Microscopic model for the ferroelectric field effect in oxide heterostructures

PHYSICAL REVIEW B 84, 155117 (2011)

Microscopic model for the ferroelectric field effect in oxide heterostructures

Shuai Dong,^1,2 Xiaotian Zhang,^3,4 Rong Yu,⁵ J.-M. Liu,^2,6 and Elbio Dagotto^3,4

1Department of Physics, Southeast University, Nanjing 211189, China

2National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

3Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA 4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831,

USA ⁵Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA

6International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China (Received 21 September 2011; published 14 October 2011)

A microscopic model Hamiltonian for the ferroelectric field effect is introduced for the study of oxide heterostructures with ferroelectric components. The long-range Coulomb interaction is incorporated as an electrostatic potential, solved self-consistently together with the charge distribution. A generic double- exchange system is used as the conducting channel, epitaxially attached to the ferroelectric gate. The observed ferroelectric screening effect, namely, the charge accumulation/depletion near the interface, is shown to drive interfacial phase transitions that give rise to robust magnetoelectric responses and bipolar resistive switching, in qualitative agreement with previous density functional theory calculations. The model can be easily adapted to other materials by modifying the Hamiltonian of the conducting channel, and it is useful in simulating ferroelectric field effect devices particularly those involving strongly correlated electronic components where ab initio techniques are difficult to apply.

DOI: 10.1103/PhysRevB.84.155117 PACS number(s): 85.30.Tv, 75.47.Lx, 77.80.−e, 85.75.Hh

I. INTRODUCTION

The research area known as oxide heterostructures con- tinues attracting considerable attention of the condensed matter community due to the rich physical properties of its constituents, often involving strongly correlated elec-tronic materials, and also for their broad potential in device applications.^1–4 Among these heterostructures, those involving ferroelectric (FE) and magnetic, or multiferroic, components are particularly interesting since they could be used in the next generation of transistors and nonvolatile memories.^5–7 From the applications perspective, the FE/magnetic heterostruc-tures could become even superior to the currently avail-able bulk multiferroics with regards to their magnetoelectric performance.^5–7 In these heterostructures, it is easier to obtain large FE polarizations and a robust magnetization, and the manifestations of the magnetoelectric coupling can be fairly diverse. For example, an exchange bias effect that can be controlled with electric fields has been recently reported in La0.7Sr 0.3MnO3/BiFeO3,^8,9 and the associated physical mechanism that produces this interesting behavior is being actively discussed.^10–13

Even without the magnetic coupling across the interface, interfacial magnetoelectric effects still generally exist in these heterostructures. A mechanism contributing to these effects involves the possibility of lattice distortions, since the oxides magnetic or FE properties are often sensitive to strain.^14–16 An additional contribution is the carrier-mediated field effect,^17,18 especially crucial in ultrathin film heterostructures. The FE field effect not only generates magnetoelectricity, but also gives rise to a bipolar resistive switching.^19–31

A heterostructure FE field-effect transistor (FE-FET) is basically composed of a FE oxide film and a thin metallic or semiconducting oxide film, as sketched in Fig. 1, similarly to traditional FETs used in the semiconductor industry. In those standard FET devices, the conductivity of the semiconducting

channel can be switched on and off by tuning the gate voltage. The FE-FETs can provide similar functions by switching the direction of the polarization of the FE gate.

Moreover, this switching, at least ideally, can be nonvolatile due to the remnant FE polarization.^19,21 Furthermore, due to the strongly correlated character of the electronic component in several oxides, the above mentioned switching in FE-FET is not limited to the conductivity, but it may also influence other physical quantities as well, such as the magnetization, orbital order, elastic distortions, etc. Therefore compared with traditional semiconductor FETs, the physics in FE- FETs can be richer, and potentially additional functionalities can be expected.

Although the FE-FETs have been experimentally studied for several years, only recently, theoretical investigations have been focused on this topic.^{17,18,32–35} These recent theoretical studies have been based on the density functional theory (DFT). In fact, studies using model Hamiltonians including strongly correlated electronic effects, beyond the reach of DFT, and focusing on the basic aspects of the FE field effect in these oxide heterostructures are rare. An important technical prob-lem in this context is how to take into account the contribution from the FE polarization on the physics of the microscopic model Hamiltonian representing the other components. In recent efforts by some of us, the FE polarization was modeled as an interfacial potential at the first layer of the conducting channel,³⁶ but this approximation must be refined to address the subtle energy balances between competing tendencies near the interface. Thus, for all these reasons, in this manuscript, the FE- FET structures will be revisited using model Hamiltonian techniques and applying new approximations to handle this problem. Our effort has the main merit of paving the way for the use of models for the study of FE-FET systems where one of the components has a strongly correlated electronic character that is difficult to study via ab initio methods.

1098-0121/2011/84(15)/155117(8) 155117-1 ©2011 American Physical Society

DONG, ZHANG, YU, LIU, AND DAGOTTO PHYSICAL REVIEW B 84, 155117 (2011)

are

−

√

t

x

ta ax

t

a b

x t0 3 3 ,

t bax

=

tbbx

= 4 √ 3 1

− √

t

y t

y

3

t0 3

t

y aa ab ,

=

t

^b

a^y t

b b y

=

√

1

4

3

t

z t z

t

z

t

0 0 0 (2)

= aa ab = ,

FIG. 1. (Color online) Sketch of an FE-FET heterostructure (S _tbaz

t

b

bz 0 1

indicates the source and D the drain). The physical properties of

the where t0 is the DE hopping amplitude scale. In the rest of channel can be switched on and off by the FE polarization of the this

gate.

publication, t0 is considered the unit of energy. Its real value is approximately 0.5 eV in wide-bandwidth manganites such

II. MODEL AND METHOD

as La0.7Sr0.3MnO3

(LSMO).^37,38

A. Model Hamiltonian

The second term in the Hamiltonian is the on-site potential energy: Vi is the actual potential at each site and ni is the eg

As discussed in the introduction, in this manuscript, the

electronic density operator at the same site. The last term is

FE field effect will be studied from the model Hamiltonian

the Heisenberg-type antiferromagnetic (AFM) superexchange

perspective. More specifically, here, the standard two- orbital

(SE) interaction between the localized NN t2g spins. Its actual

(2O) double-exchange (DE) model will be used for the typical strength is about 10% that of t0.^37,38 metallic component of the heterostructure. This 2O

DE model is well

known

to be successful

in modeling

the B. Self-consistent calculations perovskite manganites,^37–39 which are materials often

used in In the actual calculations described in this publication, a FE-FET devices. Furthermore, previous model

Hamiltonian

studies have already confirmed that the 2O DE model, cuboid lattice (L × L × L , L = L = 4, L = 12) will be

+ _JAFSi Sj .·

ij

In this expression, the ij

H = −

γ γ

with some simple modifications, is still a proper model to use

used with open boundary conditions (OBCs) along the z axis to

for manganite layers when they are in the geometry of a

avoid having two interfaces.^36,42 Twisted boundary conditions

heterostructure.^36,40–44 In addition, since the DE mechanism

(TBCs) are adopted in the x-y plane to reduce finite size effects

provides a generic framework to describe the motion of

via a 6 × 6 k mesh.

electrons in several magnetic systems, the approach

followed The FE gate will be here modeled as a surface charge (Q here, with minor modifications, could potentially be

adapted per site, in units of the elementary charge e, and located to other oxides beyond the manganites. at z = 0) coupled to the first channel layer (z = 1). This As a widely accepted simplification, the limit of an infinite

approximation has been successfully confirmed in previous

Hund coupling will be adopted in the DE model studied

here. DFT calculations.17,18,32–35 The long-range Coulomb interac- Then, more specifically, the model Hamiltonian of the

metallic tion is included via a layer-dependent potential V (z),³⁴ and

channel reads as within each layer the potential is assumed to be uniform

for simplicity. This electrostatic potential is determined via

the Poisson equation.^36,41–44 In particular, the electric field

r † H.

c.) between the zth and (z 1)th layers is determined by

the

t γ γ (

ij c iγc

j γ + ^Vi n

i net charge [Q + 1 l

z +

+ i l (−n(l) + nb )] counted from the FE

interface, where n(l) is the eg electronic density corresponding

to the lth layer, and nb is the background (positive) charge

(1)

density. Thus the electrostatic potential (with respect to the negative charge of electrons) of each layer can be calculated first term is the standard DE via the relation:

interaction. The operator ciγ (ciγ†

) annihilates (creates) an electron at the orbital γ of the eg band and at the lattice site i, with its spin perfectly parallel to the localized t_2g spin S_i . The indices i and j represent nearest-neighbor (NN) lattice sites. The Berry phase factor _ij , generated by the infinite Hund coupling limit adopted here, equals cos(θi /2) cos(θj /2) + sin(θi /2) sin(θj

/2) exp[−i(φ_i − φ_j )], where θ and φ are the polar and azimuthal angles defining the direction of the t2g spins, respectively.

When a ferromagnetic (FM) t2g back-

ground is used, then = 1. The labels γ and γ denote the two Mn eg-orbitals a (|x² − y² ) and b (|3z ² − r ² ). The NN hopping direction is denoted by r . The DE hopping

depends on the direction in which the hopping occurs, and it is orbital-dependent as well. The actual hopping amplitudes

V (z + 1) = V (z) + α Q +

1 ^{l z}[−n(l) + nb ) , (3)

l

where α is the Coulomb coefficient, which is inversely proportion to the dielectric constant ε [α = c/(εt0), where c is the lattice constant, ε is the dielectric constant, and t₀ is in unit of eVs as explained before]. In the following, n_b is fixed at the value 0.7 since typical manganites are FM metals at this doping value, e.g., LSMO and La0.7Ca0.3MnO3 (LCMO).⁴⁵

In our computational study, the 12th layer is assumed to be sufficiently far from the interface such that V (z = 12) is set to be zero as the reference point of the electrostatic potential. This choice, combined with a fixed chemical potential, restores the system to its original bulk state for layers far from the

155117-2

MICROSCOPIC MODEL FOR THE FERROELECTRIC FIELD . . .

interface. A FM t2g background is adopted to simulate the metallic channel in the FE-FET device. The DE Hamiltonian (including the term with Vi ni ) is diagonalized to obtain the charge distribution n(z), which is iterated together with V (z) until a self-consistent solution is reached. After convergence in n(z) and V (z), the total grand potential (per unit cell) can be calculated as

1

1 z

L_z 1

1 z L_z

= f − ^z V (z)n(z)

− n

b

z V

(z) 2

L

z

2 L

z

1 V

(0) Q

J

AF S S

, (4)

− 2Lz

+ Lx Ly Lz _i,j i · j

where _f is the fermionic grand potential (per site), calculated from the diagonalization eigenvalues. The second term consid- ers the reduction of the electrostatic Coulomb energy of the eg

electrons, since it is doubly-counted in the first term. The third and fourth terms are the electrostatic Coulombic energies of the positive background charge (nb ) and the FE surface charge, respectively. The last term describes the AFM SE energy, namely, the Heisenberg interaction among the localized spins.

A finite but low temperature T = 0.005t₀ ( 30 K) is used for the Fermi-Dirac distribution function smearing.

III. RESULTS AND DISCUSSION A. Charge accumulation/depletion

To investigate the screening effects in the FE-FET het- erostructure, the results for four values of α (0.5, 1, 2, and 4) were compared. For each α, the surface charge Q is initially set to zero to find the chemical potential where the average eg density equals nb . With this chemical potential, Q is then varied from +0.4 to −0.4 (in units of the elementary charge per cell). Ideally, |Q| = 0.4 corresponds to a FE polarization as large as 40 μC/cm² (if the pseudocubic lattice constant c is set as 4 A),

˚

which is a typical and reasonable value for standard FE oxide materials.

The screening effects correspond to the accumula- tion/depletion of charges near the interface. Under a positive (negative) Q, more eg electrons will be attracted to (repelled from) the interface. Since the chemical potential is fixed in our simulation, the screening effect can also be obtained from the average eg density as a function of Q, as shown in Fig. 2. This screening effect increases when α is increased, which is concomitant with a stronger electrostatic Coulomb interaction near the interface. In the rest of the manuscript, α

= 2 will be here adopted: using t₀ = 0.5 eV and c = 4 A,

˚

this α value corresponds to a relative permittivity εr ≈ 45, which is quite reasonable to represent real materials. Also note that α= 2 is already very close to the fully screened case according to the results shown before. It should be remarked that the total charge for the whole system is zero (i.e., the combined FE gate and manganite channel are neutral) although the gate and channel themselves are charge polarized.

The screening effect is better observed by studying the e

electron density profiles and their corresponding electrostatic potentials in Fig. 3. The Q = +0.4 and −0.4 cases are shown together for better comparison. When Q = +0.4, then

PHYSICAL REVIEW B 84, 155117 (2011)

FIG. 2. (Color online) (a) The average eg density n vs Q. The (red) dashed line corresponds to the fully screened case, where n = nb + Q/Lz. Here, only the electronic screening is considered, while the ionic screening⁴⁶ is neglected, since its effect can be partially expressed by the dielectric constant that enters in α and an effective Q. (b) The deviations of the eg density from the fully screened limit, where δn = ( ni − nb ) × Lz− Q. The maximum deviation (|δn|) is

<0.015 for α = 0.5, which decreases to <0.01 for α = 2 and 4.

V (z) becomes deep enough near the interface to accumulate considerably more eg electrons than in the bulk. In contrast, when Q = −0.4, then V (z) is large and positive near the interface, thus repelling those eg electrons. With α= 2, the screening of eg electrons is the most significant within a thin region near the interface, typically involving just 2–3 layers for the 2O DE model employed here.

B. Interfacial phase transitions

Since the previous results show that the interfacial elec- tronic density can be substantially modulated by the FE polarization, then it is natural to expect local phase transitions.

The reason is that the phase diagrams of oxides are usually

FIG. 3. (Color online) The eg density profiles n(z) (dots) and the electrostatic potential V (z) (lines without dots). The cases Q = −0.4 (left) and Q = 0.4 (right) are shown together for better comparison.

The FE gate is in the middle and its polarization points to the right as indicated. The original eg density (nb = 0.7) is shown as dashed lines for better reference.

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DONG, ZHANG, YU, LIU, AND DAGOTTO PHYSICAL REVIEW B 84, 155117 (2011)

FIG. 4. (Color online) The candidates for the spin order at the two interfacial layers, as described in the main text. The layer indices (1 and 2) counting from the interface are shown on the left side of the figure. The FM spin order is the original one, in the absence of the surface charge Q. In the rest of the panels, the spins pointing down are shown in red. All spins in other layers (z > 2) point “up” for these variational states. Here, the choices for the spin candidate states are not arbitrary but have clear correspondences to states already known to exist in the bulk phase diagrams. In addition, some combinations of different magnetic orders in the two layers have also been included since the interfacial region may be different from the bulk.

highly sensitive to charge density variations,⁴⁷ i.e., density- driven phase transitions are well known to occur in bulk materials when chemically doped to modify the electronic density.^32,33 To explore these possible phase transitions, the zero-temperature variational method is here employed by comparing the total ground-state energy [Eq. (4)] for a variety of spin patterns. From Fig. 3, it is clear that most of the charge accumulation/depletion occurs within the first two layers near the FE interface. Hence, for simplicity, the several non-FM (collinear) spin patterns explored here will only be proposed to exist in these two layers in our present variational calculation, while the spins in the other layers remain fixed to be FM. The candidate spin patterns in the two interfacial layers are shown in Fig. 4.

The ground-state phase diagram obtained in our calcu- lations for the interfacial layers in FE-FET is shown in Fig.

5. According to this phase diagram, the original FM metallic phase at Q = 0 is stable when JAF < 0.128, while the boundary between the FM and A-type AFM phases is at JAF

= 0.13 for the calculation representing the bulk (see Fig. 7 later in this paper). These two, almost identical values suggest that the lattice size effects and surface effects are negligible in our simulation of FE-FET.

By adjusting the FE polarization (i.e., by modifying the surface charge Q) in the FE-FET setup, in the present variational effort it has been observed that the interfacial spins have a transition to arrangements different from the original FM state. This is the main result of our publication.

For example, the CE1 and Cx1 orders are stabilized and replace the FM state in sequence with increasing negative

Q when JAF 0.12t0, as shown in Fig. 5. In contrast, the FM order remains robust under a positive Q, thus establishing an asymmetry in the response of the system to the FE polarization orientation that is of value for applications.

The FE screening effect plays an important role to deter- mine the dominant interfacial spin order that is competing with the DE mechanism that favors ferromagnetism. Considering JAF = 0.12 as example, when Q < 0, the CE1 and Cx1 orders can accommodate more holes near the interface than the original FM state, thus reducing the Coulomb potential pronouncedly, as shown in Fig. 6. In simple words, the

FIG. 5. (Color online) Ground-state phase diagram for the interfa-cial layers in FE-FET, obtained by the variational procedure described in the text.

MICROSCOPIC MODEL FOR THE FERROELECTRIC FIELD . . . PHYSICAL REVIEW B 84, 155117 (2011)

FIG. 6. (Color online) The eg density profiles (left axes) and the corresponding electrostatic potentials (right axes) of the 2O model studied here. Panel (a) is for Q = −0.17, while panel (b) is for Q =

−0.3. The ground states [CE1 state in (a) and Cx1 state in (b)]

provide the best screening effect, i.e., a smooth potential V (z) varying z.

system chooses an interfacial state which can screen the FE polarization rather well.

C. Comparison with bulk properties

For comparison, the ground state of the bulk is also calculated using the standard 2O DE model, under a similar variational approximation with states now covering the whole system. This information can be used as a guide to explore the interfacial spin orders that may be of relevance in the FE-FET setup. The results are shown in Fig. 7. Considering

FIG. 7. (Color online) The ground-state phase diagram of the 2O DE model for manganites in the bulk, which is calculated using the variational method described in the text. All DE energies are obtained from analytical band structures. The phase boundary between the FM and A states at n = 0.7 is illustrated by the vertical dashed line, while the horizontal dashed line shows the phase transition at J_AF = 0.12 obtained by changing the eg electron density. A, C, CE, and G denote the typical AFM phases found in manganites.⁴⁵

FIG. 8. (Color online) Comparison of the transitions found in the FE-FET heterostructure and in the bulk, at JAF = 0.12. According to the bulk phase diagram, with decreasing electronic density, the bulk system turns from the FM phase into an A-type AFM state, followed by a CE phase and then by a C-type AFM state. However, in the FE-FET setup, from the original FM phase and with increasing |Q|, the spins in the first interfacial layer directly jump to the CE order, and then to the Cx1 order. Moreover, the shaded regions were found to be unstable due to phase separation tendencies in the FE-FET case.

the simplicity of the model (with only two competing NN interactions: DE versus SE), this phase diagram agrees fairly well with the experimental perovskite manganite results.⁴⁵ The most typical phases found in bulk manganites, namely, the FM and various AFM states (A, C, G, and CE types), appear in the proper eg density and bandwidth regions, providing support to the qualitative accuracy of our calculations.

Considering JAF = 0.12 as an example, Fig. 8 compares the spin-order transitions in the bulk and in the FE-FET. In the bulk’s phase diagram, by reducing the e_g density from n = 0.7, the system transitions from a FM phase to an A-type AFM state at n = 0.63, then from A to CE at n = 0.5, and from CE to C- type AFM one at n = 0.42. In the FE heterostructure, on the other hand, the system changes from FM to CE1 at Q = −0.1 (n

= 0.62 in FM and n = 0.55 in CE1), and then from CE1 to Cx1 at Q = −0.19 (n = 0.51 in CE1 and n = 0.45 in Cx1). There are several interesting aspects in this interfacial phase transitions.

First, the “critical” eg densities are found to be different between the bulk and the FE-FET heterostructure. Second, the fragile A-type AFM state is absent in the heterostructure geometry. Third, in the heterostructure the interfacial electronic density jumps at the locations of the spin-order transitions, causing some density regions to be unreachable (i.e., they are unstable). Such density discontinuities originate from the well- known electronic phase separation tendencies in manganites,^37–39 a phenomenon that does not have an analog in semiconducting devices. Last but not least, the CE1 and Cx1 states predicted here have not been considered in previous DFT studies, since these states typically need larger in-plane cells than previously analyzed with DFT. These two interfacial states, CE1 and Cx1, may exist particularly in those manganite channels with relative narrow bandwidths, such as LCMO.

There are two main reasons for the differences observed here in the phase diagrams between the bulk and the het- erostructures. The first reason is the FE screening effect, as shown in Fig. 6, namely, the ground state near the interface is determined not only by the competition between the DE kinetic energy and the SE energy as in the bulk, but also by the electrostatic potential energy. Second, since the spin-order transitions occur only near the interface, the global phases shown here, except for the FM one, are actually “artificial”

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