ON MODEL OF MANUFACTURING OF A W-BAND CMOS POWER AMPLIFIER TO INCREASE DENSITY OF ELEMENTS

Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 123 ON MODEL OF MANUFACTURING OF A W-BAND CMOS POWER AMPLIFIER TO INCREASE DENSITY OF ELEMENTS E.L. Pankratov Nizhny Novgorod State University, 23 Gagarin avenue, Nizhny Novgorod, 603950, Russia Nizhny Novgorod State Agrotechnical University, 97 Gagarin avenue, Nizhny Novgorod, 603950, Russia ABSTRACT In this paper we introduce an approach to increase density of field-effect heterotransistors in the framework of a W-band CMOS power amplifier. In the framework of the approach we consider manufacturing the amplifier in heterostructure with specific configuration. Several required areas of the heterostructure should be doped by diffusion or ion implantation. After that dopant and radiation defects should by annealed in the framework of an optimized scheme. We also consider an approach to decrease value of mismatch-induced stress in the considered heterostructure. We introduce an analytical approach to analyze mass and heat transport in heterostructures during manufacturing of integrated circuits with account mismatch-induced stress. KEYWORDS heterotransistors; W-band CMOS power amplifier; optimization of manufacturing; analytical approach for prognosis. 1. INTRODUCTION In the present time several actual problems of the solid state electronics (such as increasing of performance, reliability and density of elements of integrated circuits: diodes, field-effect and bipolar transistors) are intensively solving [1-6]. To increase the performance of these devices it is attracted an interest determination of materials with higher values of charge carriers mobility [710]. One way to decrease dimensions of elements of integrated circuits is manufacturing them in thin film heterostructures [3-5,11]. In this case it is possible to use inhomogeneity of heterostructure and necessary optimization of doping of electronic materials [12] and development of epitaxial technology to improve these materials (including analysis of mismatch induced stress) [13-15]. An alternative approaches to increase dimensions of integrated circuits are using of laser and microwave types of annealing [16,17]. In the framework of the paper we introduce an approach to optimize manufacture of field-effect heterotransistors. The approach gives a possibility to decrease their dimensions with increasing their density in the framework of W-band CMOS power amplifier. We consider the possibility of decreasing mismatch-induced stress to decrease quantity of defects, generated due to the stress. In this paper we consider a heterostructure, which consist of a substrate and an epitaxial layer (see Fig. 1). We also consider a buffer layer between the substrate and the epitaxial layer. The epitaxi- Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 124 al layer includes into itself several sections, which were manufactured by using another materials. These sections have been doped by diffusion or ion implantation to manufacture the required types of conductivity (p or n). These areas became sources, drains and gates (see Fig. 1). After this doping it is required annealing of dopant and/or radiation defects. Main aim of the present paper is analysis of redistribution of dopant and radiation defects to determine conditions, which correspond to decreasing of elements of the considered amplifier and at the same time to increase their density. At the same time we consider a possibility to decrease mismatch-induced stress. Fig. 1a. Structure of considered amplifier [18] Fig. 1b. Heterostructure with a substrate, epitaxial layers and buffer layer (view from side) 2. METHOD OF SOLUTION To solve our aim we determine and analyzed spatio-temporal distribution of concentration of dopant in the considered heterostructure. We determine the distribution by solving the second Fick's law in the following form [1,19-22] Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 125                                             z tzyxC D zy tzyxC D yx tzyxC D xt tzyxC ,,,,,,,,,,,,                z L S SWdtWyxCtzyx Tk D x0 1,,,,,,                 z L S SWdtWyxCtzyx Tk D y0 1,,,,,,  (1)                           z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SCSCSC             ,,,,,,,,, 222 with boundary and initial conditions   0 ,,, 0    x x tzyxC ,   0 ,,,    x Lx x tzyxC ,   0 ,,, 0    y y tzyxC , C (x,y,z,0)=fC (x,y,z),   0 ,,,    y Lx y tzyxC ,   0 ,,, 0    z z tzyxC ,   0 ,,,    z Lx z tzyxC . Here C(x,y,z,t) is the spatio-temporal distribution of concentration of dopant;  is the atomic volume of dopant; s is the symbol of surficial gradient;   z LzdtzyxC 0,,, is the surficial concentration of dopant on interface between layers of heterostructure (in this situation we assume, that Z-axis is perpendicular to interface between layers of heterostructure);  1(x,y,z,t) and  2(x,y,z,t) are the chemical potential due to the presence of mismatch-induced stress and porosity of material; D and DS are the coefficients of volumetric and surficial diffusions. Values of dopant diffusions coefficients depends on properties of materials of heterostructure, speed of heating and cooling of materials during annealing and spatio-temporal distribution of concentration of dopant. Dependences of dopant diffusions coefficients on parameters could be approximated by the following relations [23-25]                           2 * 2 2 * 1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD LC    ,                           2 * 2 2 * 1 ,,,,,, 1 ,,, ,,, 1,,, V tzyxV V tzyxV TzyxP tzyxC TzyxDD SLSS    . (2) Here DL (x,y,z,T) and DLS (x,y,z,T) are the spatial (due to accounting all layers of heterostruicture) and temperature (due to Arrhenius law) dependences of dopant diffusion coefficients; T is the temperature of annealing; P (x,y,z,T) is the limit of solubility of dopant; parameter  depends on properties of materials and could be integer in the following interval  [1,3] [23]; V (x,y,z,t) is the spatio-temporal distribution of concentration of radiation vacancies; V* is the equilibrium distribution of vacancies. Concentrational dependence of dopant diffusion coefficient has been described in details in [23]. Spatio-temporal distributions of concentration of point radiation defects have been determined by solving the following system of equations [19-22,24,25] Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 126                         y tzyxI TzyxD yx tzyxI TzyxD xt tzyxI II           ,,, ,,, ,,, ,,, ,,,                  TzyxktzyxITzyxk z tzyxI TzyxD zVIIII ,,,,,,,,, ,,, ,,, , 2 ,                        z L S IS WdtWyxItzyx Tk D x tzyxVtzyxI 0,,,,,,,,,,,,                              x tzyx TkV D x WdtWyxItzyx Tk D y SI L S IS z,,, ,,,,,, 2 0                            z tzyx TkV D zy tzyx TkV D y SISI ,,,,,, 22  (3)                         y tzyxV TzyxD yx tzyxV TzyxD xt tzyxV VV           ,,, ,,, ,,, ,,, ,,,                  TzyxktzyxVTzyxk z tzyxV TzyxD zVIVVV ,,,,,,,,, ,,, ,,, , 2 ,                        z L S VS WdtWyxVtzyx Tk D x tzyxVtzyxI 0,,,,,,,,,,,,                              x tzyx TkV D x WdtWyxVtzyx Tk D y SV L S VS z,,, ,,,,,, 2 0                            z tzyx TkV D zy tzyx TkV D y SVSV ,,,,,, 22  with boundary and initial conditions   0 ,,, 0  x x tzyxI   ,   0 ,,,  x Lx x tzyxI   ,   0 ,,, 0  y y tzyxI   ,   0 ,,,  y Ly y tzyxI   ,   0 ,,, 0  z z tzyxI   ,   0 ,,,  z Lz z tzyxI   ,   0 ,,, 0  x x tzyxV   ,   0 ,,,  x Lx x tzyxV   ,   0 ,,, 0  y y tzyxV   , (4)   0 ,,,  y Ly y tzyxV   ,   0 ,,, 0  z z tzyxV   ,   0 ,,,  z Lz z tzyxV   , I (x,y,z,0)= =fI (x,y,z), V (x,y,z,0)=fV (x,y,z), Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 127             2 1 2 1 2 1 111 2 1,,, zyxTk VttVztVytVxV nnn   . Here I (x,y,z,t) is the spatio-temporal distribution of concentration of radiation interstitials; I* is the equilibrium distribution of interstitials; DI(x,y,z,T), DV(x,y,z,T), DIS(x,y, z,T), DVS(x,y,z,T) are the coefficients of volumetric and surficial diffusions of interstitials and vacancies, respectively; terms V2(x,y,z,t) and I2(x,y,z,t) correspond to generation of divacancies and diinterstitials, respectively (see, for example, [25] and appropriate references in this book); kI,V(x,y,z,T), kI,I(x,y,z,T) and kV,V(x,y,z,T) are the parameters of recombination of point radiation defects and generation of their complexes; k is the Boltzmann constant;  = a3, a is the interatomic distance;  is the specific surface energy. To account porosity of buffer layers we assume, that porous are approximately cylindrical with average values 2 1 2 1yxr  and z1 before annealing [22]. With time small pores decomposing on vacancies. The vacancies absorbing by larger pores [26]. With time large pores became larger due to absorbing the vacancies and became more spherical [26]. Distribution of concentration of vacancies in heterostructure, existing due to porosity, could be determined by summing on all pores, i.e.            l i m j n kptkzjyixVtzyxV 0 0 0 ,,,,,,  , 222 zyxR  . Here  ,  and  are the average distances between centers of pores in directions x, y and z; l, m and n are the quantity of pores inappropriate directions. Spatio-temporal distributions of divacancies  V (x,y,z,t) and diinterstitials  I (x,y,z, t) could be determined by solving the following system of equations [24,25]                            y tzyx TzyxD yx tzyx TzyxD xt tzyx III II           ,,, ,,, ,,, ,,, ,,,                           z I I L IS S IWdtWyxtzyx Tk D xz tzyx TzyxD z0 1,,,,,, ,,, ,,,                        tzyxITzyxkWdtWyxtzyx Tk D yII L IS Sz I,,,,,,,,,,,, 2 , 0 1                                          z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS III ,,,,,,,,, 222      tzyxITzyxkI,,,,,, (5)                            y tzyx TzyxD yx tzyx TzyxD xt tzyx VVV VV           ,,, ,,, ,,, ,,, ,,,                           z V V L VS S VWdtWyxtzyx Tk D xz tzyx TzyxD z0 1,,,,,, ,,, ,,,      Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 128                   tzyxVTzyxkWdtWyxtzyx Tk D yVV L VS Sz V,,,,,,,,,,,, 2 , 0 1                                          z tzyx TkV D zy tzyx TkV D yx tzyx TkV D x SSS VVV ,,,,,,,,, 222      tzyxVTzyxkV,,,,,, with boundary and initial conditions   0 ,,, 0   x Ix tzyx   ,   0 ,,,   x Lx Ix tzyx   ,   0 ,,, 0   y Iy tzyx   ,   0 ,,,   y Ly Iy tzyx   ,   0 ,,, 0   z Iz tzyx   ,   0 ,,,   z Lz Iz tzyx   ,   0 ,,, 0   x Vx tzyx   ,   0 ,,,   x Lx Vx tzyx   ,   0 ,,, 0   y Vy tzyx   , (6)   0 ,,,   y Ly Vy tzyx   ,   0 ,,, 0   z Vz tzyx   ,   0 ,,,   z Lz Vz tzyx   ,  I (x,y,z,0)=f  I (x,y,z),  V (x,y,z,0)=f  V (x,y,z). Here D  I(x,y,z,T), D  V(x,y,z,T), D  IS (x,y,z,T) and D  VS(x,y,z,T) are the coefficients of volumetric and surficial diffusions of complexes of radiation defects; kI(x,y,z,T) and kV(x,y,z,T) are the parameters of decay of complexes of radiation defects. Chemical potential  1 in Eq.(1) could be determine by the following relation [19]  1=E(z)  ij [uij(x,y,z,t)+uji(x,y,z,t)]/2, (7) where E(z) is the Young modulus,  ij is the stress tensor;               i j j i ij x u x u u2 1 is the deformation tensor; ui, uj are the components ux(x,y,z,t), uy(x,y,z,t) and uz(x,y,z,t) of the displacement vector   tzyxu ,,,  ; xi, xj are the coordinate x, y, z. The Eq. (3) could be transform to the following form                                           i j j i i j j ix tzyxu x tzyxu x tzyxu x tzyxu tzyx ,,, ,,, 2 1 ,,, ,,, ,,,                  zETtzyxTzzK x tzyxu z z ij k k ij ij 2 ,,,3 ,,, 21 000                    , where  is Poisson coefficient;  0 = (as-aEL)/aEL is the mismatch parameter; as, aEL are lattice distances of the substrate and the epitaxial layer; K is the modulus of uniform compression;  is the coefficient of thermal expansion; Tr is the equilibrium temperature, which coincide (for our case) with room temperature. Components of displacement vector could be obtained by solution of the following equations [20] Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 129           z tzyx y tzyx x tzyx t tzyxu zxz xy xxx           ,,, ,,, ,,,,,, 2 2               z tzyx y tzyx x tzyx t tzyxu zyzyyyxy           ,,,,,,,,,,,, 2 2             z tzyx y tzyx x tzyx t tzyxu zzz zy zx z           ,,, ,,, ,,, ,,, 2 2     , where                                 ij k k ij i j j i ij zK x tzyxu x tzyxu x tzyxu z zE     ,,, 3 ,,, ,,, 12           r k kTtzyxTzKz x tzyxu    ,,, ,,,  ,  (z) is the density of materials of heterostructure,  ij Is the Kronecker symbol. With account the relation for  ij last system of equation could be written as                                            z zE zK x tzyxu z zE zK t tzyxu zxx   13 ,,, 16 5 ,,, 2 2 2 2                                             z zE zK z tzyxu y tzyxu z zE yx tzyxu z yy  13 ,,, ,,, 12 ,,, 2 2 2 22         x tzyxT zzK zx tzyxuz      ,,,,,, 2                                     y tzyxT yx tzyxu x tzyxu z zE t tzyxu zx yy ,,, ,,, ,,, 12 ,,, 2 2 2 2 2                                          2 2,,, ,,, ,,, 12 y tzyxu y tzyxu z tzyxu z zE z zzK y z y   (8)                       yx tzyxu zK zy tzyxu z zE zKzK z zE yy                     ,,,,,, 16112 522                                 zx tzyxu y tzyxu x tzyxu z zE t tzyxu zx zzz ,,, ,,,,,, 12 ,,, 2 2 2 2 2 2 2                                         z tzyxu y tzyxu x tzyxu zK zzy tzyxu x y x y,,, ,,, ,,, ,,, 2                                      z tzyxu y tzyxu x tzyxu z tzyxu z zE zz y x z,,, ,,, ,,, ,,, 6 16 1        z tzyxT zzK   ,,,  . Conditions for the system of Eq. (8) could be written in the form Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 130   0 ,,,0    x tzyu  ;   0 ,,,    x tzyLu x  ;   0 ,,0,    y tzxu  ;   0 ,,,    y tzLxu y  ;   0 ,0,,    z tyxu  ;   0 ,,,    z tLyxu z  ;   0 0,,, uzyxu   ;   0 ,,, uzyxu   . We determine spatio-temporal distributions of concentrations of dopant and radiation defects as well as of displacement vector by solution of the appropriate equations by the standard method of averaging of function corrections [27]. In the framework of the paper we calculate concentration of dopant, concentrations of radiation defects and components of displacement vector by using the second-order approximation framework method of averaging of function corrections. This approximation is usually enough good approximation to make qualitative analysis and to obtain some quantitative results. All obtained results have been checked by comparison with results of numerical simulations. 3. DISCUSSION In this section we analyzed dynamics of redistributions of dopant and radiation defects during annealing and under influence of mismatch-induced stress and modification of porosity. Typical distributions of concentrations of dopant in heterostructures are presented on Figs. 2 and 3 for diffusion and ion types of doping, respectively. These distributions have been calculated for the case, when value of dopant diffusion coefficient in doped area is larger, than in nearest areas. The figures show, that inhomogeneity of heterostructure gives us possibility to increase compactness of concentrations of dopants and at the same time to increase homogeneity of dopant distribution in doped part of epitaxial layer. However framework this approach of manufacturing of bipolar transistor it is necessary to optimize annealing of dopant and/or radiation defects. Reason of this optimization is following. If annealing time is small, the dopant did not achieve any interfaces between materials of heterostructure. In this situation one cannot find any modifications of distribution of concentration of dopant. If annealing time is large, distribution of concentration of dopant is too homogenous. We optimize annealing time framework recently introduces approach [2836]. In the framework of the criterion we approximate real distribution of concentration of dopant by step-wise function (see Figs. 4 and 5). Farther we determine optimal values of annealing time by minimization of the following mean-squared error Fig. 2. Distributions of concentration of infused dopant in heterostructure from Fig. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Increasing of number of curve corresponds to increasing of difference between values of dopant diffusion coefficient in layers of heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate Mechatronics and Applications: An International Journal (MECHATROJ), Vol. 2, No.1 131 x 0.0 0.5 1.0 1.5 2.0 C(x,  ) 23 4 1 0L/4 L/2 3L/4 L Epitaxial layer Substrate Fig.3. Distributions of concentration of implanted dopant in heterostructure from Fig. 1 in direction, which is perpendicular to interface between epitaxial layer substrate. Curves 1 and 3 corresponds to annealing time  = 0.0048(Lx2+Ly2+Lz2)/D0. Curves 2 and 4 corresponds to annealing time  = 0.0057(Lx2+Ly2+Lz2)/D0. Curves 1 and 2 corresponds to homogenous sample. Curves 3 and 4 corresponds to heterostructure under condition, when value of dopant diffusion coefficient in epitaxial layer is larger, than value of dopant diffusion coefficient in substrate C(x,  ) 0L x 2 1 3 4 Fig. 4. Spatial distributions of dopant in heterostructure after dopant infusion. Curve 1 is idealized distribution of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. Increasing of number of curve corresponds to increasing of annealing time x C(x,  ) 1 234 0L Fig. 5. Spatial distributions of dopant in heterostructure after ion implantation. Curve 1 is idealized distribution of dopant. Curves 2-4 are real distributions of dopant for different values of annealing time. Increasing of number of curve corresponds to increasing of annealing time