^{1}

^{2}

^{3}

The dependence of the polarization
P
in Hf
_{1-x}
Zr
_{x}
O
_{2}
nanoparticles on electric field, dopant concentration
x
, size and temperature are studied using the transverse Ising model and the Green’s function method. Pure ZrO
_{2}
shows at high electric fields an antiferroelectric behavior. Pure HfO
_{2}
is a linear dielectric in the monoclinic phase. With increasing ZrO
_{2}
content the
of HZO shows a ferroelectric behavior. The composition dependence x of the remanent polarization P<sub>r</sub>(x) has a maximum for x = 0.5. For x = 0, pure HfO_{2}, and x = 1, pure ZrO_{2}, P<sub>r</sub>=0. P increases with decreasing HZO nanoparticle size. The influence of Al and La doping on P<sub>r</sub> in HfO_{2} nanoparticles is also studied. The exhibiting of the ferroelectricity in ion doped HfO_{2} is due to a phase transformation and to an internal strain effect. The observed results are in good qualitative agreement with the experimental data.

ZrO_{2} is a wide-band insulating material with a high dielectric constant. With increasing temperature in ZrO_{2} exist monoclinic, tetragonal, orthorhombic and cubic phases. Antiferroelectric (AFE)-like double-hysteresis loops are observed in ZrO_{2} thin films [_{2} thin film [

Furthermore, ferroelectricity was found in HfO_{2} thin films doped with Zr (HZO) [_{2} and ZrO_{2} are not ferroelectric. HfO_{2} exists with increasing temperature in monoclinic, tetragonal and cubic phases [_{2}1/c-dielectric), orthorhombic (o-phase, Pca2_{1}-ferroelectric) and tetragonal (t-phase, P4_{2}/nmc-AFE) phases depending on the Hf:Zr ratio [

Below a critical size of 30 nm pure ZrO_{2} is stabilized in the tetragonal phase at room temperature which is considered as a crystallite size effect [_{2} is stabilized for d < 3.6 - 3.8 nm [

The phase stability and the ferroelectricity of orthorhombic HZO ferroelectric material are theoretically investigated by Chen et al. [_{2}, ZrO_{2} and HZO. Batra et al. [

The physical origin of the AFE hysteresis in ZrO_{2} NPs and the ferroelectricity in HZO and Al, La doped HfO_{2} NPs is still under debate. The aim of the present paper is to investigate theoretically these problems using a microscopic model and the Green’s function technique.

The properties of Zr doped HfO_{2}, Hf_{1−xZrxO2,} NPs can be described by the transverse Ising model [

H = − ∑ i α Ω α x i α S i α x − 1 2 ∑ i j α β J i j α β x i α x j β S i α z S j β z − μ E ∑ i S i α z . (1)

The pseudo-spin operator S i z characterizes the two positions of the ferroelectric unit at the lattice point i. J i j is the pseudo-spin interaction between the pseudo-spins at sites i and j which is positive or negative in the ferroelectric or AFE case, respectively. The dynamics of the model with strength Ω is determined by the operator S x . E is an external electric field. Here α , β mean Zr (or Al, La) or Hf. x i Z r = 1 , x i H f = 0 for pure ZrO_{2}, and x i Z r = 0 , x i H f = 1 for pure HfO_{2}. Thus, x i Z r + x i H f = 1 . Ω α has two values— Ω Z r and Ω H f . The interaction term J i j α β has three different values— J i j Z r − Z r , J i j H f − H f and J i j Z r − H f .

The Hamiltonian (1) can be written in explicit form as ( x ≡ x Z r ):

H = − Ω Z r ∑ i S i x Z r x i − Ω H f ∑ i S i x H f ( 1 − x i ) − 1 2 ∑ i j J i j Z r − Z r S i z Z r S j z Z r x i x j − 1 2 ∑ i j J i j H f − H f S i z H f S j z H f ( 1 − x i ) ( 1 − x j ) − ∑ i j J i j Z r − H f S i z Z r S j z H f x i ( 1 − x j ) . (2)

We assume that

〈 S i z Z r x i 〉 ≈ 〈 S i z Z r 〉 x ; 〈 S i z H f ( 1 − x i ) 〉 ≈ 〈 S i z H f 〉 ( 1 − x ) , (3)

where 〈 x i 〉 = x . The factor x gives the concentration of the Zr ions which substitute the Hf ions, whereas ( 1 − x ) is the concentration of the Hf ions.

The retarded Green’s function is defined as:

G i j ( t ) = − i θ ( t ) 〈 [ B i ( t ) , B j + ] 〉 . (4)

The operator B i stands for the set S i + Z r , S i − Z r , S i + H f , S i − H f , where S − , S + are Pauli operators (S = 1/2, S z = S − S − S + ).

The polarization P of a HZO NP is obtained as:

P = 1 2 N ∑ n tanh ( ϵ n / k B T ) . (5)

The mixed transverse pseudo-spin-wave excitations ϵ i j in a given shell n are calculated from the poles of the Green’s function (4) using the method proposed by Tserkovnikov [

ϵ i j = 1 2 ( ϵ i j 11 + ϵ i j 22 ) ± 1 4 ( ϵ i j 11 − ϵ i j 22 ) 2 + ϵ i j 12 ϵ i j 21 , (6)

ϵ i j 11 = 2 x Ω Z r 〈 S i − Z r 〉 δ i j / 〈 S i z Z r 〉 δ i j − x ( 1 − x ) J i j Z r − H f 〈 S i z H f 〉 − 1 2 N ′ ∑ m ( 1 − x ) J i m H f − H f [ 2 〈 S i z H f 〉 − 4 〈 S m z H f S i z H f 〉 δ i j + 2 〈 S m − H f S i + H f 〉 ] / 2 〈 S i z H f 〉 δ i j ,

ϵ i j 22 = 2 ( 1 − x ) Ω H f 〈 S i − H f 〉 δ i j / 〈 S i z H f 〉 δ i j − x ( 1 − x ) J i j Z r − H f 〈 S i z Z r 〉 − 1 2 N ′ ∑ m x J i m Z r − Z r [ 2 〈 S i z Z r 〉 − 4 〈 S m z Z r S i z Z r 〉 δ i j + 2 〈 S m − Z r S i + Z r 〉 ] / 2 〈 S i z Z r 〉 δ i j ,

ϵ i j 12 = 2 x Ω Z r 〈 S i + Z r 〉 δ i j / 〈 S i z Z r 〉 δ i j − 1 2 N ′ ∑ m x J i m Z r − Z r 〈 S m + Z r S i + Z r 〉 / 〈 S i z Z r 〉 δ i j ,

ϵ i j 21 = 2 ( 1 − x ) Ω H f 〈 S i + H f 〉 δ i j / 〈 S i z H f 〉 δ i j − 1 2 N ′ ∑ m ( 1 − x ) J i m H f − H f 〈 S m + H f S i + H f 〉 / 〈 S i z H f 〉 δ i j ,

〈 S i z Z r 〉 = 1 2 N ′ ∑ j ϵ 11 ϵ i j tanh ϵ i j 2 k B T ,

〈 S i z H f 〉 = 1 4 N ′ ∑ j ϵ 22 ϵ i j tanh ϵ i j 2 k B T ,

〈 S i − Z r 〉 = 〈 S i + Z r 〉 = 1 4 ε 12 ε i j tanh ϵ i j 2 k B T ,

〈 S i − H f 〉 = 〈 S i + H f 〉 = 1 4 ϵ 21 ϵ i j tanh ϵ i j 2 k B T ,

where N ′ is the number of lattice sites.

Our NP has an icosahedral symmetry. A certain Hf-spin is fixed in the center of the particle and all other spins are included into shells n. n = 1 denotes the central spin and n = N represents the surface shell. Strain effects on the surface of the NP change the number of next neighbors on the surface and reduce the symmetry. Therefore the pseudo-spin interaction constants can take different values on the surface and in the bulk, denoted with the index “s” and “b”, respectively. Moreover, J is proportional to the inverse of the distance between two nearest spins, i.e. of the lattice parameters.

In order to clarify the AFE behavior in ZrO_{2} we will firstly consider the electric field dependence of the polarization in the tetragonal phase of a ZrO_{2} NP with N = 3 shells for T = 300 K. Materlik et al. [_{2} thin films is observed after stabilization of the tetragonal phase for d < 35 nm. Using the lattice parameters for ZrO_{2} from Ref. [_{3} (PZO)-like AFE one, the electric dipoles are aligned antiparallel to their nearest neighbors—analogous to the magnetic moments in antiferromagnetic materials, therefore, we chose J < 0 . The results are presented in _{2} NPs is a phase transformation from a tetragonal to an orthorhombic phase induced by an external electric field which is an intrinsic behavior. This is confirmed by the ab-initio study of Reyes-Lillo et al. [

Now we will study the electric behaviour for different electric field, temperature, crystal phase and size of Hf_{1-x}Zr_{x}O_{2} NPs. By doping of ions with different radius appear different strains which give rise to additive changes (increasing or decreasing) of the pseudo-spin interaction constant J i j = J ( r i − r j ) in the defect sizes (denoted as J d ) compared to the undoped samples. The radius of the tetravalent Zr ion (86 pm) is a little larger than that of the Hf ion (85 pm), i.e. there is a small tensile strain ( J d < J b ), in agreement with the experimental data

of Shiraishi et al. [

The electric field dependence of the polarization in Hf_{0.5}Zr_{0.5}O_{2} NPs is shown in _{2} and HfO_{2} have almost equivalent crystal phases, with almost identical lattice parameters. It is seen that pure HfO_{2} (_{2} content increases, the P ( E ) curve reaches its maximum value for doping concentration x = 0.5 (_{2} displays an AFE-behavior at high fields, where the polarization response becomes non-linear with hysteresis (_{2} NP (x = 1), the crystal phase is tetragonal. The polar phase cannot be induced when the temperature T is around T N even under an external electric field. For temperatures higher than the AFE transition temperature T N in the cubic phase remain only paraelectric properties. The monoclinic phase decreases with increasing the ZrO_{2} content. It can be seen from _{2} and ZrO_{2}. P r reaches at doping concentration x = 0.5 its maximum value.

In _{2}, P r = 0 . With increasing of x P r increases, reaches at x = 0.5 its maximum value and then in pure ZrO_{2}, x = 1, P r is again zero. The experimentally reported maximum value of the remanent polarization P r is in the interval x = 0.5 - 0.6 [_{2} thin films remain in this double-loop hysteresis starting from low temperatures.

To completely explain the ferroelectric-phase stability in HZO NPs, we want to focus now on the size dependence of the polarization P in HZO NPs which is demonstrated in

Finally, we will consider the effect of different ion doping on the electric properties of HfO_{2} NPs. Variations of Al and La doping concentration influences the crystallographic structure of the NP and therefore the polarization. The insertion of a 3+ (Al) or 4+ (La) cation in the HfO_{2} lattice leads to the appearance of oxygen vacancies to keep the charge balance. The radius of the Al ion (67.5 pm) is smaller compared to the ionic radius of the Hf ion (85 pm) (i.e. in our model we have J d > J b ). _{2} NP as a function of the Al-concentration (_{2} NP behaves as a paraelectric material. Mueller et al. [

A similar behavior for the Al concentration dependence of the dielectric constant in HfO_{2} thin films is reported by Yoo et al. [

The electric properties of La doped HfO_{2} NPs are also studied. The radius of the La ion (117.2 pm) is larger compared to the ionic radius of Hf (85 pm) (this means J d < J b ). Batra et al. [_{2} NP which starts at higher x value, x ≈ 0.05, is shifted to higher doping concentrations and is broader due to the larger ionic radius of the La ion. In addition, the remanent polarization P r is larger for the La doping than that for the Al doping (_{2} thin films. Our results confirm the experimental data of Ref. [_{2} thin films. It must be noted that the observed here maximum values of the ion doped HfO_{2} NPs are comparable to the values reported for Al-doped (x = 0.025 - 0.03 [_{2} epitaxial thin films.

The properties of HZO are theoretically investigated till now with DFT computations. In this paper for the first time is used the microscopic transverse Ising model in order to clarify the physical origin of the AFE hysteresis in ZrO_{2} NPs and the ferroelectricity in HZO and Al, La doped HfO_{2} NPs which is still under debate. Therefore, we have investigated the dependence of the polarization P in ion doped HfO_{2} NPs on electric field, dopant concentration x, size and temperature. Different from the DFT we study the behavior of the material at finite temperatures. To that aim we use a Green’s function technique for T ≠ 0 . It can be concluded that the change in the polarization P r with respect to the doping concentration in HfO_{2} NPs is the result of the transformation of the crystalline phase due to the internal stress, of the appearance of an orthorhombic phase exhibiting ferroelectricity. Moreover, we try to clarify some discrepancies in the literature, for example about the appearing strain in HZO NPs (it is tensile and not compressible).

We obtain that pure ZrO_{2} displays in the tetragonal phase an AFE-behavior ( J < 0 ) at high fields inducing a t-o phase transformation. Pure HfO_{2} is a linear dielectric in the monoclinic phase. With increasing the ZrO_{2} content in HZO the hysteresis loop is consistent with that for ferroelectric materials ( J > 0 ). P r ( x ) shows a maximum for x = 0.5. For x = 0 and x = 1 P_{r} = 0. It is shown that the properties of these three compounds—ZrO_{2}, HfO_{2} and HZO—are changed with ion doping and size. The polarization P increases with decreasing NP size, i.e. the non-ferroelectric m-phase disappears with decreasing size. We show that strain can be used in very small NPs of HZO to induce a ferroelectric phase with large P and P r .

The influence of Al and La doping on P r ( x ) in HfO_{2} NPs is also studied. Stress due to the different ionic radii of the doping ions compared to the host ones (which cause different pseudo-spin interaction constants in the defect states) as well as the distribution of oxygen vacancies play a key role for the phase transformations in doped HfO_{2} nanostructures. Both remanent polarizations have a maximum value at x ≈ 0.03 and 0.14, respectively. The P r curve for La doping is shifted to higher doping concentrations and is broader due to the larger radius of the La ion. Moreover, P r is larger for La-doped compared with that of Al-doped HfO_{2} NPs.

There are some differences in the electric properties of ion doped HfO_{2} and ZrO_{2} nanostructures [_{2} thin films undergoes a maximum whereas in Al doped ZrO_{2} thin films it decreases. The electric properties of ion doped HZO and ZrO_{2} NPs will be considered in the next paper.

One of us (A. A.) acknowledges financial support by the Bulgarian National Fund “Scientific Studies” (contract number KP-06-OPR 03/9).

The authors declare no conflicts of interest regarding the publication of this paper.

Apostolov, A.T., Apostolova, I.N. and Wesselinowa, J.M. (2020) Antiferroelectricity in ZrO_{2} and Ferroelectricity in Zr, Al, La Doped HfO_{2} Nanoparticles. Advances in Materials Physics and Chemistry, 10, 27-38. https://doi.org/10.4236/ampc.2019.102003