Tek esnekli ferroelektrik kristalde sınır tabakası denklemlerinin çözümleri
The Solution of boundry layer Equations in the uniaxial crystal
Yazar: SALİM CEYHAN
Danışman: PROF.DR. ERDOĞAN ŞUHUBİ
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Yüksek Lisans, 1994
Konu: Mühendislik Bilimleri = Engineering Sciences
Dizin: Ferroelektrik = Ferroelectrics ; Sınır tabaka denklemleri = Boundary layer equations

Özet:
ÖZET Dış elektrik alanın yokluğunda doymuş bir ferroelektrik rijid kristalde polarizasyon, p birim vektör alanı ve Pq kristalin karakteristik bir sabiti olmak üzere ^P=P0p vektörel alanı ile ifade edilir. Gözlemler böyle bir durumdaki kristalde oluşan Weiss bölgelerinde polarizasyon vektörünün parça parça sabit kaldığım göstermektedir. Ferroelektrik kristallerde oluşan Weiss bölgeleri arasındaki bölge duvarlarının düzlemsel olduğu gösterilmiştir [3]. Kristalin sınır yüzeyi, yüzeye teğet olan polarizasyon vektörünü taşıya bilen düzlemsel yüzeylerin birleşiminden oluşmuyorsa Weiss bölgelerinin sınıra kadar genişleyemeyeceği açıktır. Bu yüzden kristalin keyfi bir şekli için polarizasyon vektörünün boyunun sabit olmasına karşın doğrultusunun sürekli olarak değişebildiğim kabul ettiğimiz, kristalin sınırına yakın ince bir tabaka olduğunu varsaydık. Bu tabaka içindeki polarizasyon vektörünün Weiss bölgelerindeki parça parça sabit duruma asimptotik olarak geçişini sağlayacak içteki sınır eğrisini veren ifadenin asimptotik açılımındaki sıfırına dereceden terimi verecek olan lineer olmayan bir diferansiyel denklem elde ettik. Sonuçta örnek olarak sınırı dairesel olan bir kristali aldığımızda sınır tabakasında oluşacak durumu inceledik. iv

Abstract:
SUMMARY THE SOLUTION OF BOUNDARY LAYER EQUATIONS IN THE UNIAXIAL CRYSTAL 1. Introduction If a dielectric crystal can get a very high level of a polarization up to saturation point with the application of very low level of electric fields than it is named as ferroelectric material. In a ferroelectric crystal, even in the absence of external fields, its state of electric polarization is described by a vectorial field 'P=PoP where Pq is a characteristic scalar of the crsytal and p is a differentiable vector field of unit magnitude. The latter is piecewise constant in almost all the volume D of the crystal except in thin layers across which it varies from one con stant direction to another. In general, the regions Da, a = 1,2,. -.,m in which "P is constant (Weiss domains) have at least one microscopic dimen sion (a few hundred microns) with other dimensions comparable to those of crystal whereas the transition layers (domain walls) have a tickness of a few microns. The theoretical justification of this complex distribution is found in the combined effects of three factors: the electric energy due to the anisotropy of the crystal, dipole-dipole exchange quantum forces and the form of the crystal. The presence of an external field can only modify the weight of the effects of these three factors on the distribution of the domains. From a classical viewpoint one takes the first two factors into accpunt by associating to the crystal an energy e(p, Vp) per unit polar ization and per unit volume in which the contribution due to the presence of Vp is significant only for very large of ||Vp||. Then the equilibrium, equations become the extremals of a functional expressing the total energy of the crystal in the presence of external field. These equations are in general so complex that it is very difficult to deduce some general results from them and little information on the distributions of the domains or on the energy contained in a transition layer was obtained only by resorting to rather drastic approximations. Finally, when the forms of the domains are known their dimensions are determined by imposing that total energy be an absolute minimum. In order to obtain a description sufficiently accurate but also to simplify the situation illustrated above, we suggest in the present work to substitude the domain walls by surfaces of discontinuity Sb, h = 1, 2,..., q for p and to take into account the energy contained in a transition layer with a surface energy ea(n, p~,p+) depending a priori on the orientation of the normal n of Si, with respect to crystallographic axes and on the constant vectors p in two domains adjacent to the domain wall under consideration. However inaddition to this surface energy we still go on to consider an energy per unit polarization and per unit volume of the crystal depending on Vp besides p. In fact, expect for very particular forms of D associated with the symmetry class of the ferroelectric crystal, the satisfaction of the boundary conditions on dD may require the formation of small regions adherent to the external walls in the. ferroelectric crystal where the polarization field is not uniform. The equilibrium configuration of a rigid ferroelectric crystal in the presence of an external field is then derived by imposing the requirement that the total energy becomes stationary with respect to varitions of the polarization and electric fields as well as the surfaces Sb which engender the Weiss domains under the constraints that p is a unit vector and the volume of each domains remains constant. Starting from equations so obtained we are able to prove that the do main walls in the interior of D are necessarily plane and the resultant electric field in each domain is uniform. Moreover, it is straightforward to see that if p verifies the equation eiP(p) = A(jp) = Ap (A is real), that is to say, if p is, in a sense, an eigenvector of the operator A then electric and polariza tion fields have the same direction in each Weiss domain. Otherwise, if p is not such an eigenvector the two fields have different directions. If there is no external electric field p proves tobe necessarily an eigenvector of the p-gradient of the polarization energy of anisotropy. These results are in perfect agreement with experiments since it is verified that the distribution of domains in absence of an external field is modified with the presence of the latter. The external field causes either to increase the volume of the domains in which the polarization vector is directed along the external field (weak fields) or to rotate the polarization in domains where it is not directed along the external field without increasing their volume (strong fields). 2. Basic Equations For a ferroelectric crystal the total energy is given by the functional below [3]. J7 = / P0e(p,Vp)c2u + -e0 / (j>,i(j>,idv Jd z Jr3 -Po / 4>,iPidv + ^2p0 / e"(n,p~,p+)da (1) Jd i_i J St It is assumed that the ferroelectric crystal has m Weiss domains Da, a = 1,2,.?.,m. The boundaries that seperates these domains are Sb, b = 1,2,..-, ,i\ni + PoPim = 0, [eo0.i-PoPi]ni = O, onSb (4) where A(x) is a Lagrange multiplier. Because the polarization field is a piecewise constant in the Weiss do mains Eg.(2,a-b)-(4) simplify very much for these domains. e,Pi- ,i + \pi = 0, /km in Da (5, a - b) ,H - 0, [e04>,i - PoPi]n~i = 0, on Sb (6) Also electric field in a Weiss domain can only be a constant field. For this reason it is shown that the boundaries seperating Weiss domains can only be planes, [3]. 3. The structure of the Weiss Domains in the Absence of Electric Field In the absence of an external electric field and also if an electric field induced by the variation of the piecewise constant polarization field is not exist equations (2,a-b),(3,a-b) and (4) take the following simpler forms. e,Pi = -^Ph in Da (7) pim = 0 on dD (8) \pi]rii = 0 on Sb. (9) If the boundary surface of the crystal is not made up the union of planar surfaces which can carry an admissible polarization vector it is obvious that internal Weiss domains cannot be extended up to the boundary. Therefore for an arbitrary shape of crystal we are compelled to consider a thin layer ad jacent to the boundary of the crystal in which the polarization vector cannot be assumed anymore to be piecewise uniform. We expect that the solution for the polarization vector in this layer Dl approaches constant states which are prevalent in the interior parts of the crystal at places sufficently far from viithe boundary dD. However the polarization field must vary very rapidly from these constant states to a field which is tangent to the boundary. Hence in a very thin boundary layer adjacent to the boundary we have to assume that polarization gradients are very large but in the rest of Dl it approaches asymptotically to a piecewise constant state. 4. The Solutions in Boundary Layer The ferroelectric crystal that we deal with is barium titanate (BaTi03) of which the one and only polarization axis is the easiest to find. The energy of the crystal in the first approximation is taken as follows Po e ^[atr[PPr]+>(|p|2-(h.p)2)] (10) We will study in x - z plane (py =0). The energy of the crystal is taken as follow : e = y {a {(Px,x)2 + (Px,z)2 + (Pz,x)2 + (pz,zf) + fol) (11) For this purpose many parametrizations are experimented. But even though considerable effort are spent, because of the scarseness of the number of the unknown field variables, unfortunately a suitable solution that satisfy- all the boundary conditions could not be found. Then it is observed that the extra unknown, arised when an electric potential is considered inside this layer close to the boundary, overcome this difficulty. In this case the local field equation that we use in the thin layer are, co ,u - PoPi,i = 0, And the boundary conditions become, Piiii = 0, in DL (13, a - b) an dD (14, a-b) In addition, on the boundary between this layer and the neighbouring Weiss domain, the jump condition, eo[4>,i]ni + Popini = 0, _ on dD (15, a-b) e,p>,j nj = "> viii[eo<£,« - PoPi]n-i - 0, on 5j (16) must also be satifiecl. The following conditions must be valid everywhere |p| = l, [ ] = 0. Let us define a generalized curvilinear coordinate frame in the thin layer. Position of a point on the plane is defined in this coordinate frame as r(x\x2) = r0(x2)+x1n(x2). (17) The base vectors and the matrix from of the metric tensor are given as gi = n(z2) (18) x g2 = ( i - - 1 1 (19) bu] = (i-*)' (20) In this, coordinate frame the equations (13,a-b) become for » = 1,2 (!-t) pV-y) PV-f) P (l - 2^) (ı-£) p2{^-xİ) P2{^-Xi) * İ> i 1 3 P,2 ; ttP - = /c(/i + nxa)p* - k ( 1 - - ) nztzp2 x' (21) P'.n + 2. 2î 0 o d o TP,22 + - ; ~^P,2 - - :ryP,1 /,\2^,22 '.,\3^,2 /,\ T73^,2 + P "1 P + SP2(2-Î) xlp x' p' + H1-*)' '^-t) J 0-t) -2^,2 (x-t) (22) IX1 ı XV, 0,11 + -T2 (!??) W = 0 (23) where the dimensionless quantities £ = x1 fh and r; = x2/Z- are used. We also define a scale parameter that is assumed to be very small e = h/L < 1. Using this parameter and defining p1 = eP,p2 = Q and = e2<&, the following asymptotic expansions are considered, ^(e,r?;o = E^n(e^) CO n=0 oo n=0 At the end, the zeroth order solutions of P, Q and $ that satisfy the boundary and the jump conditions, Po(v) = Afo) *o(e,)7) = MA(i7)e whereM = P0L/e0. The curve xl = ] = 0 on the separating botmdary curve x1 = d(x2; e), $o =C similar to the constant field inside. *otf,!7) = MA(rj)S(rj) = C =? Afo) = M6(ri) is found. Then the solution of $0 M(,v) = c *(v) is found. (26) XI