Enhancement of photovoltaic effect in nanoscale polarization graded ferroelectrics

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Solar Energy 86 (2012) 811–815 www.elsevier.com/locate/solener

Enhancement of photovoltaic effect in nanoscale polarization graded ferroelectrics Yan Zhou Department of Information Technology, Hong Kong Institute of Technology, 2 Breezy Path, Mid-levels West, Hong Kong Received 18 July 2011; received in revised form 7 October 2011; accepted 14 October 2011 Available online 4 January 2012 Communicated by: Associate Editor Arturo Morales-Acevedo

Abstract In this work, we propose the compositionally graded ferroelectrics (CGF) as solar energy harvesting device. Such a novel geometrically frustrated system exhibits an intrinsic built-in potential which can be used to separate the hole and electron currents. It is shown that the CGF based photovoltaic devices can achieve orders of higher efficiency than bulk/non-graded ferroelectric thin films. The dependence of the photovoltaic effect on various device parameters have also been investigated. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Common photovoltaic devices are referred to siliconbased semiconductor junction, in which electron–hole pairs are excited by the photon energy higher than the bandgap of the junction and then subsequently separated by the internal electric field within the space-charge region. Recently, the photovoltaic effect of ferroelectric materials has aroused intensive research interests. In ferroelectric structure, an effective built-in electric field originating from the incomplete screening of the polarization can dissociate the electrons-holes and increase the drift length of the charge carriers (Ji et al., 2010; Qin et al., 2009a,b,c; Qin et al., 2008; Qin et al., 2007a,b,c,d). It has been shown that even single-phase (non-graded) ferroelectric material can exhibit photovoltaic effect under light illumination (Ji et al., 2010; Qin et al., 2008; Gan et al., 2008; Pintilie et al., 2011; Huang, 2010; Yang et al., 2010; Ichiki et al., 2005). In this regard, intensive efforts have been made to increase the light-to-electricity conversion efficiency of ferroelectric structure by implementing sophisticated device

E-mail address: [email protected] 0038-092X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2011.10.014

design and material properties optimization (Huang, 2010; Yang et al., 2010; Yuan et al., 2011). Graded ferroelectric materials have been intensively studied in the past thirty years due to the many intriguing features of such system, such as polarization offset and imprint, the enormous piezo- and pyro-electric responses, tunable built-in potential and effective bandgap, enhanced thermoelectric and energy harvesting functionalities (Zhou et al., 2005; Alpay et al., 2003; Ban et al., 2003; Mantese et al., 2002; Zhou et al., 2005; Zhou and Shin, 2006a,b,c; Zhang and Ponomareva, 2010). In a more recent paper, it has been demonstrated that the CGF system exhibits all “fingerprint” properties of fundamental geometric frustration effect (Choudhury et al., 2011). Such revelation has brought graded ferroelectric materials to the novel class of geometric frustration system and may lead to many ground-breaking applications in diverse fields. In this work, we adopt the Landau-type free-energy functional approach to study the photovoltaic effect of compositionally modulated ferroelectric nanodevices. We show that the intrinsic built-in electric field can significantly enhance the charge extraction process and thereby the photocurrent generation efficiency. While the amplitude of the polarization gradient (PG) has a direct impact on the

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photocurrent amplitude, the photocurrent direction can be controlled by switching the polarization of the graded ferroelectric device. The various material and size dependence of the photocurrent generation efficiency has also been studied by numerical simulations. In addition, the bandgap of graded ferroelectrics can be continuously tuned so that such system will be suitable candidate for photovoltaic solar cell, ultraviolet (UV) sensors, and a wide range of novel optoelectronic devices. The PG (and thus the photovoltaic current) can be modulated by applied mechanical stress and environment temperature, making such system appealing for applications of nanoscale energy harvesting devices. 2. Model 2.1. Review of the Landau–Ginzburg–Devonshire theory for ferroelectric CGF nano-cylinder The continuum Landau–Ginzeburg–Devonshire (LGD) phenomenalogical theory was originally proposed to describe the phase transition behavior of bulk materials based on the functional Taylor expansion and has been extensively employed for investigating the properties of ferroelectrics. In recent years, it has been extended to various systems at nanoscale such as nanotube and nanowire. However, modeling the nanosized structure is non-trivial and requires the careful treatment of surface stress, depolarization effect, correlation (gradient) energy, near-surface effect and other surface related energy terms, which is correlated with the size of the system such as the wire radius of nanowire ferroelectrics. The surface energy contribution may even become dominant for certain nano-systems and must be taken into account to give an accurate account of the ferroelectrics devices of interest (e.g., nanorod, nanotube and nanowire etc). Thereby the formulism of thermodynamic LGD free energy model has been continuously evolving to incorporate various coupling mechanisms of the order parameters and contributions arising from the size effects of the nano-system. We consider a CGF nanocylinder with length h and radius of R sandwiched between two metal electrodes, as in Fig. 1. It is assumed all the vector fields (such as electrical polarization P, etc.) are along the wire axial direction, i.e. the z-axis. For such system, the contribution of the surface stress is inversely proportional with the wire radius. The intrinsic surface stress leads to strains in both the radial and wire axes directions. Taking the surface energy term FS into consideration, the Landau–Ginzburg–Devonshire free energy can be expressed in terms of spontaneous polarization as (Ban et al., 2003; Zhou, 2011; Morozovska et al., 2008; Eliseev and Morozovska, 2009; Morozovska et al., 2010; Morozovska et al., 2010; Zhou, 2010):    Z a b c g Ed 2 F ¼ d 3 r P 2 þ P 4 þ P 6 þ ðrP Þ  P Ea þ 2 4 6 2 2 V i Z ha i c ijkl S 2 uij ukl þ d 2 r P þ lSij uij ; ð1Þ qij33 þ P 2 þ 2 2 S

Fig. 1. Schematic illustration of CGF nano-cylinder under illumination. In this illustrative example, the polarization points downward, i.e., z direction.

where a = a0(T  T0) with a0 being proportional with the inverse Curie constant and T0 the curie temperature of the corresponding bulk materials (field-free and tensionless). b, c and g are temperature-independent parameters. lij and uij are the intrinsic surface stress tensor and mechanical strain, respectively. cijkl and qijkl represent the elastic stiffness tensor and electrostriction stress tensor, respectively. Ea and Ed are the applied electric field, and depolarization field, respectively. In the expression of the surface energy term, aS is the surface energy coefficient and the surface energy need to include the energy associated with the cylindrical sidewall and the two electrode/nano-cylinder interfaces. The total energy in Eq. (1) is integrated over the volume V for bulk part (the first term) and the whole surface S for the surface part (the second term). In the absence of applied electric field, the total electric field has the sole contribution from the depolarization field, i.e. E = Ed, which mainly arises from the inhomogeneous polarization profile in CGF structure. For the CGF nano-cylinder, the polarization gradient results in a nonlinear spatial and temporal evolution of various parameters including the dipole moment, electric field and displacement. Such equations involving high nonlinearity must be solved with the photon-generated charge carrier motions with full consistency.

2.2. The photovoltaic effect Considering the above CGF nanostructure, the photonexcited electron–hole generation can be calculated by the current continuity equations (Qin et al., 2007a), dn 1 dJ n ¼ þ G n  An ; dt q dz dp 1 dJ p ¼ þ Gp  A p : dt q dz

ð2aÞ ð2bÞ

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813

where q is elementary charge, n and p are the electron and hole concentrations under illumination. Jn and Jp are the electron and hole current densities, which comprises of drift and diffusion components (Qin et al., 2007a),

dP P ðz ¼ 0Þ  ¼ 0; dz d dP P ðz ¼ hÞ þ ¼ 0: dz dþ

dn ; dz dp J p ¼ qlp pE  qDp : dz

where d± represents the extrapolation length, differentiating the polarization between the surface and the interior. For the thin ferroelectric barrier at the top and bottom interfaces. It should be noted that the extrapolation length might be different to take into account the asymmetric surface effect on the polarization, i.e., d – d+ for the two interfaces due to different growth conditions and asymmetric adjacent electrodes (Zhou, 2011).

J n ¼ qln nE þ qDn

ð3aÞ ð3bÞ

where Dn, p are the electrons and holes diffusion coefficients, and ln, p are the mobilities of electrons and holes, respectively. Gn, p and An, p are the electron–hole pair generation and annihilation rate and can be expressed as (Qin et al., 2007a; Zhou et al., 2004), Gp ¼ Gn ¼ bl al ul T e eal z

ð4Þ

where al is the optical absorption coefficient, ul is the incident photon flux density per second, and bl the quantum efficiency representing the number of electron–hole pairs that per photon can produce (0 < bl < 1). Te is the transmittance of the light for the top electrode. The charge carrier (electrons, holes) re-combination rate are (Qin et al., 2007a; Zhou et al., 2004) An ¼ n=sn ; AP ¼ p=sp ;

ð5aÞ ð5bÞ

where sn and sp are the carriers’ recombination lifetimes. At steady state d(p, n)/dt = 0, the spatial distribution of the electron and hole densities can be obtained by solving the steady state continuity equations (Qin et al., 2007a; Zhou et al., 2004) ln kT q lp kT q

d 2n dn n þ ln E þ bl al ul T e eal z  ¼ 0; 2 dz sn dz d 2p dp p  lp E þ bl al ul T e eal z  ¼ 0: 2 dz sp dz

ð6aÞ ð6bÞ

Under the short-circuited condition, the conservation of charge gives the continuity of the total current Jt across the circuit dJ t dJ n dJ p ¼ þ ¼ 0: dz dz dz

ð7Þ

The conservation of energy gives the vanishing of voltage sum across the circuit Z

h

EðzÞdz ¼ 0:

ð8Þ

0

At the two electrode/cylinder interfaces z = 0 and z = h, the equilibrium conditions are given by the stationary solutions of the Euler–Lagrange equation (Qin et al., 2007a; Morozovska et al., 2010; Zhou et al., 2004),

ð9aÞ ð9bÞ

3. Results and discussion For numerical simulation, we follow a similar procedure as described in Refs. Zhou et al. (2005) and Zhou et al. (2004). The nanowire is modeled as a stack of discretized layers. At t = 0, an initial spatial distribution profile of electric field, charge carrier, and polarization are given by the Maxwell equation without considering the light illumination. After switching on the light illumination, the system will undergo a transient dynamics to an equilibrium state following Eqs. (1) and (2), which is subject to the circuit boundary condition of Eqs. (7) and (8). The polarization must be solved and governed by the time dependent Ginzburg - Landau equations derived from the thermodynamic Landau free energy functional in Eq. (1) with full self-consistency, and the boundary condition should be determined by Eq. (9). Also, the spatial and temporal evolution of electric field, current, and charge carrier density must be solved in a self-consistent way together with the electrical polarization. A wide range of ferroelectric materials have been shown to exhibit reasonable photovoltaic effects such as BaFeO3 (BFO), PbZrTiO3 (PZT), and (PbLa)(ZrTi)O3 (PLZT). We take graded PZT for example to illustrate the photovoltaic effect in this paper. We consider a compositionally graded PZT-based nanocylinder subject to gold electrodes under short-circuited condition (Qin et al., 2007a). The material parameters such as the expansion coefficients of the free energy, electrostrictive coefficients, elastic properties, and background dielectric constant used in the calculation are the same as in the literature (Qin et al., 2007a; Zhou and Shin, 2006c; Zhou et al., 2004; Chan et al., 2004). The other related material parameters are listed as follows (Qin et al., 2007a): sp = sn = 200 ps, al = 2  106 m1, bl = 2  106 m1, Te = 60%, T = 300 K, q = 1.6  1016, k = 1.38  1023 J/K, ln = 3  104 m2/V s, lp = 2.93  1010 m2/ V s. It should be noted that both our model and simulations will give qualitatively similar results if we choose different sets of material parameters such as graded BFO. Also we should keep in mind that even the two ferromagnetic electrodes are the same, the screening at the two interfaces can still be quite different due to different growth condition etc (Duan et al., 2006). However, it is assumed in this work that the screening at the top- and bottom-electrodes are the same

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for simplicity. In addition, the photocurrent, electron and hole density throughout the work is normalized by their diffusion components when Ed = 0. Fig. 2 shows (a) the calculated steady-state polarization profile, (b) electric field spatial distribution, (c) photon induced charge carrier density p(z) and (d) n(z) as a function of the specimen length h at room temperature. It can be seen that the steady state polarization profile is graded and the direction of the gradient is along +z direction. The polarization profile becomes quite flat at z = 0 and the slope of the polarization increases with z when approaching the top electrode z = h. The spatial distribution of electric field E follows somehow a similar trend with P except that it exhibits a negative slope. Also it reaches a maximum (absolute value) when approaching to z = h, where the top electrode is illuminated by the incident light. The photon induced charge carrier density are plotted versus z in Fig. 2c and d. There are both positive and negative charge carriers (electrons and holes) accumulated at the top electrode, whereas the negative charge carrier density is noticeably larger than that of positive charge so that the overall charge density is negative. It also shows that the photon induced charge carriers are mainly exited within very small thickness of a few nanometers below the top electrode, where the built-in electric field reaches its maximum. The total current is mainly dominated by the electrons current and is uniform across the structure at the steady state and hence will not be shown here. In Fig. 3, we plot the dependence of photocurrent on the length of the nanocylinder for R = 35 nm. The photocurrent is almost zero below a critical thickness, above which it increases sharply with h and then slowly decreases to zero again. This dependence can be partially explained by the polarization dependence of h as shown in the inset. Below the critical thickness hcr, the system is paraelectric and net polarization is zero. Just above hcr, it increases sharply and gradually approach to the saturation value. The decreasing of photocurrent with h at higher end in Fig. 3 is due to the decrement of depolarization field with h. It should be noted

0.6

6 5

0.4

0.2

4

0

0

Jn

10

1

h (nm)

10

3 2 1 0

0

1

10

2

10

10

h (nm) Fig. 3. The photocurrent vs. nanocylinder length h with radius R = 35 nm.

that the direction of the photocurrent is switchable by switching the polarization vector as seen in the inset of Fig. 3. Fig. 3 gives a clear guidance for the experimentalists to design the most efficient photovoltaic nanodevice based on graded ferroelectrics materials. The power conversion efficiency is calculated by the photocurrent produced electric power divided by the power of the incident light illumination, and such conversion efficiency should be a function of the polarization gradient dP/dz. To compare the photovoltaic effect of the compositionally graded ferroelectrics photovoltaic device with the non-graded device, we plot the ratio of the energy conversion efficiency of graded device to that of non-graded ferroelectric device S vs. polarization gradient dP/dz in logarithmic scale in Fig. 4. It can be seen that the energy conversion efficiency S increases and then saturates and the maximum value of S for experimentally accessible range of dP/dz can be as large as 50, which means the energy conversion efficiency of CGFbased photovoltaic devices can be dramatically improved compared with the non-graded ferroelectric material. While

1.0

(a)

-0.55

(c)

2

P (C/m )

7

(C/m2)

814

n

0.8 -0.60

0.6 0.4

0

1.0

(b)

-1

p

8

E (10 V/m)

-0.65

0.8

(d)

0.6

-2

0.4

-3 0

10

20

30

z (nm)

40

50

60

10

20

30

40

50

60

z (nm)

Fig. 2. The spatial profile of (a) electric polarization, (b) electric field, (c) normalized negative charge carrier density and (d) normalized positive charge carrier density of graded PZT at room temperature for h = 20 nm and R = 20 nm.

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Acknowledgement

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The author gratefully acknowledges the many stimulating discussions with Prof. K. H. Chew.

30

References

S

50

20

10

0

0

2

4

6

dpdz

8

10 7

x 10

Fig. 4. The enhancement coefficient S vs. polarization gradient dP/dz in logarithmic scale.

Fig. 4 is obtained at h = 200 nm and R = 50 nm, we have also calculated S at different wire length and radius and it is found that the efficiency can be noticeably enhanced with the enhancement coefficient depending on the specific geometry of the devices. Finally, we briefly discuss the possible shortcoming of CGF solar cell. The major disadvantage of utilizing compositionally graded ferroelectric system in photovoltaic solar cell is the long-term reliability of such system (typically required for more than 10 years operation). As reported earlier by some experiments and theoretical analysis (Zhou et al., 2005,; Mantese et al., 2002; Jin et al., 1998; Zheng et al., 2009; Misirlioglu et al., 2010; Okatan et al., 2010), the graded ferroelectric system may suffer strong polarization offset (vertical displacement along P axis of the P–E hysteresis loop) and imprint effect (horizontal shift along E axis of the hysteresis loop). These uncontrollable shifts as well as the fatigue problems in the graded ferroelectric system may limit its application in photovoltaic solar cell to small scale and short lifetime. In addition to the compositionally graded ferroelectrics, the polarization gradient in nanoscale ferroelectric system can also be achieved via surface effects and doping etc. Thereby our approach and conclusion can be generalized to any kind of ferroelectric structures with polarization gradient. 4. Conclusions In summary, we have theoretically studied the feasibility of utilizing compositionally graded ferroelectric system for photovoltaic solar cell in the framework of Landau freeenergy functional approach, through strict adherence to the thermodynamic principle. We have studied the various materials and size dependence of such CGF nanocylinder and it is found that the efficiency can be dramatically enhanced compared with that of the non-graded ferroelectric device. We believe such study will be very significant for photovoltaic device design and optimization based on ferroelectric materials.

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