywords: Longitudinal acoustic waves; Semiconductor heterostructure; Sound generation
1. Introduction The modulation } doping concept [1] has enabled not only essentially to increase mobility of charged carriers in semiconductor heterostructures (HS), but also has allowed to put forward an idea [2] about an opportunity of observation of linear generation of sound by an external electromagnetic "eld. In experiments [3] it was shown that the high-frequency electric "eld applied in the growth direction of Si/SiGe HS generates longitudinal acoustic waves. In this paper we consider sound generation caused by spatially separated charged regions in HS. By solving the sound wave equation and the Poisson equation we have derived the amplitude of sound generated as a function of HS parameters and temperature.
2. Sound amplitude Let us consider the model of HS with the single quantum well and a Schottky barrier (SB) at the surface of the HS. The space charge region of the SB should not extend to the quantum well under small AC bias voltage [4]. Let the width of SB region be ¸ and the width of the E-mail address: [email protected] (V.I. Khizhnyi)
charged depletion region at the side of the quantum well be ¸ . These regions are separated by a neutral region with free carriers (at some temperature ¹O0) and with the width of ¸. The lengths of all these layers are determined by band parameters. The valence band of such HS is presented in Ref. [4]. Let us consider the case when uq;1, where u is the angular frequency of the sound wave and q\ is the relaxation frequency of carriers in region ¸, and the sound wavelength j<¸ , ¸ , ¸. (1) The phenomenon of sound excitation by an external electric "eld E""x (x is the crystal growth axis) can be described (without attenuation) by the wave equation: *u 1 #ku"! f . *x c
(2)
Here u(x, t)"u(x) exp(iut) is the displacement of the lattice, k"u/< is the sound wave vector, <"(c/o ) , c is the longitudinal elastic modulus, o is the density of the material, f is the density of volume forces acting on the lattice. The f can be determined as f (x)"o(x)E (x), (3) where o(x) is the charge density distribution along the x direction and E (x) is the variable electric "eld distribution caused by an external applied AC bias voltage. By
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 5 6 2 - 4
Phyb=21574=GV=VVC=BG
1154
V.I. Khizhnyi / Physica B 284}288 (2000) 1153}1154
neglecting the secondary e!ect of acoustic-wave transformation into electric "eld and assuming that the HS is loaded by a semi-in"nite acoustic-bu!er rod with the same acoustic impedance, one can obtain the solution of Eq. (2) for the acoustic-wave amplitude u"u(x):
ie " u"! e IV cos(kx )E (x )o(x ) dx , (4) kc where e is the electron charge, D is the sum of all `activea region lengths. The "eld distribution E (x) is de"ned as the "rst approximation that the small variable potential <(x) in the structure causes small charge variation do(x);o(x) in ¸ layer only. By using the boundary conditions of potential and electric "eld at the interfaces x"¸ and x"¸ #¸, the electrical "eld distribution could be determined by solving the Poisson equation do(x) d<(x) "! , dx e
ieN (E ¸ #E ¸ ), u+! kc
(6)
where N is the doping concentration, and E "< (¸ #¸ )\, (7) " E "¸\(E ¸ #< )e\* *" , (8) " where < is the external AC potential at the surface and ¸ is the screening length for Maxwell}Boltzmann statis" tics
E n +(n N ) exp ! , k ¹;E , 2k ¹ n "2(mk ¹/2p ) .
Acknowledgements This work was partially supported by the STCU Project N346. The author thanks V. Litvinov for aquaintance with results of an unpublished work.
(5)
where e is the dielectric constant. Thus, from Eqs. (2)}(4) and according to approximation (1), the "nal result for the sound wave amplitude could be shown to be
ek ¹ ¸ " , " en
Here k is the Boltzmann constant, E is the ionization energy of acceptors and m is the e!ective hole mass. As visible from Eq. (8) the temperature dependence of sound generation e$ciency is determined by the parameter ¸ . " For the conditions of experiment [3] the formulae give the deep fall of sound amplitude at ¹ less than 20 K. This temperature is de"ned by the acceptors ionization energy E . Thus, the above-described mechanism of sound exci tation can be useful for study of dynamical electrical properties of HS by acoustic methods.
(9)
References [1] L. Esaki, R. Tsu, IBM Res. Rep. RC (1969) 2418. [2] J.J. Quinn, U. Strom, L.L. Chang, Solid State Commun. 45 (1983) 111. [3] V.I. Khizhnyi, O.A. Mironov, E.H.C. Parker et al., Appl. Phys. Lett. 69 (1996) 960. [4] J.B. Wang, F. Lu, S.K. Zhang et al., Phys. Rev. 54 (1996) 7979.