Controlling domain wall motion in ferroelectric thin films · where 𝜖0=8.85×10−12 F/m is...

Controlling domain wall motion in ferroelectric thin films

L. J. McGilly, P. Yudin, L. Feigl, A. K. Tagantsev and N. Setter

Contents

A. Electrode Characterisation

B. Stefan model for DW motion along a linear electrode

B.1 Governing equations from charge conservation and energy balance

B.1.1 Equations in Domains

B.1.2 Conditions on the fixed boundaries

B.1.3 Conditions on the moving boundary

B.1.4 Full set of equations

B.2 Similarity to the Stefan problem for heat

B.3 Solution for stepwise boundary conditions

B.4 Solution for square-pulse boundary conditions

C. Experimental data for DW motion and Stefan problem fits

D. Experimental error and uncertainties

E. References


In order to assess the quality of the Pt electrodes deposited by electron-beam induced deposition (EBID)

a series of switching experiments were carried out. A total of 10,000 hysteresis loops were measured

via piezoresponse force microscopy (PFM) and the positive and negative coercive biases extracted as

can be seen in Supplementary Fig. 1a. From this data it can be seen that the coercive biases (both

positive and negative) change monotonically from initial values of ~ +3.5 V, -3 V to ~ ±4.5 V. This

means that over a range relevant for experiments i.e. <1000 switching cycles the coercive bias

(Supplementary Fig. 1b) and by extension the domain wall (DW) behaviour changes very little. This

validates the experiments where many switching cycles are used to assess DW motion e.g. Fig. 3 in the

main text.



L. J. McGilly, P. Yudin, L. Feigl, A. K. Tagantsev and N. Setter

Contents













E. References


In order to assess the quality of the Pt electrodes deposited by electron-beam induced deposition (EBID)

a series of switching experiments were carried out. A total of 10,000 hysteresis loops were measured

via piezoresponse force microscopy (PFM) and the positive and negative coercive biases extracted as

can be seen in Supplementary Fig. 1a. From this data it can be seen that the coercive biases (both

positive and negative) change monotonically from initial values of ~ +3.5 V, -3 V to ~ ±4.5 V. This

means that over a range relevant for experiments i.e. <1000 switching cycles the coercive bias

(Supplementary Fig. 1b) and by extension the domain wall (DW) behaviour changes very little. This

validates the experiments where many switching cycles are used to assess DW motion e.g. Fig. 3 in the

main text.

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The repetitive nature of measurement of DW positions requires many scans to produce the images.

Supplementary Fig. 1c shows the electrode thickness as a function of the number of scans and shows

that the thickness changes little over several hundred scans. It is clear that the EBID Pt top electrodes

are suitably durable and resilient structures to allow for investigation of DW motion.

Supplementary Figure 1 Characterisation of EBID Pt top electrode. (a) The evolution of positive

and negative coercive biases up to 104 switching cycles of which the first 1000 cycles can be seen

clearly in (b). (c) The electrode thickness as a function of the number of PFM scans. (d) The effect of

PFM tip loading force on the coercive bias

The displacement of the DW is limited by the ability of the system to deliver charge to the area of DW

propagation. It is clear that the role of the top electrodes’ conductivity is crucial. Another conductive

bottleneck here may be the contact between the conductive probe tip (Ti/Ir-coated Si) to the EBID Pt.

Supplementary Fig. 1d shows that this interface is neither a limit or critical to the system. As the probe

loading force is increased, the contact area changes correspondingly, achieving a large contact radius

and overall ‘better’ contact however the coercive bias is not altered by this change in the contact area

a b

d c


leading to the assumption that the interface is not a limiting factor. Hereafter we assume perfect

electrical contact of the PFM tip with the electrode. This determines the Dirichlet boundary condition

for voltage in the charge transport problem. As we will demonstrate below this charge transport problem

is mathematically equivalent to propagation of heat in the classical Stefan problem [1].

In the classic case the Stefan problem is used to describe the motion of a pre-existing (i.e. meaning

nucleation is not considered) phase front that separates a liquid from a solid phase. In particular we will

draw an analogy from a heat problem of a rod initially resting at a constant temperature, 𝑇𝑖𝑛𝑖𝑡 below the

melting temperature, 𝑇𝑚 when one end is instantaneously subjected to a temperature 𝑇𝐿 > 𝑇𝑚. The

thermal conductivity of the material (in the liquid phase) determines the amount of heat that reaches the

phase front, thus allowing it to propagate further due to the subsequent further melting. It is clear that

far from the melting front the temperature in the solid phase will be a minimum i.e. the initial value

before introduction of heat. As such there will be a large temperature distribution along the rod, from a

maximum at the heating end to a minimum at the opposite end (for the case of a finite rod).

The Stefan model is sufficient to describe the motion of DWs in ferroelectric thin films with EBID Pt

top electrodes. Supplementary Fig. 2 shows a comparison between the classic form of the model and

the DW motion analogue. Each quantity in the classic form of the model can be found to have an

analogue as outlined in Tables S1 & S2. As can be seen from Table S2 there are several quantities that

describe the temperature and their analogues for voltage within the Stefan model. This clearly

demonstrates that there is a voltage distribution where the maximum value must be 𝑉𝑡 and the minimum,

far away from the tip position is 𝑉0 = 0.


Here we develop a mathematical description for the DW motion in a thin ferroelectric film along a

linear electrode driven by a voltage pulse applied to one end of the electrode. This problem allows for


a one-dimensional (1D) treatment, where the voltage distribution 𝑢(𝑥, 𝑡) and DW position 𝑋(𝑡) are

found as functions of the applied voltage 𝑉𝑡(𝑡), coordinate 𝑥 (distance along the electrode measured

from the point of contact with the PFM tip) and time 𝑡. Essentially, we address the propagation of charge

from the PFM tip into and along the electrode (see Supplementary Fig. 2a). As we will demonstrate, a

direct similarity can be drawn to the classical Stefan problem describing propagation of heat into a

material, where the position of a melting front is found as a function of the surface temperature 𝑇𝐿

(Supplementary Fig. 2b).

Supplementary Figure 2 Schematics of DW motion along a linear electrode. (a) A cross-sectional

view of the experimental setup showing partial switching of the capacitor structure. Polarisation is given

by red arrows, free charges are pluses and minuses whilst bound charges are circled. (b) The analogue

to the DW motion example in (a) for heat conduction in a rod which is described by the classical Stefan

problem. The meanings of the symbols in bold are given in Tables S1 & S2.


In this section we develop equations for the voltage distribution from basic laws: charge conservation,

Ohm’s law and energy balance. First we will develop differential equation for the voltage inside

(a)

(b)

Domain wall Domain1Domain2

PFM tip

Pt

PbZr0.1Ti0.9O3

SrRuO3

SrTiO3

Phase front Phase1Phase2


domains then we will discuss boundary conditions including conditions on the DW (phase front)

between domains (phases) 1 & 2.


Inside the domains the charge conservation law yields

−𝐽′ = 𝑐�̇�

(S1)

where 𝐽 is the electric current in the electrode, 𝑐 is the system’s capacitance per unit length. Hereafter

(∎′) and (∎̇) denote derivation with respect to coordinate (𝑥) and time (𝑡) respectively. The capacitance

per unit length may be estimated as

𝑐 =𝜖𝜖0𝑤

ℎ

(S2)

where 𝜖0 = 8.85 × 10−12 F/m is electric permittivity of vacuum, 𝜖 is relative permittivity of the

ferroelectric material, ℎ is the thickness of the ferroelectric layer and 𝑤 is the width of the electrode.

Equation (S2) is applicable for wide electrodes, where 𝑤 ≫ ℎ. Using ℎ = 7 × 10−8 m, 𝑤 =

3 × 10−7 m, 𝜖 = 80 one obtains 𝑐 = 3 × 10−9 F/m.

Additionally Ohm’s law states

𝑢′ = −𝑟𝐽

(S3)

𝑟 = 𝜌/𝑤𝑑

(S4)

where 𝑟 is electric resistance of the electrode per unit length, 𝑑 is the thickness of the electrode, 𝜌 is the

resistivity of the electrode. In our system the resistivity of the electrode is strongly dependent on the

thickness of the electrode. We expect for electrodes with thickness 𝑑 around 10 nm to have resistivity

𝜌 in the range from 1 to 10 Ωm, thus (for 𝑤 = 3 × 10−7m) leading to 𝑟 ~1014 − 1016 Ω/m.


Eliminating electric current between (S1) and (S3) one obtains

�̇� =𝑢′′

𝑟𝑐

(S5)


We assume that the probe tip has a perfect electrical contact with the electrode, thus the voltage under

the tip (at 𝑥 = 0) is equal to the tip voltage 𝑉𝑡. The other edge of the electrode is assumed to be far from

the zone of interest and voltage on it equal to zero. This implies boundary conditions

𝑢(0, 𝑡) = 𝑉𝑡(𝑡)

(S6)

𝑢(∞, 𝑡) = 0

(S7)


From the requirement that electric current must be finite, one obtains that voltage is a continuous

function, implying

𝑢|𝑋− = 𝑢|𝑋+ ≡ 𝑉

(S8)

Here |𝑋∓ denotes the limit at the DW approaching from the left (domain 2) and right (domain 1)

respectively. At the same time the current undergoes a jump at the domain wall. This jump is defined

from the charge balance at the DW. Consider infinitesimal displacement of the DW, 𝛿𝑋 which is

essentially switching of the area 𝑤𝛿𝑋 under the electrode. The bound charge in this area will increase

by 𝛿𝑄𝑏 = −2𝑃𝑠𝑤𝛿𝑋, where 𝑃𝑠 is spontaneous polarization. Assuming perfect screening one obtains

𝛿𝑄𝑏 = −𝛿𝑄𝑓 = (𝐽2 − 𝐽1)𝛿𝑡. Here 𝐽2 and 𝐽1 are electric currents on the left and right hand sides (LHS

& RHS respectively) of the DW respectively.


Thus from charge conservation one obtains:

𝐽2 − 𝐽1 = 2𝑃𝑠𝑤𝑋 ̇

(S9)

𝐽2 =𝑢′

𝑟|𝑋−

(S10)

𝐽1 =𝑢′

𝑟|𝑋+

(S11)

The voltage 𝑉 above the DW may be found from the energy balance for the DW having infinitesimal

displacement 𝛿𝑋. The work done by the current source will be 𝐴𝑐 = 2𝑃𝑆𝑉𝑠𝑤𝛿𝑋. In the frame of our

model, one part of this work will be spent on elongation of the DW, the other part is dissipated by

different friction mechanisms mainly related to the Peierls potential, pinning by defects and switching

dynamics. The work done against DW surface tension may be written as 2𝜏ℎ𝛿𝑋, while the work against

dissipative forces may be combined into a term 𝑃𝑓ℎ𝑤𝛿𝑋, where 𝜏 is the domain wall energy per unit

area, and 𝑃𝑓 is a pressure on the moving domain wall exerted from different friction mechanisms. In the

general case 𝑃𝑓 is a function of the DW speed �̇�. For sufficiently slow motion a “dry friction” model is

applicable where 𝑃𝑓 = 𝑃𝑓𝑚𝑎𝑥 ∙

�̇�

|𝑋|̇ on a moving DW, while 𝑃𝑓 compensates other forces on the DW at

rest. The energy balance (after eliminating the displacement 𝛿𝑋) yields:

2𝑃𝑆𝑉𝑤 = 2𝜏ℎ + 𝑃𝑓ℎ𝑤

(S12)

From equation (S12) one derives the equilibrium voltage above the DW

𝑉 = {

𝑉𝑠−, 𝑋 ̇ < 0

𝑉 ∈ (𝑉𝑠−, 𝑉𝑠

+), 𝑋 ̇ = 0

𝑉𝑠+, 𝑋 ̇ > 0

(S13)

𝑉𝑠± =

𝜏ℎ

𝑤𝑃𝑆±𝑃𝑓𝑚𝑎𝑥ℎ

2𝑃𝑆

(S14)


Here we introduce switching voltages 𝑉𝑠±, where “+” and “-” stand for growing and shrinking domain

2. Note that under the assumption of validity of the dry friction model the switching voltage does not

depend on the DW velocity. As a criterion of applicability of the dry friction model we suggest

𝑡𝑐𝑖𝑟 = 𝑟𝑐𝑋2 ≫ 𝑡𝑝𝑜𝑙

(S15)

where 𝑡𝑐𝑖𝑟 & 𝑡𝑝𝑜𝑙 are the characteristic times of the electrical circuit and the load (ferroelectric capacitor

structure) and where 𝑡𝑝𝑜𝑙 ~10−11 − 10−12 s is the polarization switching time limited by polarization

dynamics (estimated as inverse soft-mode frequency). Using 𝑟 = 1014 Ω/m and 𝑐 = 3 × 10−9 F/m one

checks that the dry friction model is safely applicable for our system for 𝑋 >10 nm.


In full we obtain the following equations for the pair of functions, 𝑢(𝑥, 𝑡) & 𝑋(𝑡):

�̇�(𝑥, 𝑡) = (𝑟𝑐)−1𝑢′′(𝑥, 𝑡), 0 < 𝑥 ≠ 𝑋

(S16)

𝑢(0, 𝑡) = 𝑉𝑡 (𝑡)

(S17)

𝑢(∞, 𝑡) = 0

(S18)

𝑢|𝑋− = 𝑢|𝑋+ = 𝑢(𝑋, 𝑡) = {

𝑉𝑠−, 𝑋 ̇ < 0

𝑉 ∈ (𝑉𝑠−, 𝑉𝑠

+), 𝑋 ̇ = 0

𝑉𝑠+, 𝑋 ̇ > 0

(S19)

𝑢′

𝑟|𝑋− −

𝑢′

𝑟|𝑋+ = 2𝑃𝑠𝑤𝑋 ̇

(S20)

From equations (S16)-(S20) we can describe the voltage evolution along the length of the electrode.



The classical Stefan problem [1] describes the evolution of temperature and the position of the melting

front in a semi-infinite material. We will draw an analogy between our problem and the melting of a

rod by heating from the LHS (Supplementary Fig. 2b).

The governing equations for the problem [1] are:

�̇�(𝑥, 𝑡) =𝐾

𝜌𝑠𝐶𝑇′′(𝑥, 𝑡), 0 < 𝑥 ≠ 𝑋

(S21)

𝑇(0, 𝑡) = 𝑇𝐿 (𝑡) > 𝑇𝑚

(S22)

𝑇(∞, 𝑡) = 𝑇𝑖𝑛𝑖𝑡 < 𝑇𝑚

(S23)

𝑇(𝑋, 𝑡) = 𝑇𝑚

(S24)

𝐾𝑇′|𝑋− − 𝐾𝑇′|𝑋+ = 𝜌𝑠𝐶𝐿𝑋 ̇

(S25)

The physical values that describe the heat problem are listed in Table S1, where we also introduce

values and complexes that we will need for conversion to our charge transport analogue; our problem

corresponds to the case where physical properties (specific heat, thermal conductivity and density) in

the liquid and solid phases are the same.


Temperature, melting temperature , [K]

Density 𝝆 [Kg/m3]

Cross-sectional area of the rod 𝑺 [m2]

Specific heat 𝑪 [J/Kg K]

Thermal conductivity 𝑲 [J/Kms]

Latent heat [J/kg]

Heat capacitance per length 𝑪∗ = 𝝆 𝑺𝑪 [J/mK]

Latent heat per length ∗ = 𝝆 𝑺 [J/m]

Supplementary Table 1: Some physical values from the classical Stefan problem and their

relevant complexes

Analysis of equations (S16)-(S20) & (S21)-(S25) reveals direct similarity of the problem with the

classical Stefan problem [1] for heat. Thus the existing solutions can be used with respect to

substitutions reflected in Table S2.

Classical Stefan problem DW motion analogue

Temperature. 𝐿, , [K] , , =

[V]

Voltage

Heat [J] [C] Charge

Heat Flux [J/s] [A] Current

Latent heat per length ∗ [J/m] = 𝟐𝑷 𝒘[C/m] Switching charge

Thermal conductivity 𝑲 [J/Kms] 𝟏 𝝆⁄ [C/Vms] Electrical conductivity

Heat capacitance per length 𝑪∗ [J/mK] 𝒄 [F/m] Capacitance per length

Stefan no.- solid phase 𝑺 =𝑪( − )

𝑺𝟏 =

𝒄 𝟐𝑷 𝒘

Stefan no.- domain1

Stefan no.- liquid phase 𝑺𝒍 =𝑪( − )

𝑺𝟐 =

𝒄( − )

𝟐𝑷 𝒘 Stefan no.- domain2

Supplementary Table 2: Correspondence between physical values describing the classical Stefan

problem (left column) and the present problem (right column)


However there is one essential difference between the models. This difference is reflected in equation

(S19) which is modified with respect to the corresponding equation (S24) for the classical Stefan

problem. This modification rises from the fact that in the classical Stefan problem the difference

between temperatures for freezing and melting is typically negligible1. In our case, since we are

applying voltages comparable with the coercive voltage, the difference needed for switching and back-

switching is crucial. However for the DW travelling in only one direction our problem is fully equivalent

to the classical heat-related Stefan problem.


Consider the problem with the initial condition

𝑢(𝑥, 0) = 0

(S26)

and boundary condition

𝑢(0, 𝑡) = 𝑉𝑡𝜃(𝑡)

(S27)

Here 𝜃(𝑡) is Heaviside theta-function, which is 1 if 𝑡 ≥ 0 and 0 if 𝑡 < 0. The solution to this problem

may be adapted from Ref. [1] with substitutions in accordance with Table S1 to obtain for the DW

position

𝑋(𝑡) = 2𝛾√𝑡

𝑟𝑐

(S28)

and for the voltage distribution in domain 2 (0 < 𝑥 < 𝑋)

𝑢(𝑥, 𝑡) = 𝑉𝑡 − (𝑉𝑡 − 𝑉𝑠) [

erf (𝑥

2√𝑡/𝑟𝑐)

erf (𝛾)]

(S29)

1 The difference in melting and freezing temperatures becomes essential for impure materials


and in domain 1 (𝑥 > 𝑋)

𝑢(𝑥, 𝑡) = 𝑉𝑠 [

erfc (𝑥

2√𝑡/𝑟𝑐)

erfc (𝛾)]

(S30)

where 𝛾 is the only positive root for the transcendental equation (obtained from requirement that

solutions (S29) and (S30) satisfy (S9)-(S11)

𝑆2𝑒𝑟𝑓 (𝛾)

−𝑆1

𝑒𝑟𝑓𝑐(𝛾)= √𝜋𝑒𝛾

2

(S31)

𝑆2 =𝑐(𝑉𝑡 − 𝑉𝑠)

2𝑃𝑠𝑤

(S32)

𝑆1 =𝑐𝑉𝑠2𝑃𝑠𝑤

(S33)

Here 𝑆1 and 𝑆2 are Stefan numbers for the first and second domains respectively, as introduced in Table

S2. It is instructive to rewrite these numbers in terms of thermodynamic coercive field

𝐸𝑐 ≈ 0.2𝑃𝑠𝜖𝜖0

(S34)

Here we used the approximate relationship for 2nd order ferroelectrics from Ref. [2]. Eliminating 𝑐 and

𝑃𝑠 in (S32) and (S33) using (S34) and (S2) one derives

𝑆2 ≈ 0.1(𝑉𝑡 − 𝑉𝑠)/ℎ

𝐸𝑐= 0.1

(𝑉𝑡 − 𝑉𝑠)

𝑉𝑐

(S35)


𝑆1 ≈ 0.1𝑉𝑠/ℎ

𝐸𝑐= 0.1

𝑉𝑠𝑉𝑐

(S36)

Here we introduced coercive voltage 𝑉𝑐 = 𝐸𝑐ℎ for the given film thickness. In view of the numerical

factor 0.1 in expressions (S35) and (S36), in case if the applied voltage is of the order of coercive

voltage, Stefan numbers may be considered as small parameters. For typical values of our experiment

using 𝑐 = 3 × 10−9 F/m, 𝑉𝑡 = 8 V, 𝑉𝑠 = 3 V, 𝑃𝑠 = 0.8 C/m2, 𝑤 = 3 × 10−7 m one evaluates 𝑆2 ≈ 0.03

and 𝑆1 ≈ 0.02.

The voltage distribution along the electrode, given by equations (S28)-(S30) is used to show the

distribution for different times in Fig. 2h in the main text.

In the case of small Stefan numbers (when charge needed for charging the capacitor is negligible

compared with the switching charge needed for the displacement of the DW) 𝛾 becomes small and by

decomposing equation (S31) into a Taylor series and keeping only the most important terms one obtains

𝛾 ≈S1 +√2𝜋S2 + S1

2

2√𝜋≈ √

S22

(S37)

Combining equation (S28) with equation (S37) one obtains expressions for the domain wall

displacement

𝑋 =√𝑡𝑐𝑟 (

√𝑐(𝑐V𝑠2 + 4𝜋Ps(Vt − Vs)𝑤) − 𝑐V𝑠)

2√𝜋Ps𝑤≈ √

(𝑉𝑡 − 𝑉𝑠)𝑡

𝑟𝑃𝑠𝑤

(S38)


This expression for the DW displacement as a function of time has been used to fit the experimental

data as seen in Fig. 2e and appears in the main text as equation 1. The DW velocity, 𝑣 can be determined

from differentiation of equation (S38) so that

𝑣 =1

2√(𝑉𝑡 − 𝑉𝑠)

𝑟𝑃𝑠𝑤𝑡

(S39)

and expressed in terms of DW position along the electrode by use of equation (S38) to give

𝑣 =(𝑉𝑡 − 𝑉𝑠)

2𝑟𝑃𝑠𝑤∙1

𝑋

(S40)

which is used as the fit to describe the experimental results of Fig. 2f in the main text.


The experimental measurement of the DW position corresponded to a square pulse of voltage. In our

simplest approximation where electric capacitance of the system is fully neglected, the domain wall is

expected to stop abruptly when the voltage is off. However in case of small but finite capacitance a

further motion “by inertia” (driven by charge stored in the capacitance of domain 2) will occur. The

displacement of the domain wall 𝛿𝑋 caused by this effect may be evaluated using

𝛿𝑋 =𝛿𝑄

2𝑃𝑆𝑤

(S41)

𝛿𝑄 = 𝑘(𝑉𝑡 − 𝑉𝑠)𝑐𝑋

2

(S42)


Here 𝛿𝑄 is the charge stored in the capacitance of the 2nd domain2, 𝑘 < 1 ≈ 1/3 is the fraction of the

charge that will flow to the DW but not to the tip, approximate relationship holds for the limiting case

{ 𝑉𝑠

𝑉𝑡→ 0, 𝑆2 → 0}. From (S41) and (S42) one obtains

𝛿𝑋

𝑋=𝑘

2

(𝑉𝑡 − 𝑉𝑠)𝑐

2𝑃𝑆𝑤≈1

6S2

(S43)

One concludes that the relative displacement of the wall after the voltage is off is of the order of Stefan

number. This validates use of (S38) to evaluate the displacement of the DW under the square voltage

pulse with duration 𝑡. For the parameters of our system the relative error will be of the order of 1% as

controlled by equation (S43).

2 Here we use only part of the charge from charging the capacitor above 𝑉𝑠 (subtracting the charge 𝑐𝑉𝑠𝑋) since

this part is “trapped” under the barrier of the DW and will flow into the tip.



In Supplementary Fig. 3 the full data set for both positive and negative switching is shown. The DW

displacement and calculated DW velocity are shown for several different tip voltages.

Supplementary Figure 3 DW motion and fits from the Stefan problem. In (a) the DW displacement

as a function of time and negative applied voltage is shown and (b) is the associated DW velocity as a

function of displacement along the line electrode. (c) DW displacement as a function of time for positive

applied voltages and the associated DW velocities (d). The inset in (d) is a magnified view. The solid

lines in all charts are the Stefan model fits to the experiment data.


In this section we will discuss the uncertainty in the measurements and calculations presented in the

main text.

a b

c d


The most basic measurement and one of the most vital is the position of the DW after a voltage pulse

i.e. 𝑋(𝑡). The AR Cypher atomic force microscope used in this study is an extremely accurate and stable

machine. Therefore we assume that the major source of uncertainty in our measurement of DW position

is the resolution of the image captured i.e. the pixel size. For the experiments shown in Fig. 2-4 of the

main text we know that the pixel size is 21.5 nm therefore we assign an error of ∆𝑋 = ±22 nm. In fact

this error in the DW position is much smaller than the size of the data points presented in Fig. 2e, 3c-e

4e-g.

In a similar approach the error in the voltage pulse duration will be equal to half the smallest increment

of the signal generator so that ∆𝑡 = ±0.25 ms; again smaller than the data point size. For Fig. 3d,e the

time increment was changed for precise DW placement and the associated error is ∆𝑡 = ±0.02 ms.

Therefore the error in the velocity is ∆𝑣 = |𝑋

𝑡| √(

∆𝑋

𝑋)2+ (

∆𝑡

𝑡)2. This equation has been used to

calculate the uncertainty in velocity and gives the error bars in Fig. 2f in the main text. The uncertainty

in the velocity will decrease as the fractional uncertainty (∆𝑡

𝑡) decreases as 𝑡 → 𝑡𝑚𝑎𝑥.

For Fig. 4f,g the values ‘d’ and ‘DW regression’ are simply 𝑑 = 𝑋1 − 𝑋2 so ∆𝑑 = √(∆𝑋1)2 + (∆𝑋2)

2

= 31 nm. This is smaller than the size of the data points.

For the average DW velocity data presented in Fig. 2g the error is propagated from the error in the DW

velocity such that ∆�̅� = 1

2(𝑣𝑚𝑎𝑥̅̅ ̅̅ ̅̅ ̅ − 𝑣𝑚𝑖𝑛̅̅ ̅̅ ̅̅ ) where 𝑣𝑚𝑎𝑥 = 𝑣 + ∆𝑣 and 𝑣𝑚𝑖𝑛 = 𝑣 − ∆𝑣. This has been

calculated for each voltage and gives the error bars visible in Fig. 2g for positive and negative voltages.


The voltage required to move an existing DW, 𝑉𝑠 is taken from the x-intercept of the line fits from the

Stefan model seen in Fig. 2g. The uncertainty, Δ𝑉𝑠, is obtained from fitting to the �̅� ± ∆�̅� data sets.

Calculation of the values of the resistance per unit length, 𝑟 and resistivity Ω both depend on the

propagation of the error in the value of 𝑉𝑠.

E. References

[1] Gupta, S. C. The classical Stefan problem: basic concepts, modelling and analysis. Elsevier,

Amsterdam (2003)

[2] Tagantsev, A. K., Cross, L. E. & Fousek, J. Domains in Ferroic Crystals and Thin Films. Springer

(2010)


Controlling domain wall motion in ferroelectric thin films · where 𝜖0=8.85×10−12 F/m is...

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