Thermophoresis/diffusion as a plausible mechanism for unipolar resistive switching in metal–oxide–metal memristors

Abstract

We show that the SET operation of a unipolar memristor could be explained by thermophoresis, or the Soret effect, which is the diffusion of atoms, ions or vacancies in a steep temperature gradient. This mechanism explains the observed resistance switching via conducting channel formation and dissolution reported for TiO₂ and other metal-oxide-based unipolar resistance switches. Depending on the temperature profile in a device, dilute vacancies can preferentially diffuse radially inward toward higher temperatures caused by the Joule heating of an electronic current to essentially condense and form a conducting channel. The RESET operation occurs via radial diffusion of vacancies away from the channel when the temperature is elevated but the gradient is small.

Appendices

Appendix A1: Derivation of Eqs. (1)–(3)

In the spirit of the microscopic hopping model [12, 26, 40, 49], we consider a one-dimensional billiard ball model with thermal, concentration and potential gradients and derive Eqs. (1)–(3) here to place them all on an equal footing. The ion fluxes in the positive (J ⁺) and negative (J ⁻) directions crossing an imaginary plane in the potential profile at position x (Fig. 6) are given by the following:

(7a)

(7b)

(7c)

(8a)

(8b)

(9a)

(9b)

Here J is the total current, while N _R, N _L, T _R, T _L are the concentrations and temperatures on the right and on the left of the imaginary plane, respectively. Other notation is similar to that introduced in Sect. 2 of the paper. In this derivation, we use the ad hoc formula for the excluded volume effect (last term in Eqs. (7a) and (7b)), similar to Ref. [26].

Fig. 6

Notation used for deriving Eqs. (1)–(3) for simplified billiard ball model

Next, we simplify Eq. (7c) by deriving the first-order approximation (i.e. neglecting higher order terms) of Eqs. (7a)–(7c) and show that the current in this case can be decomposed into three independent terms—Soret, Fick and drift components (Eqs. (1)–(3) of the paper). In particular, combining Eqs. (7a, 7b, 7c)–(9a, 9b) together we obtain the following:

(10)

where the approximation is due to the omission of the high-order terms and

(11a)

(11b)

To further simplify Eq. (10), we define the (exp[+]±exp[−]) terms, i.e.

(12)

where for convenience we denote

(13a)

(13b)

(13c)

Noting that β≪1 is always true for any practical range of thermal gradients and temperatures, we use Taylor’s expansion about β=0 so that

(14a)

(14b)

Therefore, substituting Eqs. (14a) ands (14b) into Eq. (12), we obtain

(15a)

(15b)

where the approximation sign is due to the omission of the higher order terms in β ². Substituting Eqs. (15a) and (15b) into Eq. (4), we obtain

(16)

where

$$ D=\frac{1}{2}a^2f\exp \bigl[-U^{\prime}\bigr]$$

(17)

is a typical equation for thermally activated diffusion.

Equation (16) is the general form of the ionic current in the presence of thermal, concentration and potential gradients. For small electric fields, W≪1 so that sinhW≈W and coshW≈1, and for sufficiently low concentrations N≪N _MAX. In this case, Eq. (16) simplifies to

(18)

which corresponds to classical drift, Fick diffusion and thermophoresis, i.e. Eqs. (1)–() where

(19a)

(19b)

are ion mobility and Soret coefficient, respectively. Note that the approximation in Eq. (18) is due to the fact that W ² β and (1+U′)Wβ are higher order terms and much smaller than the other terms. For example, W ² β≪U′β because of the small electric field assumption, so this term is much smaller than the thermophoresis current. Likewise, it is easy to show that $(1+U')W \beta\approx\frac {U}{kT}\frac{Eqa}{2kT}\frac{a}{2T}\frac{\mathrm{d}T}{\mathrm{d}x}\ll1$.

Appendix A2: On the assumption of the neutrality of mobile defects

In our analysis for Fig. 2a, we assumed that mobile defects, i.e. oxygen vacancies, might be neutral so that we could neglect the drift of the charged defects in the electric field. The validity of this assumption largely depends on the position of energy levels of mobile defects inside the band gap. In titanium dioxide, some density functional theory calculations predicted very shallow (or even in the conduction band) electronic states induced by oxygen vacancies [50], while early experimental work [51] and other theoretical investigations predicted rather deep states, i.e. with up to 1.1 eV for neutral vacancies, and high sensitivity on the morphology, nonstoichiometry and the presence of other dopants [52–55]. Similarly, relatively deep electronic states of ∼0.3 eV associated with oxygen vacancy defects are predicted for bulk NiO [56]. In the case of such deep energy levels, the neutrality assumption should be valid even for high temperatures. For example, Fig. 7 shows the fraction of ionized dopants as a function of the energy difference between the defects and the bottom of the conduction band for several values of temperature. Here we used simple Poisson–Boltzmann statistics to estimate the probability of the electron detachment to the conduction band, i.e.

$$ \mbox{Fraction of ionized dopants} \approx \exp \bigl[-(E_{\mathrm{C}} - E_{\mathrm{D}})/ k_{\mathrm{B}}T\bigr],$$

(20)

which is a good approximation for n-type materials with a large band gap.

Fig. 7

Approximate fraction of ionized donors as a function of their energy levels for several temperatures

Even if the neutrality assumption does not hold, the thermophoresis effect could still play the dominant role in defect transport given that the radial component of the electric field is much smaller than the axial component [57]. The switching dynamics, however, in that case will be most likely defined by the axial drift [31], similar to bipolar memristive devices—see Fig. 2b and its discussion.