Ferroelectrics Characterization and Modeling Part 15 pptx

A General Conductivity Expression for Space-charge-limited Conduction in Ferroelectrics and Other Solid Dielectrics 479 Using an ideal-gas approximation for each of the two types of free charge-carriers, i.e. (,) (,) (,) pBp Pxt kTxtC xt  (56) and (,) (,) (,) nBn Pxt kTxtC xt  (57) where k B is the Boltzmann constant and T(x,t) is a generally position- and time-dependent local temperature, the diffusion-current density can be expressed as (,) () (,) (,) (,) [()] (,) () (,) [ ()] (,) p p dB n n Bppnn Cxt x x Jxt kTxt Cxt x x Txt k xCxt xCxt x                       (58) where , () (,) (,) pB op xkTxt Dxt q   (59) and , [()](,) (,) () nB on xkTxt Dxt q     (60) are the Einstein relations for the local diffusion coefficients of p-type and n-type free charge-carriers, respectively. Using the definitions in Eqs. (49) and (52), Eq. (58) can be rewritten as () [() ()] (,) (,) (,) (,) () () [() ()]() (,) () (,) () (,) in pn dB pn pnin B pn dC x xx dx Jxt kTxt pxt nxt xx xx xxCx Txt k xpxt xnxt x                                      (61) From Eqs. (21) and (22), we obtain 2 2 1(,) (,) () 2 1(,) () 2 in in Dxt pxt C x qx Dxt Cx qx           (62) Ferroelectrics - Characterization and Modeling 480 and 2 2 1(,) (,) () 2 1(,) () 2 in in Dxt nxt C x qx Dxt Cx qx           (63) Differentiating Eqs. (62) and (63) with respect to x yields 2 2 1 2 2 2 22 (,) () 1(,) 2 1(,) () 2 () 1(,)(,) () 4 in in in in pxt dC x Dxt xq dx x Dxt Cx qx dC x Dxt Dxt Cx xdx qx                              (64) and 2 2 1 2 2 2 22 () (,) 1 (,) 2 1(,) () 2 () 1(,)(,) () 4 in in in in dC x nxt Dxt xq dx x Dxt Cx qx dC x Dxt Dxt Cx xdx qx                               (65) respectively. Putting Eqs. (62) to (65) into Eq. (61), we obtain a general expression for the local diffusion-current density: 12 (,) (,) (,) dd d Jxt J xt J xt   (66) where 2 1 2 1 2 2 2 22 (,)[ () ()] (,) (,) 2 1(,) [() ()] (,) () 2 () 1(,)(,) () 4 Bpn d pnB in in in kTxt x x Dxt Jxt q x Dxt xxkTxt Cx qx dC x Dxt Dxt Cx xdx qx                                 (67) A General Conductivity Expression for Space-charge-limited Conduction in Ferroelectrics and Other Solid Dielectrics 481 and 2 2 2 [() ()] (,) 2 (,) (,) [ () ()] 1(,) () 2 pn dB pn in xx Dxt qx Txt Jxt k x x x Dxt Cx qx                                     (68) Eq. (67) denotes a contribution to the diffusion current from the presence of a gradient in the space-charge density or in the intrinsic free-carrier concentration, while Eq. (68) denotes a contribution from the presence of a temperature gradient. At the limit of zero intrinsic conductivity, we have C in (x)  0 as explained in the beginning of the previous section. Eqs. (67) and (68) are then reduced to 2 1 2 2 2 [() ()] (,) (,) (,) 2 (,) (,) [() ()] (,) (,) 2 pnB d pnB xxkTxt Dxt Jxt q x Dxt Dxt xxkTxt x x Dxt q x              (69) and 2 [() ()] (,) 2 (,) (,) [() ()] (,) 2 pn dB pn xx Dxt qx Txt Jxt k xx x Dxt qx                           (70) respectively. Similar to the case of Eq. (44), for Eqs. (69) and (70) we can also verify that in the case of σ in (x)  0 the charge mobility is correctly equal to that of the dominant type of free carriers. Following Eqs. (69) and (70), if Eq. (42) is satisfied, we have 2 x )t,x(D 2 q )t,x(T B k)x( p x )t,x(D 2 x )t,x(D 2 x )t,x(D q2 )t,x(T B k)]x( n )x( p [ 2 x )t,x(D 2 q2 )t,x(T B k)]x( n )x( p [ )t,x( 1d J                          (71) and Ferroelectrics - Characterization and Modeling 482 2 [() ()] (,) 2 (,) (,) [() ()] (,) 2 () (,) (,) pn dB pn pB xx Dxt qx Txt Jxt k xx x Dxt qx xk Dxt Txt qxx                                     (72) Else if Eq. (43) is satisfied, we have 2 1 2 2 2 2 2 [() ()] (,) (,) (,) 2 (,) (,) [() ()] (,) (,) 2 () (,) (,) pnB d pnB nB xxkTxt Dxt Jxt q x Dxt Dxt xxkTxt x x Dxt q x xkTxt Dxt q x                         (73) and 2 [() ()] (,) 2 (,) (,) [() ()] (,) 2 () (,) (,) pn dB pn nB xx Dxt qx Txt Jxt k xx x Dxt qx xk Dxt Txt qxx                                      (74) 7. Alternative derivation of the general local conductivity expression We begin our alternative derivation of the general local conductivity expression in Eq. (34) by identifying the following quantities that appear in the conductivity expression: () () '( ) 2 pn xx x     (75) and () () "( ) 2 pn xx x     (76) The drift velocities of p-type and n-type free carriers can then be expressed as (,) ()(,) '()(,) "()(,) pp v xt xExt xExt xExt     (77) and A General Conductivity Expression for Space-charge-limited Conduction in Ferroelectrics and Other Solid Dielectrics 483 (,) ()(,) '()(,) "()( ,) nn v xt xExt xExt xExt     (78) where "( ) '( ) 0xx   (79) and µ’(x) can be positive or negative. In this description, both p-type and n-type free carriers share the same velocity component µ’(x)E(x,t), with the presence of the additional velocity components µ“(x)E(x,t) and -µ“(x)E(x,t) for p-type and n-type free carriers, respectively. The generally time-dependent local electrical conductivity can then be expressed as a sum of contributions from the velocity components µ’(x)E(x,t) and ±µ“(x)E(x,t): (,) '()[ (,) (,)] "( )[ ( , ) ( , )] pn pn xt q x C xt C xt q xC xt C xt     (80) According to Gauss’ law, the density of free space-charge is given by (,) (,) [ (,) (,)] qpn Dxt xt qC xt C xt x      (81) In the absence of free space-charge, i.e. ρ q (x,t) = 0, both C p (x,t) and C n (x,t) are by definition equal to the intrinsic free-carrier concentration C in (x), and the electrical conductivity σ(x,t) would then be equal to the intrinsic conductivity () 2 "() () in in xqxCx   (82) according to Eq. (80). Consider the reversible generation and recombination of p-type and n-type free carriers: 1 source particle  1 p-type free carrier + 1 n-type free carrier As described right below Eq. (18), the rate of free-carrier generation is assumed to be equal to the rate of free-carrier recombination due to a “heat balance“ condition, and the rate of each of these processes is assumed to be proportional to the product of the “reactants“. Following these, for C s (x,t) being the concentration of the source particles for free-carrier generation (e.g. valence electrons or molecules) we have (,) (,) (,) gs rp n KC xt KC xtC xt  (83) where K g and K r are, respectively, the rate constants for the generation and recombination of free carriers. If the conditions (,) (,) ps Cxt Cxt   (84) and (,) (,) ns Cxt Cxt   (85) hold for a dielectric insulator such that Ferroelectrics - Characterization and Modeling 484 (,) () ss Cxt Cx  (86) i.e. the concentration of source particles for free-carrier generation has an insignificant fluctuation with time and is practically a material-pertaining property, we have 2 () (,) (,) () gs rp n rin KC x KC xtC xt KC x (87) which implies 2 (,) (,) () pn in CxtCxt C x (88) As an example, we show that this mass-action approximation is valid for a dielectric insulator which has holes and free electrons as its p-type and n-type free charge-carriers, respectively, and which has valence electrons as its source particles: A hole is by definition equivalent to a missing valence electron. At anywhere inside the dielectric sample, the generation and annihilation of a hole correspond, by definition, to the annihilation and generation of a valence electron, respectively, and the flow-in and flow-out of a hole are, respectively, by definition equivalent to the flow-out and flow-in of a valence electron in the opposite directions. Therefore, (,) (,) p s Cxt Cxt tt     (89) so that the total concentration of holes and valence electrons is given by () (,) (,) ps p s CxCxtCxt    (90) Eq. (83) can then be written as [ () (,)] (,) (,) gps p rp n KC x Cxt KCxtCxt    (91) For the case of zero space charge where C p (x,t) = C n (x,t) = C in (x), we have 2 [() ()] () gps in rin KC x C x KC x   (92) Define a material paramter () () 1 () in ps Cx x Cx     (93) and consider the limit of (x)  0 for the case of a dielectric insulator. Combining Eqs. (91) to (93), we obtain the mass-action relation in Eq. (88): () 0 2 (,) (,) ()[ () () (,)] () [1 ( )] x pn in in p in CxtCxt CxCx xCxt Cx x limit        (94) Going back to our derivation of the conductivity expression, we notice that Eqs. (81) and (88) together imply A General Conductivity Expression for Space-charge-limited Conduction in Ferroelectrics and Other Solid Dielectrics 485 22 (,) (,) (,) () 0 q ppin xt Cxt Cxt C x q    (95) and 22 (,) (,) (,) () 0 q nnin xt Cxt Cxt C x q    (96) from which we obtain 2 2 (,) (,) 1 (,) 4 () 0 2 qq pin xt xt Cxt C x qq            (97) 2 2 (,) (,) 1 (,) 4 () 0 2 qq nin xt xt Cxt C x qq              (98) and 2 2 (,) (,) (,) 4 () q pn in xt Cxt Cxt C x q       (99) Using Eqs. (80) to (82) as well as Eq. (99), we obtain the following expression for the generally time-dependent local electrical conductivity: 22 (,) '() (,) [ "() (,)] () qqin xt x xt x xt x      (100) By defining the reduced paramters (,) (,) () in xt xt x      (101)  '( ) () 1,1 "( ) x x x        (102) and (,) (,) 2() q q in xt xt qC x     (103) Eq. (100) can be expressed in a simpler form: 2 (,) () (,) [ (,)] 1 qq xt x xt xt       (104) For the limiting case of zero intrinsic conductivity with C in (x)  0, Eqs. (97) and (98) can be rewritten as Ferroelectrics - Characterization and Modeling 486 (,) (,) 1 (,) 2 qq p xt xt Cxt qq           (105) and (,) (,) 1 (,) 2 qq n xt xt Cxt qq            (106) respectively, which imply the dominance of either type of free carriers: If ρ q (x,t) > 0, we have (,) (,) (,) 0 q pn xt Cxt andCxt q    (107) Else if ρ q (x,t) < 0, we have (,) (,) 0 (,) q pn xt Cxt andCxt q   (108) 8. Conclusions and future work In this Chapter, a generalized theory for space-charge-limited conduction (SCLC) in ferroelectrics and other solid dielectrics, which we have originally developed to account for the peculiar observation of polarization offsets in compositionally graded ferroelectric films, is presented in full. The theory is a generalization of the conventional steady-state trap-free SCLC model, as described by the Mott-Gurney law, to include (i) the presence of two opposite types of free carriers: p-type and n-type, (ii) the presence of a finite intrinsic (Ohmic) conductivity, (iii) any possible field- and time-dependence of the dielectric permittivity, and (iv) any possible time dependence of the dielectric system under study. Expressions for the local conductivity as well as for the local diffusion-current density were derived through a mass-action approximation for which a detailed theoretical justification is provided in this Chapter. It was found that, in the presence of a finite intrinsic conductivity, both the local conductivity and the local diffusion-current density are related to the space- charge density in a nonlinear fashion, as described by Eqs. (34), (66), (67) and (68), where the local diffusion-current density is generally described as a sum of contributions from the presence of a charge-density gradient and of a temperature gradient. At the limit of zero intrinsic conductivity, it was found that either p-type or n-type free carriers are dominant. This conclusion provides a linkage between the independent assumptions of (i) a single carrier type and (ii) a negligible intrinsic conductivity in the conventional steady-state SCLC model. For any given space-charge density, it was also verified that the expressions we have derived correctly predict the dominant type of free carriers at the limit of zero intrinsic conductivity. Future work should be carried out along at least three possible directions: (i) As a further application of this general local conductivity expression, further numerical investigations should be carried out on how charge actually flows inside a compositionally graded ferroelectric film. This would provide answers to interesting questions like: Does a graded ferroelectric system exhibit any kind of charge-density waves upon excitation by an A General Conductivity Expression for Space-charge-limited Conduction in Ferroelectrics and Other Solid Dielectrics 487 alternating electric field? What are the physical factors (dielectric permittivity, carrier mobility, etc.) that could limit or enhance the degree of asymmetry in the SCLC currents of a graded ferroelectric film? The latter question has been partially answered by ourselves (Zhou et al., 2005b), where we have theoretically found that the observation of polarization offsets, i.e. the onset of asymmetric SCLC, in a compositionally graded ferroelectric film is conditional upon the presence of relatively large gradients in the polarization and in the dielectric permittivity. Certainly, a detailed understanding of the mechanism of asymmetric electrical conduction in such a graded ferroelectric film would also provide insights into the designing of new types of electrical diodes or rectifiers. The recently derived expression for the local diffusion-current density, as first presented in this Book Chapter (Eqs. (66) to (68)), has also opened up a new dimension for further theoretical investigations: Using this expression, the effect of charge diffusion in the presence of a charge-density gradient or a temperature gradient can be taken into account as well, and a whole new range of problems can be studied. For example, it would be interesting to know whether asymmetric electrical conduction would also occur if a compositionally graded ferroelectric film is driven by a sinusoidal applied temperature difference instead of a sinusoidal applied voltage. In this case, one also needs to take into account the temperature dependence of the various system parameters like the remanent polarization and the dielectric permittivity. The theoretical predictions should then be compared against any available experimental results. (ii) Going back to the generalized SCLC theory itself, it would be important to look for possible experimental verifications of the general local conductivity expression, and to establish a set of physical conditions under which the conductivity expression and the corresponding mass-action approximation are valid. Theoretical predictions from the conductivity expression should be made for real experimental systems and then be compared with available experimental results. It would also be worthwhile to generalize the mass-action approximation, and hence the corresponding local conductivity expression, to other cases where the charge of the free carriers, or the stoichiometric ratio between the concentrations of p-type and n-type free carriers in the generation- recombination processes, is different. (iii) In the derivation of the Mott-Gurney law J ~ V², the boundary conditions E p (0) = 0 and E n (L) = 0 were employed to describe the cases of conduction by p-type and n-type free carriers, respectively. If we keep E p (0) or E n (L) as a variable throughout the derivation, an expression of J as a function of E p (0) or E n (L) can be obtained and it can be shown that both the boundary conditions E p (0) = 0 and E n (L) = 0 correspond to a state of maximum current density. As an example, for the case of conduction by p-type free carriers, we have (Fig. 2) 32 2 2 2 3 912(0) 16 912(0) 3 (0) [1 (0)] 16 4 pp p p p pp JL e j V e ee              (109) where e p (0) ≡ E p (0)L/V. If we consider our general local conductivity expression which takes into account the presence of a finite intrinsic conductivity and the simultaneous presence of Ferroelectrics - Characterization and Modeling 488 p-type and n-type free carriers, it would be important to know whether this maximum- current principle can be generally applied to obtain the system’s boundary conditions. 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.25 0.50 0.75 1.00 1.25 j p e p (0) Fig. 2. Plot of j p against e p (0), showing a maximum of j p at e p (0) = 0. 9. Acknowledgments Stimulating discussions with Prof. Franklin G. Shin, Dr. Chi-Hang Lam and Dr. Yan Zhou are gratefully acknowledged. 10. References Alpay, S. P.; Ban, Z. G. & Mantese, J. V. (2003). Thermodynamic Analysis of Temperature- graded Ferroelectrics. Applied Physics Letters, Vol. 82, pp. 1269 – 1271 (February 2003), ISSN 1077-3118 Bao, D.; Mizutani, N.; Zhang, L. & Yao, X. (2001). Composition Gradient Optimization and Electrical Characterization of (Pb,Ca)TiO 3 Thin Films. Journal of Applied Physics, Vol. 89, pp. 801 – 803 (January 2001), ISSN 1089-7550 Bao, D.; Wakiya, N.; Shinozaki, K.; Mizutani, N. & Yao, X. (2001). Abnormal Ferroelectric Properties of Compositionally Graded Pb(Zr,Ti)O 3 Thin Films with LaNiO 3 Bottom Electrodes. Journal of Applied Physics, Vol. 90, pp. 506 – 508 (July 2001), ISSN 1089- 7550 Bao, D.; Mizutani, N.; Yao, X. & Zhang, L. (2000). Structural, Dielectric, and Ferroelectric Properties of Compositionally Graded (Pb,La)TiO 3 Thin Films with Conductive LaNiO 3 Bottom Electrodes. Applied Physics Letters, Vol. 77, pp. 1041 – 1043 (August 2000), ISSN 1077-3118 Bao, D.; Mizutani, N.; Yao, X. & Zhang, L. (2000). Dielectric and Ferroelectric Properties of Compositionally Graded (Pb,La)TiO 3 Thin Films on Pt/Ti/SiO 2 /Si Substrates. Applied Physics Letters, Vol. 77, pp. 1203 – 1205 (August 2000), ISSN 1077-3118 Bao, D.; Yao, X. & Zhang, L. (2000). Dielectric Enhancement and Ferroelectric Anomaly of Compositionally Graded (Pb,Ca)TiO 3 Thin Films Derived by a Modified Sol-gel Technique. Applied Physics Letters, Vol. 76, pp. 2779 – 2781 (May 2000), ISSN 1077- 3118 [...]... Characterization and Modeling P(E, T0 ) = P(E,θ0 ) αΔT ≡ 2 βΔθ With; ΔE ≡ T − T0 = T and Δθ = θ − θ 0 (29) (θ0 = 298K ) Thus T ≡ δ × Δθ (30) As illustrated in figure 8, we determine P(T) and P(θ) from P(E) (steps 1 and 3), P(E) and P(θ) from P(T) (steps 1 and 2), and at the end we can determine P(E) and P(T) from P(θ) (steps 2 and 3), Fig 8 Schematic illustration of the scaling laws Nonlinearity and Scaling... of temperature negative and positive(step 2) and stress (compressive and tensile stress) (step 1) In conclusion, we could determine P(T) and P(θ) from P(E) (steps 1 and 3),P(E) and P(θ) from P(T) (steps 1 and 2), and finally P(E) and P(T) from P(θ) (steps 2 and 3), Fig 13 Schematic illustration of the material behavior under excitations 6 Relationship between the coefficients d33 and ε33 The proposed scaling... Mater 56, 215 (2008) 512 Ferroelectrics - Characterization and Modeling [9] vA E Glazounov and M J Hoffmann, J Eur Ceram Soc 21, 1417 (2001) [10] D Berlincourt, H Helmut, and H A Krueger, J Appl Phys 30, 1804 (1959) [11] A Hajjaji, S Pruvost, G Sebald, L Lebrun, D Guyomar, and K Benkhouja, Acta Mater 57, 2243 (2009) [12] S C Hwang, J E Huber, R M McMeeking, and N A Fleck, J Appl Phys 84, 153 0 (1998)... (E, T )dE (1) 494 Ferroelectrics - Characterization and Modeling where E, T, and S represent the electric field, the mechanical stress, and the strain, respectively T E The constants ε 33 , s33 , and d33 correspond to the dielectric permittivity, the elastic compliance, and the piezoelectric constant, respectively Here, the superscripts signify the variable that is held constant, and the subscript 3... dΓ = c dθ θ T dθ + p.dE and dD = ε 33 dE + pdθ D = ε 0E + P (12) (13) D, P, E, θ and Γ and G represent the electric displacement, the polarization, the electric field, the temperature and the entropy, respectively, and where c and p, respectively, correspond to the heat capacity and the pyroelectric coefficient Here, the superscripts signify the variable that is held constant, and the subscript 3 indicates... soft PZT 498 Ferroelectrics - Characterization and Modeling 3 Temperature/electric field scaling in ferroelectrics 3.1 Presentation of the scaling law In order to determine a scaling law between the electric field and the temperature, one should start by following the piezoelectric constrictive equations, restricting them in one dimension These equations can be formulated with the temperature and the electric... Polarization and Permittivity Gradients and Other Parameters on the Anomalous Vertical Shift Behavior of Graded Ferroelectric Thin Films Journal of Applied Physics, Vol 98, No 034105 (August 2005), ISSN 1089-7550 Part 5 Modeling: Nonlinearities 25 Nonlinearity and Scaling Behavior in a Ferroelectric Materials 1Ecole Abdelowahed Hajjaji1, Mohamed Rguiti2, Daniel Guyomar3, Yahia Boughaleb4 and Christan... field and the temperature [16] This law is expressed by the expression (25) ( ΔE ≡ (2 β × P(E ,θ 0 )).Δθ ) 506 Ferroelectrics - Characterization and Modeling ΔE and Δθ where good linear fits are apparent P( E , θ 0 ) (R close to 1) This implies a power-law relation between the temperature and electric field, i.e., ( ΔE ≡ (2 β × P(E ,θ 0 )).Δθ ), the exponent β can be extracted from the slope, i.e Figure... 490 Ferroelectrics - Characterization and Modeling Pope, M & Swemberg, C E (1998) Electronic Processes in Organic Crystals and Polymers, Oxford University Press, ISBN 0195129636, Oxford, United Kingdom Poullain, G.; Bouregba, R.; Vilquin, B.; Le Rhun, G & Murray, H (2002) Graded Ferroelectric Thin Films: Possible Origin of the Shift Along the Polarization Axis Applied Physics Letters, Vol 81, pp 5 015. .. θ0=298K) and Δθ is the temperature variation In most cases the coefficient χ is negligible compared to 2 β × P(E ,θ 0 ) Thus, the expression (26) becomes: ΔE ≡ (2 β × P(E ,θ0 )) × Δθ and Δθ ≡ ΔE (2 × β P( E,θ 0 )) (27) With; ΔE ≡ E − E0 = E and Δθ = θ − θ 0 According to equations (25) and (27) we find the following expression: ΔE ≡ αΔT × P( E , T0 ) ≡ 2 β × P( E ,θ 0 ) (28) 504 Ferroelectrics - Characterization . Ferroelectrics - Characterization and Modeling 494 where E, T, and S represent the electric field, the mechanical stress, and the strain, respectively. The constants 33 T ε , 33 E s , and. C in (x)  0, Eqs. (97) and (98) can be rewritten as Ferroelectrics - Characterization and Modeling 486 (,) (,) 1 (,) 2 qq p xt xt Cxt qq           (105) and (,) (,) 1 (,) 2 qq n xt. the presence of a finite intrinsic conductivity and the simultaneous presence of Ferroelectrics - Characterization and Modeling 488 p-type and n-type free carriers, it would be important