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Piezoelectric excitation of semiconductor plates
Ultrasonics 38 (2000) 849–851 www.elsevier.nl/locate/ultras
Piezoelectric excitation of semiconductor plates Arthur Ballato * US Army Communications-Electronics Command, Fort Monmouth, NJ, USA
Abstract The microelectronics revolution is due largely to the electronic properties of semiconducting crystals. Almost all binary semiconductors are piezoelectric, and can be fashioned with high resistivity, permitting incorporation of mechanical motion as an additional variable, leading to the possibility of mixed-effect devices with novel properties. In order to further develop these new structures, this paper provides newly derived expressions for piezoelectric excitation of simple thickness plate modes of binary semiconductors, by both thickness and lateral electric fields. © 2000 Published by Elsevier Science B.V. Keywords: Piezoelectricity; Plates; Resonators; Semiconductors; Vibrations
1. Introduction Modern communications are an offshoot of the microelectronics revolution. This, in turn, is due largely to the electronic properties of semiconducting crystals. Major drivers in the future will be cost and packing density, which can be impacted by providing both increased functionality and integration. One way of approaching this is by making use of the fact that virtually all III–V and II–VI compound semiconductors are piezoelectric, and many can be fashioned to have areas with high resistivity. This allows one to integrate, on one substrate, all the ‘smarts’ associated with the usual micro/nanoelectronics circuitry, along with highQ acoustomechanical devices [1]. Resonant membranes etched in a binary semiconductor chip can provide stable frequencies, cheaply, directly in the gigahertz range. Network means of compensation for temperature and other effects are readily accomplished because the resonant structure is integral with the circuitry. Binary semiconductors exhibit, primarily, either the wurtzite or the zincblende crystal structure. Wurtzite belongs to the 6mm point group, whereas zincblende has point group symmetry 4: 3m. Both of these classes exhibit piezoelectricity. As with the Group IV elements Ge and Si, the III–V intermetallic semiconductors form tetrahedral bonds with four valence electrons. Because
* Tel.: +1-732-427-4308; fax: +1-732-427-3733. E-mail address: [email protected] (A. Ballato) 0041-624X/00/$ - see front matter © 2000 Published by Elsevier Science B.V. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 10 0 - 6
the zincblende and wurtzite structures are both tetrahedrally coordinated, they are closely related [2–4]. Table 1 lists the crystal class of some of the III–V binaries; entry ‘c’ denotes cubic zincblende, and ‘h’ stands for hexagonal wurtzite. Table 2 similarly gives a listing of some II–VI compounds. The tables are composed by pairing non-metals with metals that form covalent solids. The important Group IV binary SiC exists in both forms: a-h and b-c. Piezoelectric semiconducting resonator-type devices have been used in the past, in various acoustic and ultrasonic applications [5,6 ]. Interest is currently growTable 1 Binary III–V semiconductorsa
B Al Ga In
N
P
As
Sb
c h h h
c c c c
c c c c
c c c
a c, h denote zincblende, wurtzite structures. Table 2 Binary II–VI semiconductorsa
Be Zn Cd
O
S
Se
Te
h h
c a-h; b-c c
c c c
c c c
a c, h denote zincblende, wurtzite structures.
850
A. Ballato / Ultrasonics 38 (2000) 849–851
ing at an explosive rate (IEEE Journal of Microelectromechanical Systems, Sensors and Actuators, and a host of other journals are filled with papers relevant to our topic), and new developments in this topic are presented here. In particular, the prototypical example of simple thickness modes of plates and their piezoelectric excitation are considered. These motions are independent of the plate lateral coordinates, so the problem is one-dimensional [7,8]. Plates are most often driven piezoelectrically by electrodes placed on the major surfaces, creating an electric field in the thickness direction; this is referred to as thickness excitation ( TE ). When the exciting field lies instead in the plane of the plate, the situation is referred to as lateral excitation (LE ) [9]. A plate resonator of thickness t, mass density r, and modal elastic stiffness c supports m three simple thickness modes (m=a, b, c; resp. quasilongitudinal, fast and slow quasi-shear) with fundamental eigenfrequencies f =v /2t, where v =Ec /r are the m m m m acoustic velocities. Apart from the eigenstiffnesses c , m the most important practical quantities associated with the vibrational modes are the values of the piezoelectric coupling factors, k( TE ) and k(LE ) for the two types m m of excitation. These determine the circuit functions of devices fashioned from the resonators. For example, the input immittance of each type of resonator, simplified for illustration to the case of a single active mode, varies as follows [with h =(p/2)( f/f )=vt/2v ]: m m m Z ( TE ) 3[1−k2( TE ) · tan h /h ] in m m m m Y (LE) 3[1+k2(LE ) · tan h /h ]. in m m m m
2. Zincblende structure All possible independent orientations for this class are circumscribed within the primitive region bounded by great-circle arcs joining the three principal directions [100], [110], and [111] corresponding, respectively, to the cube axis, the face diagonal, and the body diagonal. Space constraints limit discussion to the principal directions, for which are given newly derived results relating to piezoelectric coupling coefficients. $ For the [100] cut, all three k( TE ) are zero. Stiffnesses m are: c =cE ; c =c =cE . A lateral field at any azia 11 b c 44 muth drives shear motion with particle displacement along the field direction. The coupling coefficient is k(LE ) =|e |/Ee cE . 11 44 shear 14 $ For the [110] cut, k( TE) and k( TE) are zero. a c Stiffnesses are: c =(cE +cE )/2+cE ; c =c : = a 11 12 44 b 44 (cE +e2 /e ); c =(cE −cE )/2. The fast shear mode 44 14 11 c 11 12 is uncoupled, and driven with coupling factor k( TE ) =|e |/Ee c : . The quasi-extensional mode b 14 11 44 may be driven with LE, having coupling
k(LE) =|e sin y|/Ee [(cE +cE )/2+cE ], where y a 14 11 11 12 44 is reckoned from [110] toward [001]. $ For the [111] cut, k( TE ) =0. Stiffnesses are: shear c =(cE +2cE +4c : )/3; c =c =(cE −cE +cE )/3. a 11 12 44 b c 11 12 44 The pure extensional mode is driven with coupling factor k(TE ) =2|e |/Ee (cE +2cE +4c : ). A lata 14 11 11 12 44 eral field at any azimuth drives shear motion with particle displacement along the field direction. The coupling coefficient is k(LE ) =|e |/ shear 14 Ee (cE −cE +cE ). 11 11 12 44 It may be shown that for all simple thickness modes of this crystal class, the azimuthal dependence of the lateral coupling at any orientation is simply given by k(LE;y) =|k(LE ) | · |sin(y−y )|, where k(LE) is m mo mo mo the maximum value of k(LE ) for mode m, and y is m mo an offset. The strictly sinusoidal azimuthal dependence is a result of the isotropy of the permittivity in cubic crystals.
3. Wurtzite structure Cubic materials possess a family of doubly rotated orientations within the primitive region discussed above, but hexagonal crystal plate orientations are limited in their variety to a single rotation about any axis normal to the polar axis. This despite having richer elastic, piezoelectric, and dielectric matrix arrays. A synopsis of the behavior of plates of class 6mm material piezoelectrically driven in simple thickness modes has been derived for this paper, and is given below. The unique polar axis is denoted P; P∞ is the direction of projection of P onto the plane of the plate, and n˜ is the direction of the plate normal; y is the in-plane azimuthal angle, reckoned from the basal plane. $ For P parallel to the plate normal ( Z cut), all modes are pure. Stiffnesses are: c =c : =(cE +e2 /e ), a 33 33 33 33 c =c =c =cE . TE drives only the extensional b c shear 44 mode with coupling coefficient k( TE ) =|e |/ a 33 Ee c . LE drives only the shear mode with coupling 33 33: coefficient k(LE ) =|e |/Ee cE , and displaceshear 15 11 44 ment is along the LE field direction. $ For P in the plane of the plate ( Y cut), all three modes are pure, with one shear mode having displacement along P, and the other having displacement in the plane, but normal to P. TE drives only the shear mode having motion along P; its effective stiffness is c : =(cE +e2 /e ), and coupling is k( TE) = 44 44 15 11 shear |e |/Ee c : . The undriven shear has stiffness cE . 15 11 44 66 LE drives only the extensional mode, providing there is a component of E along P; its effective stiffness lateral is cE , and coupling is k(LE ) =|e sin y|/Ee ∞ cE , 11 a 31 11 11 where e ∞ =(e cos2 y+e sin2 y). 11 11 33 $ When the unique axis P is neither parallel nor perpendicular to the plate normal, one shear mode remains
A. Ballato / Ultrasonics 38 (2000) 849–851
$
$
pure, and inert to TE (polarized perpendicular to both P and to the plate normal ). It may be either the ‘b’ or ‘c’ mode, depending on the material and angle. The other two modes are coupled, and TE drives both. All three modes, the pure shear and the two coupled modes, are driven by LE if E is lateral neither normal nor parallel to P∞. If E is normal lateral to P∞, only the pure shear mode, polarized normal to P∞ (and therefore along E ), is driven by LE. If lateral E is parallel to P∞, only the two coupled modes lateral are driven by LE (since they both have components of displacement along E ), and the motions are lateral normal to that of the inert pure shear mode, i.e. in the plane containing P∞ and the plate normal. For a rotated cut, with rotation angle h, and P therefore oblique to the plate normal, the pure shear perpendicular to both P∞ and n˜ is driven by LE (for a rotated Y cut, E is the required component), 1 although k(LE ) goes to zero at the Y cut (h=0°). For h between 0° and 90°, the stiffness for this pure mode is cE =(cE cos2 h+cE sin2 h). The coupureshear 66 44 pling factor is k(LE ) =|e sin h cos y|/ pureshear 15 Ee °cE , where e °=e +D cos2 h sin2 y(1− 11 pureshear 11 11 D sin2h/e ∞ ), D=(e −e ), and e ∞ is given above. 11 33 11 11 The coupled shear and extensional modes (in the plane of P∞ and n˜), for a rotated cut, are driven by LE (for a rotated Y cut, the required component is E∞ ). At the Y cut, the mode coupling ceases, and 3 k(LE ) drives the a mode only. At the Z cut, the mode coupling likewise ceases, but now k(LE ) drives the degenerate shear polarized normal to P with isotropic azimuthal piezocoupling; the displacement is along the applied field. For h between 0° and 90°, the stiffnesses and coupling factors for these modes transform in algebraically cumbersome fashions, that space precludes elaborating here.
851
4. Conclusions The piezoelectric effect in high resistivity binary semiconductors may be utilized to provide frequency control and selection functions in modern communications devices. Electromechanical coupling factor relations are given, for driving electric fields in both the thickness and lateral directions, of prototypical plate resonators belonging to the two crystal classes in which these substances occur.
References [1] A. Ballato, Piezoelectricity: old effect, new thrusts, IEEE Trans. Ultrason., Ferroelect., Freq. Control 42 (5) (1995) 916–926. [2] H. Welker, Discovery and development of III–V compounds, IEEE Trans. Electron Dev. ED-23 (1976) 664–674. [3] W.A. Harrison, Electronic Structure and the Properties of Solids, Dover, New York, 1989. [4] W.R.L. Lambrecht, B. Segall, General remarks and notations on the band structure of pure Group III nitrides, Data Review 4.1, Properties of Group III Nitrides, EMIS Data Reviews Series No. 11, IEE, London, 1994, pp. 125–128. [5] A. Ballato, G.J. Iafrate, Contactless resistivity measurement method and apparatus, US Patent 4,353,027, issued October 5, 1982. [6 ] J. So¨derkvist, K. Hjort, The piezoelectric effect of GaAs used for resonators and resonant sensors, J. Micromech. Microeng. 4 (1994) 28–34. [7] T.R. Meeker, Thickness mode piezoelectric transducers, Ultrasonics 10 (1) (1972) 26–36. [8] A. Ballato, Doubly rotated thickness mode plate vibrators, in: W.P. Mason, R.N. Thurston ( Eds.), Physical Acoustics: Principles and Methods Vol. 13, Academic Press, New York, 1977, pp. 115–181, Chapter 5. [9] A. Ballato, E.R. Hatch, M. Mizan, T.J. Lukaszek, Lateral field equivalent networks and piezocoupling factors of quartz plates driven in simple thickness modes, IEEE Trans. Ultrason., Ferroelect., Freq. Control 33 (4) (1986) 385–392.
Piezoelectric excitation of semiconductor plates Arthur Ballato * US Army Communications-Electronics Command, Fort Monmouth, NJ, USA
Abstract The microelectronics revolution is due largely to the electronic properties of semiconducting crystals. Almost all binary semiconductors are piezoelectric, and can be fashioned with high resistivity, permitting incorporation of mechanical motion as an additional variable, leading to the possibility of mixed-effect devices with novel properties. In order to further develop these new structures, this paper provides newly derived expressions for piezoelectric excitation of simple thickness plate modes of binary semiconductors, by both thickness and lateral electric fields. © 2000 Published by Elsevier Science B.V. Keywords: Piezoelectricity; Plates; Resonators; Semiconductors; Vibrations
1. Introduction Modern communications are an offshoot of the microelectronics revolution. This, in turn, is due largely to the electronic properties of semiconducting crystals. Major drivers in the future will be cost and packing density, which can be impacted by providing both increased functionality and integration. One way of approaching this is by making use of the fact that virtually all III–V and II–VI compound semiconductors are piezoelectric, and many can be fashioned to have areas with high resistivity. This allows one to integrate, on one substrate, all the ‘smarts’ associated with the usual micro/nanoelectronics circuitry, along with highQ acoustomechanical devices [1]. Resonant membranes etched in a binary semiconductor chip can provide stable frequencies, cheaply, directly in the gigahertz range. Network means of compensation for temperature and other effects are readily accomplished because the resonant structure is integral with the circuitry. Binary semiconductors exhibit, primarily, either the wurtzite or the zincblende crystal structure. Wurtzite belongs to the 6mm point group, whereas zincblende has point group symmetry 4: 3m. Both of these classes exhibit piezoelectricity. As with the Group IV elements Ge and Si, the III–V intermetallic semiconductors form tetrahedral bonds with four valence electrons. Because
* Tel.: +1-732-427-4308; fax: +1-732-427-3733. E-mail address: [email protected] (A. Ballato) 0041-624X/00/$ - see front matter © 2000 Published by Elsevier Science B.V. PII: S0 0 4 1 -6 2 4 X ( 9 9 ) 0 0 10 0 - 6
the zincblende and wurtzite structures are both tetrahedrally coordinated, they are closely related [2–4]. Table 1 lists the crystal class of some of the III–V binaries; entry ‘c’ denotes cubic zincblende, and ‘h’ stands for hexagonal wurtzite. Table 2 similarly gives a listing of some II–VI compounds. The tables are composed by pairing non-metals with metals that form covalent solids. The important Group IV binary SiC exists in both forms: a-h and b-c. Piezoelectric semiconducting resonator-type devices have been used in the past, in various acoustic and ultrasonic applications [5,6 ]. Interest is currently growTable 1 Binary III–V semiconductorsa
B Al Ga In
N
P
As
Sb
c h h h
c c c c
c c c c
c c c
a c, h denote zincblende, wurtzite structures. Table 2 Binary II–VI semiconductorsa
Be Zn Cd
O
S
Se
Te
h h
c a-h; b-c c
c c c
c c c
a c, h denote zincblende, wurtzite structures.
850
A. Ballato / Ultrasonics 38 (2000) 849–851
ing at an explosive rate (IEEE Journal of Microelectromechanical Systems, Sensors and Actuators, and a host of other journals are filled with papers relevant to our topic), and new developments in this topic are presented here. In particular, the prototypical example of simple thickness modes of plates and their piezoelectric excitation are considered. These motions are independent of the plate lateral coordinates, so the problem is one-dimensional [7,8]. Plates are most often driven piezoelectrically by electrodes placed on the major surfaces, creating an electric field in the thickness direction; this is referred to as thickness excitation ( TE ). When the exciting field lies instead in the plane of the plate, the situation is referred to as lateral excitation (LE ) [9]. A plate resonator of thickness t, mass density r, and modal elastic stiffness c supports m three simple thickness modes (m=a, b, c; resp. quasilongitudinal, fast and slow quasi-shear) with fundamental eigenfrequencies f =v /2t, where v =Ec /r are the m m m m acoustic velocities. Apart from the eigenstiffnesses c , m the most important practical quantities associated with the vibrational modes are the values of the piezoelectric coupling factors, k( TE ) and k(LE ) for the two types m m of excitation. These determine the circuit functions of devices fashioned from the resonators. For example, the input immittance of each type of resonator, simplified for illustration to the case of a single active mode, varies as follows [with h =(p/2)( f/f )=vt/2v ]: m m m Z ( TE ) 3[1−k2( TE ) · tan h /h ] in m m m m Y (LE) 3[1+k2(LE ) · tan h /h ]. in m m m m
2. Zincblende structure All possible independent orientations for this class are circumscribed within the primitive region bounded by great-circle arcs joining the three principal directions [100], [110], and [111] corresponding, respectively, to the cube axis, the face diagonal, and the body diagonal. Space constraints limit discussion to the principal directions, for which are given newly derived results relating to piezoelectric coupling coefficients. $ For the [100] cut, all three k( TE ) are zero. Stiffnesses m are: c =cE ; c =c =cE . A lateral field at any azia 11 b c 44 muth drives shear motion with particle displacement along the field direction. The coupling coefficient is k(LE ) =|e |/Ee cE . 11 44 shear 14 $ For the [110] cut, k( TE) and k( TE) are zero. a c Stiffnesses are: c =(cE +cE )/2+cE ; c =c : = a 11 12 44 b 44 (cE +e2 /e ); c =(cE −cE )/2. The fast shear mode 44 14 11 c 11 12 is uncoupled, and driven with coupling factor k( TE ) =|e |/Ee c : . The quasi-extensional mode b 14 11 44 may be driven with LE, having coupling
k(LE) =|e sin y|/Ee [(cE +cE )/2+cE ], where y a 14 11 11 12 44 is reckoned from [110] toward [001]. $ For the [111] cut, k( TE ) =0. Stiffnesses are: shear c =(cE +2cE +4c : )/3; c =c =(cE −cE +cE )/3. a 11 12 44 b c 11 12 44 The pure extensional mode is driven with coupling factor k(TE ) =2|e |/Ee (cE +2cE +4c : ). A lata 14 11 11 12 44 eral field at any azimuth drives shear motion with particle displacement along the field direction. The coupling coefficient is k(LE ) =|e |/ shear 14 Ee (cE −cE +cE ). 11 11 12 44 It may be shown that for all simple thickness modes of this crystal class, the azimuthal dependence of the lateral coupling at any orientation is simply given by k(LE;y) =|k(LE ) | · |sin(y−y )|, where k(LE) is m mo mo mo the maximum value of k(LE ) for mode m, and y is m mo an offset. The strictly sinusoidal azimuthal dependence is a result of the isotropy of the permittivity in cubic crystals.
3. Wurtzite structure Cubic materials possess a family of doubly rotated orientations within the primitive region discussed above, but hexagonal crystal plate orientations are limited in their variety to a single rotation about any axis normal to the polar axis. This despite having richer elastic, piezoelectric, and dielectric matrix arrays. A synopsis of the behavior of plates of class 6mm material piezoelectrically driven in simple thickness modes has been derived for this paper, and is given below. The unique polar axis is denoted P; P∞ is the direction of projection of P onto the plane of the plate, and n˜ is the direction of the plate normal; y is the in-plane azimuthal angle, reckoned from the basal plane. $ For P parallel to the plate normal ( Z cut), all modes are pure. Stiffnesses are: c =c : =(cE +e2 /e ), a 33 33 33 33 c =c =c =cE . TE drives only the extensional b c shear 44 mode with coupling coefficient k( TE ) =|e |/ a 33 Ee c . LE drives only the shear mode with coupling 33 33: coefficient k(LE ) =|e |/Ee cE , and displaceshear 15 11 44 ment is along the LE field direction. $ For P in the plane of the plate ( Y cut), all three modes are pure, with one shear mode having displacement along P, and the other having displacement in the plane, but normal to P. TE drives only the shear mode having motion along P; its effective stiffness is c : =(cE +e2 /e ), and coupling is k( TE) = 44 44 15 11 shear |e |/Ee c : . The undriven shear has stiffness cE . 15 11 44 66 LE drives only the extensional mode, providing there is a component of E along P; its effective stiffness lateral is cE , and coupling is k(LE ) =|e sin y|/Ee ∞ cE , 11 a 31 11 11 where e ∞ =(e cos2 y+e sin2 y). 11 11 33 $ When the unique axis P is neither parallel nor perpendicular to the plate normal, one shear mode remains
A. Ballato / Ultrasonics 38 (2000) 849–851
$
$
pure, and inert to TE (polarized perpendicular to both P and to the plate normal ). It may be either the ‘b’ or ‘c’ mode, depending on the material and angle. The other two modes are coupled, and TE drives both. All three modes, the pure shear and the two coupled modes, are driven by LE if E is lateral neither normal nor parallel to P∞. If E is normal lateral to P∞, only the pure shear mode, polarized normal to P∞ (and therefore along E ), is driven by LE. If lateral E is parallel to P∞, only the two coupled modes lateral are driven by LE (since they both have components of displacement along E ), and the motions are lateral normal to that of the inert pure shear mode, i.e. in the plane containing P∞ and the plate normal. For a rotated cut, with rotation angle h, and P therefore oblique to the plate normal, the pure shear perpendicular to both P∞ and n˜ is driven by LE (for a rotated Y cut, E is the required component), 1 although k(LE ) goes to zero at the Y cut (h=0°). For h between 0° and 90°, the stiffness for this pure mode is cE =(cE cos2 h+cE sin2 h). The coupureshear 66 44 pling factor is k(LE ) =|e sin h cos y|/ pureshear 15 Ee °cE , where e °=e +D cos2 h sin2 y(1− 11 pureshear 11 11 D sin2h/e ∞ ), D=(e −e ), and e ∞ is given above. 11 33 11 11 The coupled shear and extensional modes (in the plane of P∞ and n˜), for a rotated cut, are driven by LE (for a rotated Y cut, the required component is E∞ ). At the Y cut, the mode coupling ceases, and 3 k(LE ) drives the a mode only. At the Z cut, the mode coupling likewise ceases, but now k(LE ) drives the degenerate shear polarized normal to P with isotropic azimuthal piezocoupling; the displacement is along the applied field. For h between 0° and 90°, the stiffnesses and coupling factors for these modes transform in algebraically cumbersome fashions, that space precludes elaborating here.
851
4. Conclusions The piezoelectric effect in high resistivity binary semiconductors may be utilized to provide frequency control and selection functions in modern communications devices. Electromechanical coupling factor relations are given, for driving electric fields in both the thickness and lateral directions, of prototypical plate resonators belonging to the two crystal classes in which these substances occur.
References [1] A. Ballato, Piezoelectricity: old effect, new thrusts, IEEE Trans. Ultrason., Ferroelect., Freq. Control 42 (5) (1995) 916–926. [2] H. Welker, Discovery and development of III–V compounds, IEEE Trans. Electron Dev. ED-23 (1976) 664–674. [3] W.A. Harrison, Electronic Structure and the Properties of Solids, Dover, New York, 1989. [4] W.R.L. Lambrecht, B. Segall, General remarks and notations on the band structure of pure Group III nitrides, Data Review 4.1, Properties of Group III Nitrides, EMIS Data Reviews Series No. 11, IEE, London, 1994, pp. 125–128. [5] A. Ballato, G.J. Iafrate, Contactless resistivity measurement method and apparatus, US Patent 4,353,027, issued October 5, 1982. [6 ] J. So¨derkvist, K. Hjort, The piezoelectric effect of GaAs used for resonators and resonant sensors, J. Micromech. Microeng. 4 (1994) 28–34. [7] T.R. Meeker, Thickness mode piezoelectric transducers, Ultrasonics 10 (1) (1972) 26–36. [8] A. Ballato, Doubly rotated thickness mode plate vibrators, in: W.P. Mason, R.N. Thurston ( Eds.), Physical Acoustics: Principles and Methods Vol. 13, Academic Press, New York, 1977, pp. 115–181, Chapter 5. [9] A. Ballato, E.R. Hatch, M. Mizan, T.J. Lukaszek, Lateral field equivalent networks and piezocoupling factors of quartz plates driven in simple thickness modes, IEEE Trans. Ultrason., Ferroelect., Freq. Control 33 (4) (1986) 385–392.