Preprint available at http://arxiv.org/abs/1012.4304. Journal reference will also be added there once known.

Joule heating in nanowires

Hans Fangohr,1, ∗ Dmitri S. Chernyshenko,1 Matteo Franchin,1 Thomas Fischbacher,1 and Guido Meier2

1School of Engineering Sciences, University of Southampton, SO17 1BJ, Southampton, United Kingdom2Institut fur Angewandte Physik und Zentrum fur Mikrostrukturforschung,

Universitat Hamburg, Jungiusstrasse 11, 20355 Hamburg, Germany

We study the increase of temperature of ferromagnetic nanostructures due to Joule heating fromelectric currents. The spatial current density distribution, the associated heat generation, anddiffusion of heat is simulated. A number of nanowire geometries on substrates is investigated.In particular, we study different constriction geometries including a symmetric constriction in ananowire with square corners, and an asymmetric notch where a triangle-shaped part of the wirehas been removed. Using different substrates such as silicon nitride membranes, silicon wafers, anddiamond wafers, we compute the spatially resolved increase in temperature as a function of time.

These multi-physics simulation results are compared with the prediction from the approximateanalytical description for the nanowire substrate as proposed by You et al.1. We provide an expres-sion to estimate the maximum pulse duration tc for which the You model is accurate. For a threedimensional substrate, the temperature in the nanowire stays approximately constant for interme-diate pulse durations (t > tc), whereas for thin membrane substrates the asymptotic temperatureincrease is proportional to the logarithm of the time, and the You model cannot be applied.

We provide an analytical expression that allows to compute the temperature increase for ananowire on a membrane substrate.

I. INTRODUCTION

Recently, there has been much interest both in funda-mental studies of spin torque transfer2–6 and in effortsto realize devices such as the race-track memory exploit-ing the spin torque transfer7,8. In either case, at thepresent very large current densities have to be used tomove domain walls and, more generally, to modify theferromagnetic patterns. Associated with these large cur-rent densities is a substantial amount of Joule heatingthat increases the temperature of the sample or device. Itis a crucial question to understand how strongly the tem-perature increases as this can affect the observed physicsconsiderably, e.g. Refs. 9 and 10, and may even lead toa temporary breakdown of ferromagnetism if the Curietemperature is exceeded. The depinning of a domain wallcould be due to a strong spin-current torque transfer, oras a result of the extreme heating of the material due toreduced magnetic pinning at elevated temperatures, ordue to the intermittent suppression of ferromagnetism.

You et al.1 have derived an analytical expression that

allows to compute the increase of temperature for ananowire (extending to plus and minus infinity in y-direction) with height h (in z-direction) and width w (inx-direction). The nanowire is attached to a semi-infinitesubstrate (which fills all space for z ≤ 0). The heating isdue to an explicit term S(x, t) which can vary across thewidth of the wire and as a function of time.

Meier et al.5 use energy considerations to estimate thetotal amount of energy deposited into the nanowire andsubstrate system to show that – for their particular pa-rameters – the heating and associated temperature in-crease stays clearly below the Curie temperature.

In this work, we use a numerical multi-physics simu-lation approach which allows to determine the temper-ature distribution T (r, t) for all times t and positionsr. Starting from a given geometry and an applied volt-age (or current), we compute the resulting current den-sity, the associated heat generation, and the temperaturedistribution. While such a numerical approach providesless insight than an analytical approximation, it allowsus to exactly determine the temperature distribution fornanowires of finite length, nanowires with constrictionsand thin substrates for which the assumption of an infi-nite thickness is inappropriate. Thus it evinces the limitsof applicability of analytical approximations.

Section II introduces the method underlying the work.Section III reports results from a number of case stud-ies. Starting from the heating of a nanowire of infinitelength without constrictions (Sec. IIIA), we introducea symmetric constriction in a finite-length wire where acuboidal part of material has been removed (Sec. III B)to demonstrate the additional heating that results froman increased current density in close proximity to theconstriction and in the constriction (as in Ref. 11). Sec-tion III C studies a nanowire with a notch-like constric-tion (triangular shape removed from the wire on oneside only) that is placed on a silicon nitride substrateof 100 nm thickness (as in Ref. 12). The same system isstudied in Sec. IIID where the silicon nitride substrate isreplaced with a silicon wafer with a thickness of the or-der of 500 µm. A zigzag wire on the same silicon wafer,as experimentally investigated in Ref. 4, is studied inSec. III E. We simulate a straight nanowire without con-strictions placed on a diamond substrate as in Ref. 13 inSec. III F, and investigate and discuss the applicability of

2

the temperature calculation model of Ref. 1 in Sec. IVA,deduce a model valid for quasi two-dimensional systemsin Sec. IVB, before we close with a summary in Sec. V.

II. METHOD

A current density j(r, t) can be related to the change oftemperature T (r, t) of a material as a function of positionr and time t using

∂T

∂t=

k

ρC∆T +

Q

ρC=

1

ρC(k∆T +Q) (1)

with k the thermal conductivity (W/(K· m)), ρ the den-sity (kg/m3), C the specific heat capacity (J/kg· K),

∆ = ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 the Laplace operator, T the

temperature (K), and Q a heating term (W/m3). TheJoule heating of a current density j in an electrical fieldE is given by Q = j · E = 1

σj2 where σ is the electrical

conductivity (S/m = 1/(Ω·m)) and we have used j = σE.Equation (1) becomes trivial to solve if we assume a

uniform current density in a slab of one material withconstant density, constant thermal conductivity, and con-stant specific heat capacity (see Sec. IIIA). In general,for samples with geometrical features or spatially inho-mogeneous material parameters, the problem becomesquite complex and can often only be solved using nu-merical methods. For the work presented here we haveused the simulation software suites ANSYS 12.014, Com-sol multi-physics,15 and the Nsim multiphysics simula-tion library16 that underpins the Nmag17 micromagneticsimulation package. All three tools were used for casestudy 2 (Sec. III B) and produce identical results for agiven desired accuracy within their error tolerance set-tings. The majority of the other case studies was simu-lated using ANSYS. We have taken material parameters(see Tab. I) appropriate for room temperature, and havetreated each material parameter as a constant for eachsimulation, i.e. here a temperature dependence of thematerial parameters is not taken into account.

III. SIMULATION RESULTS

A. Case study 1: Uniform current density

Initially, we study the extreme case of no cooling ofthe ferromagnetic conductor: neither through heat trans-fer to the surrounding air, nor to the substrate and thecontacts. This allows to estimate an upper limit of theheating rate and consequent change in temperature overtime.Assuming a uniform current density j, uniform conduc-

tivity σ and initially uniform temperature distribution T ,equation (1) simplifies to

dT

dt=

j2

ρCσ(2)

σ−1 C k ρ

[µΩcm] [J/(gK)] [W/(mK)] [g/cm3]

Py 25 0.43 45 8.7

Si3N4 n/a 0.5 18 3.21

Si n/a 0.7 148 2.33

Diamond n/a 0.75 1400 3.5

TABLE I. Material parameters resistivity σ−1, specific heatcapacity C, thermal conductivity k, and density ρ. Parame-ters for permalloy (Py) and silicon nitride (Si3N4) have beenchosen to match those of Ref. 12. For silicon, silicon nitride,and diamond no electric conduction was allowed in the sim-ulations (corresponding to an infinitely high resistivity). Fordetailed data on permalloy resistivity, see Refs. 18 and 19,permalloy thermal conductivity Ref. 20, and silicon thermalconductivity Ref. 21.

All the parameters on the right hand side are constant,and thus the temperature T will change at a constantrate of j2/(ρCσ). As only the ferromagnetic conductingnanowire can store the heat from the Joule heating, thetemperature has to increase proportionally to the heatingterm Q which is proportional to j2.Using material parameters for Permalloy (C =

430 J/(kgK), ρ = 8700 kg/m3, σ = 1/(25 · 10−8 Ωm) =4 · 106 (Ωm)−1, j = 1012 A/m2), we obtain a change oftemperature with time

dT

dt=

j2

ρCσ= 6.683 · 1010 K/s = 66.83K/ns. (3)

The immediate conclusion from this is that the temper-ature of the sample cannot increase by more than 66.83K per nanosecond if the current density of 1012 A/m2 isnot exceeded and if the current density is uniform withinthe whole sample for the chosen material parameters.A current pulse over 15 nanoseconds has the potential

to push the temperature up by just over 1000 degreesKelvin, and thus potentially beyond the Curie tempera-ture.The substrate on which the ferromagnetic conductor

has been grown will absorb a significant fraction of theheat generated in the conductor, and thus reduce theeffective temperature of the magnetic material. The con-tacts play a similar role. On the other hand, any con-strictions will result in a locally increased current density,which – through the j2 term in Q in Eq. (1) – results insignificantly increased local heating. We study the bal-ance of these additional heating and cooling terms in thefollowing sections in detail.

B. Case study 2: Constrictions

The effect of a constriction will vary strongly depend-ing on the given geometry. The resulting current and

3

FIG. 1. Sample geometry case study 2: slab with constriction(Sec. III B).

FIG. 2. Temperature profile ∆T (x) in the constricted geom-etry at positions [x,0,0] for t = 1ns and a current density of1012 A/m2 in the unconstricted ends of the slab.

temperature distributions as a function of temperatureand time are non-trivial.Fig. 1 shows the geometry used for this study (as in

Refs. 11 and 22): a bar with dimensions Lx = 1000 nm,Ly = 50nm, and Lz = 20nm. The origin is located inthe center of the slab, i.e. the two opposite corners of thegeometry are at [−500,−25,−10] nm and [500, 25, 10] nm.The constriction is placed at the center of the bar, andreduces the dimensions to Lconstrict

y = 20nm over a length

of Lconstrictx = 50nm. In comparison to the study shown

in Sec. IIIA, the current distribution is non-uniform inthis geometry and therefore the local Joule heating andthe resulting temperature field will be non-uniform. Wethus need the thermal conductivity k = 45 W/(K·m) forthese calculations because the ∆T term in equation (1)is non-zero.Fig. 2 shows a temperature profile ∆T (x) through the

constricted shape after application of a current density of1012A/m2 at ten different times t as a function of positionx. The top thick black line shows the temperature distri-bution after 1 ns, the other 9 lines show earlier momentsin time in successive time steps of 0.1 ns.The peak around x = 0 is due to the constriction: the

reduced cross section (in the y-z plane) results in an in-creased current density in the constriction. The Jouleheating term — which scales proportional to j2 — is in-creased accordingly in the constriction. During the 1 ns,diffusion of heat takes place and results in the increase of

temperature outside the constriction, and simultaneouslya reduction of the speed of increase of the temperaturein the constriction. The importance of heat diffusion canbe seen by the width of the peak around x = 0 in Fig. 2.The increase of temperature ∆T after t = 1ns at the

end of the nanowire (i.e. x = 500 nm and x = −500 nm)is 66.89K. This is about 0.05K higher than in the previ-ous case study in Sec. IIIA where a nanowire without aconstriction was studied and the corresponding value oftemperature increase is T = 66.83K. This difference of0.05K after 1 ns at the end of the wire originates fromthe extra heating in the constriction and diffusion of thisheat through the wire.We can also estimate the maximum possible tempera-

ture increase in the constriction by using equation (3)under the assumption that there was no diffusion ofheat (i.e. k = 0), and that we obtain a current den-sity of j = 2.5 · 1012A/m2 in the constriction (due tothe reduced cross section from LzLy = 1000 nm2 toLzL

constricty = 400 nm2). The resulting rate for tem-

perature increase in the constriction is here 417 K/ns,and thus much higher than the maximum temperatureof 102.8 K that is reached after 1ns when taking into ac-count the diffusion of heat from the constriction into theunconstricted parts of the nanowire. This comparisonunderlines the drastic influence on the actual tempera-tures in the nanowire when diffusion of heat is consid-ered.

C. Case study 3: Nanowire with a notch on asilicon nitride substrate

In this section we investigate a more realistic systemfollowing the work by Im et al.

12 for which we take intoaccount the heat dissipation through the substrate. Im et

al. study critical external fields for domain wall pinningfrom constrictions, and subsequent works23,24 study spin-torque driven domain wall motion for this geometry.Here, we choose the geometry where the domain wall

de-pinning field was most clearly defined: a permalloynanowire with dimensions Lx = 5000 nm, Ly = 150 nm,Lz = 30nm (see Fig. 1(b) in Ref. 12 and lower leftsubplot of Fig. 3 therein).This nanowire is lithographically defined centrally on

a silicon nitride membrane that is 100 nm thick (as inRef. 12). For the membrane we assume the shape ofa disk with 0.5mm radius. Preparation of nanostruc-tures on membranes is required for experiments usingsynchrotron light, that has to transmit through the sam-ple. Such experiments give access to simultaneous time-and space resolution on the nanometer and the sub-nanosecond scale2,12. Perfect thermal contact betweenthe wire and the silicon nitride substrate disk is assumed.The center of the wire contains a 45 nm wide triangu-lar notch on one side, and the geometry is sketched inFig. 3(b). A current density of 1012 A/m2 is applied atthe ends of the wire over a time of 20 ns, similar to ex-

4

(a) (b)

(c)

(d)

FIG. 3. Joule heating in a permalloy nanowire with a notch on a silicon nitride membrane as discussed in Sec. III C (geometryfrom Ref. 12): (a) temperature distribution T (r) in the nanowire and substrate after t = 20ns, (b) geometry of the model (notto scale) and the plotting path through the nanowire (dashed line) used in Fig. 3c, (c) temperature profile ∆T (x) after t = 20nsalong the length of the wire (following the plotting path shown in Fig. 3(b)) with a silicon nitride membrane substrate, (d)maximum (dotted line) and minimum (dashed line) temperature in the silicon nitride membrane and maximum temperature(solid line) in the wire as a function of current pulse length; the dash-dotted line is the prediction of the analytical You modeldiscussed in Sec IVA. The membrane is modelled as a disk with height 100 nm and radius 0.5mm. The current density isj = 1012 A/m2; for material parameters see Tab. I.

periments such as in Refs. 23–25.

Figure 3(a) shows an overview of the geometry and thecomputed temperature distribution after 20 ns. The 30%notch is just about visible on the right hand side of thenanowire halfway between the ends of the wire. The cut-plane shown in the right hand side of Fig. 3(a) as an insetshows the temperature distribution in the nanowire in they-z plane in the center of the constriction (as indicatedby a white line and semi-transparent plane in the mainplot). Figure 3(b) shows the notch geometry in moredetail (not to scale).

Fig. 3(c) shows the temperature profile ∆T (x) after20 ns taken along a line at the top of the nanowire (thesame data is encoded in the colours of Fig. 3(a) althoughmore difficult to read quantitatively). We see that mostof the wire is at an increased temperature around 150 K.The maximum temperature is found at the constriction:as in Sec. III B the current density is increased here due

to the reduced cross section in the y-z plane.

In contrast to the previous example (Sec. III Band Fig. 2), the temperature peak at the constrictionis less pronounced, as a smaller amount of material isabsent and consequently the current density and the as-sociated increase in Joule heating is smaller.

The maximum temperature increase reaches 163K.The zero level in the simulation corresponds to the tem-perature at which the experiment is started: if the ex-periment is carried out at 300 K we expect the maximumtemperature to be 463 K after 20 ns, which is well belowthe Curie temperature (≈ 840K) for Permalloy.26

From Figs. 3(a) and 3(c) we can see that the temper-ature at the ends of the wire is lower than near the mid-dle: for x . 1µm and x & 4µm temperature decreasestowards ≈ 100 K at the ends of the wire. This is due tomore efficient cooling through the substrate: at the endsof the wire there is substrate to three sides rather than

5

two as in the middle parts of the wire. Experimentallythis effect is typically not relevant because the contactwiring will follow at the end of the actual nanowire.

The importance of the substrate in cooling thenanowire can be seen if we use equation (3) to computethe temperature after 20 ns for a wire with the same ge-ometry but without the notch and without the substrate:the heating rate is dT/dt = j2/(ρCσ) = 66.83K/ns asin Sec. IIIA because the geometry does not enter thatcalculation. Without the substrate, we would have atemperature increase of ≈ 1336K after 20 nano seconds(even without taking the extra heating from the notchconstriction into account).

In Fig. 3(d), the solid line shows the maximum tem-perature in the nanowire, the dotted (dashed) line showsthe maximum (minimum) value of the temperature in thesilicon nitride membrane substrate and the dash-dottedline shows data computed using the model by You et al.

1

as a function of time over which the current pulse is ap-plied. Note that the You model has not been derived tobe used for such a thin substrate and that the large de-viation for large values of t is thus expected. We discussthis in detail in Sec. IVA.

We can use the maximum wire temperature graphto determine the length of the current pulse that canbe maintained until the temperature is pushed up tothe Curie temperature, or to the material’s evaporationtemperature (note that the plot shows the temperatureincrease since the start of the experiment, not abso-lute temperature). If the experiment is carried out atroom temperature (≈ 300K), then the Curie tempera-ture (≈ 850) is attained with a pulse duration of approx-imately 200 ns.

Asymptotically, the maximum temperature in the wireis proportional to the logarithm of pulse duration for a2d substrate and a point-like heating source. This regimeis entered when the nanowire and the constriction havebeen heated up to a steady state, and from there onthe temperature in the nanowire increases logarithmi-cally with time while the heat front propagates from thecenter of the substrate disk towards the disk’s boundary.

The maximum temperature in the substrate (dottedline) is assumed at the interface between the nanowireand the substrate, in the location where the nanowire ishottest. Because of the assumption of perfect thermalcontact between wire and the membrane substrate, thetemperature at the top of the membrane is the same asthe temperature at the bottom of the wire. The differ-ence between the maximum wire and maximum substratetemperature thus provides an indication for the temper-ature gradient found in the wire.

The dashed line in Fig. 3(d) shows the minimum valueof the temperature taken across the combined system ofnanowire and substrate. It starts to deviate from zerowhen the heat front has propagated from the center ofthe silicon nitride substrate disk to the boundary. Inour example, that happens after 1ms when the simu-lated temperature of the wire would be over 3000K. The

asymptotic logarithmic behaviour of the temperature in-crease in the wire is maintained only until the heat frontreaches the boundary of the substrate.

D. Case study 4: Nanowire with a notch on asilicon wafer

A possible way to increase the maximum pulse lengthis to use a silicon wafer instead of the silicon nitride mem-brane, which is typically done in high frequency transportexperiments. Silicon is a better conductor of heat, andthe extra thickness of the wafer gives more space for heatto dissipate. In this section, we model the same nanowirewith a notch geometry as above in Sec. III C but placedon a silicon half-sphere of radius 0.5mm instead of a sil-icon nitride disk membrane.Fig. 4(c) shows the temperature profile of the nanowire

with notch placed on a silicon wafer after a 20 ns pulse(current density at the wire ends is 1012 A/m2). Themaximum temperature increase is 16K and should becompared with Fig. 3(c) where the nanowire was placedon a (much thinner) silicon nitride substrate and themaximum temperature increase was 163K.Clearly, compared to the silicon nitride membrane the

silicon wafer is much more efficient at diffusing heat:the temperature increase is approximately a factor 10smaller.Fig. 4(d) shows the minimum and maximum temper-

ature for the silicon wafer substrate, and the maximumtemperature of the nanowire, analog to Fig. 3(d) for thenanowire on the silicon nitride membrane. We can iden-tify three regimes: (i) for small t the maximum tempera-ture in the wire increases approximately proportional tothe logarithm of time. (ii) For 1µs . t . 10, 000µs, themaximum temperature stays constant in the wire. Thisis the (3d) steady-state regime where the heat front prop-agates from the center of the silicon substrate half-sphereto the surface of the sphere. (iii) Approximately fort & 10, 000µs, the maximum wire and substrate tempera-tures and the minimum substrate temperature start to in-crease again simultaneously. This indicates that the heatfront has reached the surface of the half-sphere shapedwafer substrate, and that the heat from the nanowire can-not be carried away through the propagating heat frontanymore.The maximum temperature increase of ≈ 20K is

reached after ∼1µs and is maintained until t = 1ms,when the heat front reaches the boundary of the waferand the whole wafer begins to heat up. The tempera-ture gradient from top to bottom of the wire is of theorder of 5K (difference of maximum wire and maximumsubstrate temperature).For a system with an infinite 3d substrate we expect

in general that the nanowire temperature stays constantonce regime (ii) of the heat front propagation has beenreached. Regime (iii) would not exist for an infinitesubstrate, or a substrate that is efficiently cooled at its

6

(a) (b)

(c)

(d)

FIG. 4. Joule heating in a permalloy nanowire with a notch on a silicon wafer as discussed in Sec. IIID (geometry of wire fromRefs. 23 and 24): (a) temperature distribution T (r) in the nanowire and substrate after t = 20ns, (b) geometry of the model(not to scale) and the plotting path (dashed line) used in Fig. 4(c), (c) temperature profile ∆T (x) after t = 20ns along thelength of the wire on a silicon wafer substrate, (d) maximum and minimum temsperatures of the silicon substrate, maximumtemperature of the wire, and the prediction of the You model as a function of pulse length; tc is the characteristic time asdescribed in Sec. IVA. The silicon wafer is modelled as a half sphere (flat side attached to nanowire with notch) with radius0.5 mm. The current density is again j = 1012 A/m2; for material parameters see Tab. I.

boundaries.

E. Case study 5: Zigzag-shaped nanowire

In this section we investigate a nanowire geometry andexperimental set up as reported in Ref. 4 where currentdensities of 2.2× 1012 A/m2 are applied for 10µs.

Figure 5(b) shows the permalloy nanowire consisting ofthree straight 20µm segments connected by 45 bends ofradius 2µm (see also Fig. 1 in Ref. 4). The wire is 500 nmwide and 10 nm thick, and is placed on a silicon wafer sub-strate which is modelled as a silicon half-sphere of radius0.5 mm as in the previous example in Sec. IIID. Theresistivity of the wire material is adjusted to 42µΩcm sothat the total resistance of the wire is 5 kΩ (as reportedin Ref. 4).Figure 5(a) and 5(c) shows the temperature profile af-

ter 10µs. The maximum temperature increase does not

exceed 134K despite the large current density and thelong pulse duration. The ends of the zig-zag wire are sig-nificantly cooler (below 80K) than the center which canbe attributed to more silicon wafer substrate accessibleto carry away the heat that emerges from the nanowire.

Figure 5(d) shows how the maximum wire temperature(which is located in the middle segment of the zig-zagwire) and substrate minimum and maximum tempera-tures change over time. As before, we can identify threeregimes: (i) initial heating of wire and substrate, (ii)steady state with heat front propagating through sub-strate and (iii) general heating of wire and substratewhen heat front reaches the substrate boundary in thesimulation model. Due to the larger nanowire geometry,region (ii) is less pronounced here than in Fig. 4(d).

7

(a) (b)

(c)

(d)

FIG. 5. Joule heating in zigzag permalloy nanowire on a silicon wafer (Sec. III E): (a) temperature distribution T (r) in thenanowire and substrate after t = 10µs, (b) geometry of the model (following Ref. 4) and the plotting path (dashed) for Fig. 5(c),(c) temperature profile ∆T (x) after t = 10µs along the length of the wire (following the plotting path shown in Fig. 5(b)), (d)maximum and minimum temperatures of the silicon substrate, maximum temperature of the wire, and the prediction of the Youmodel as a function of pulse length; tc is calculated based on the distance between the opposite ends of the wire L = 56nm. Thecurrent density is j = 2.2× 1012 A/m2; the resistivity of permalloy σ−1 = 42µΩcm; for other material parameters see Tab. I.

F. Case study 6: Diamond substrate

Recently, realizations of very high current densitieshave been reported when the permalloy nanowire isplaced on a diamond substrate13 instead of silicon ni-tride or silicon. This finding is useful for time-integratingexperiments where current excitations exceeding the mi-crosecond timescale are required27,28. In this section,we investigate the geometry and experiment describedin Ref. 13.

Figure 6(b) shows a rectangular permalloy nanowirewith dimensions 25µm×650 nm×22.5 nm which is grownon a diamond crystal substrate. As before, perfect ther-mal contact between the wire and the substrate is as-sumed. In the original experiment13, the whole devicewas placed in a cryo bath. The area of contact betweenthe diamond crystal substrate and the bath is large com-pared to the size of the nanowire, therefore the temper-ature difference at the contact layer is likely to be small.In the simulation, the effect of the bath can be repre-

sented by using an infinite medium of diamond. In linewith the studies in the previous Sec., we use half-sphereshape for the diamond substrate (radius of 0.5mm). Theresistivity of the wire material is adjusted to 39µΩcm sothat the total resistance of the wire is 675Ω as in Ref. 13.Figure 6(a) and 6(c) show the temperature profile after

application of a current density of 1.5 × 1012 A/m2 for1µs, and Fig. 6(d) shows the maximum and minimumtemperature in the system as a function of time.From Fig. 6(d) we can see that the maximum temper-

ature increase in the wire is not exceeding 21K for timessmaller than 1000µs, and that the substrate tempera-ture increase stays below ≈ 16K in that period. Theplot shows that the precise current density pulse dura-tion is not so critical: if t is between 1µs and 1000µsthe nanowire attains approximately the same tempera-ture. It is this steady state temperature increase valuethat should be compared with the experiment.The difference between the maximum substrate tem-

perature and the maximum wire temperature reflectsa temperature gradient from top to bottom in the

8

nanowire. This is just about visible in the inset inFig. 6(a).For larger times than 1000µs, we find that the maxi-

mum and minimum temperature of the substrate and themaximum wire temperature start to increase simultane-ously. This is an effect of the finite size of the substrate inthe model, or, equivalently, the lack of the modelling ofheat transfer away from the diamond substrate throughthe cryo bath. There are other, in comparison to the cryobath less important cooling contributions, such as elec-tric contacts, substrate holder, and surrounding gas, thathave not been considered here. For the interpretation ofthe simulation results for the experiment in Ref. 13, weneed to ignore the regime for t & 1000µs.

The shape of the temperature profile (Fig. 6(c)) is verysimilar to the zigzag wire temperature profile (Fig. 5(c)).In the original experiment, a continuous current corre-

sponding to a current density of 1.5× 1012 A/m2 heatedthe wire by about 230K (Fig. 4 in Ref. 13).In our simulation the nanowire changed temperature

by less than 21K. A possible explanation for this discrep-ancy is that the contact between the wire and the dia-mond was imperfect. If so, then improving this contactwill lead to even better heat dissipation and concomi-tantly higher possible current densities.

IV. ANALYTICAL MODELS

A. Model by You, Sung, and Joe for a nanowire ona (3d) substrate

You, Sung, and Joe1 have provided an analytic expres-sion TYou2006(t) to approximate the temperature T (t) ofthe current-heated nanowire as a function of time t (Eq.(16) in Ref. 1):

T 3d(t) = TYou2006(t) =whj2

πkσarcsinh

(

2√

tk/(ρC)

αw

)

(4)where w and h are the width and height of the wire,σ is the wire conductivity, j is the current density, andk, ρ, and C are the thermal conductivity, mass density,and specific heat capacity of the substrate. We have usedtheir adjustable parameter α = 0.5 for calculations shownin Sec. III.In this section, we introduce the name T 3d(t) as a

synonym for TYou2006(t) to emphasize the difference tothe similar looking equation for T 2d(t) that is derived inSec. IVB.For derivation of Eq. (4), it is assumed that the

nanowire is infinitely long, and attached to a semi-infinitesubstrate. While the thickness h and width w of the wireenter the derivation to compute the Joule heating due toa given current density, the model does not allow for atemperature variation within the nanowire nor does thenanowire have a heat capacity in the model. Within thismodel, a heat front of (half-)cylindrical shape (cylinder

axis aligned with the wire) will propagate within the sub-strate when the wire is heated. Thus, there is transla-tional invariance along the direction of the wire.

We start our discussion with the zigzag nanowire asshown in Fig. 5. Figure 5(d) shows the temperature pre-diction of the You model as a dash-dotted line. It followsthe maximum temperature in the substrate very closelyfor times up to approximately 2µs.

At short times t below 1 ns we can see the You modelslightly overestimating the temperature in the nanowirein Fig. 5(d). As the model does not allow for a finiteheat capacity of the wire, this is expected. As the heatcapacity of the nanowire is insignificant in comparison tothe substrate, this overestimation disappears if sufficientheat has been pumped into the system.The difference between the maximum temperature in

the wire and the maximum temperature in the substratecomes from a temperature gradient within the wire: themaximum temperature in the wire is at the top of thewire (which is furthest away from the cooling substrate)and the maximum temperature of the substrate is foundat the top of the substrate just where the wire reaches itsmaximum temperature. Due to the assumption of perfectthermal contact, the bottom of the wire is exactly at thesame temperature as the top of the substrate within themodel description.Since the You model does not allow for a temperature

gradient within the wire, we expect its temperature pre-diction to follow the maximum temperature increase inthe substrate. This is visible in Fig. 5(d) for t & 2 ns.

Regarding the deviation between the You model andthe simulation results for t & 2µs, we need to estab-lish whether the required assumptions for the model arefulfilled. The You model is derived for an infinitely longwire on an infinite substrate, whereas the segments of thezigzag wire studied here have finite length. In the initialstage of heating, the temperature front in the substratewill move away from the wire sections with heat frontsaligned parallel with the wire. The heat front formsa half-cylinder (for each zigzag segment) whose axis isaligned with the wire. As long as the diameter of thishalf cylinder is small relative to the segment length, thewire appears locally to be infinitely long and the heatfront propagates in a direction perpendicular to the wire.This is the regime where the You model is applicable.When the heat front has propagated sufficiently far fromthe nanowire to change its shape from a cylinder surfaceto a spherical surface, the You model is not applicableanymore. This happens approximately afters t & 2µs.We have referred to the spherical heat front propagationin the discussion in previous sections as “regime (ii)”.For the zigzag wire study the agreement of the model

by You et al.1 with the simulation is thus very good

within the time range where the model is applicable. Wenote that the wire is relatively long (20µm per segment)and has no constriction. The You model cannot be ap-plied for t & 4µs because the finite wire length becomesimportant.

9

(a) (b)

(c)

(d)

FIG. 6. Joule heating in a permalloy nanowire on a diamond crystal substrate (Sec. III F): (a) temperature distribution in thenanowire and substrate T after t = 1µs, (b) geometry of the model and the plotting path, (c) temperature profile ∆T (x) aftert = 1µs along the plotting path, (d) maximum and minimum temperatures of the diamond substrate, maximum temperatureof the wire, and the prediction of the You model as a function of pulse length. The current density is j = 1.5× 1012 A/m2; theresistivity of permalloy σ−1 = 39µΩcm; for other material parameters see Tab. I.

For the nanowire without a constriction on the dia-mond substrate as studied in Sec. III F and shown inFig. 6, the agreement is similarly good; the You modeltemperature follows the substrate temperature very ac-curately up to t ≈ 0.1µs. For larger t, the model becomesinaccurate as the finite length of the wire becomes im-portant at that point.

Figure 4(d) shows for the nanowire with a notch onthe silicon substrate that for t & 0.1µs the gradients ofboth maximum temperature curves approach zero whichindicates the onset of regime (ii) and implies that theYou model cannot be applied for t & 0.1µs. For smallert . 0.1µs, the You model temperature roughly followsthe maximum substrate temperature with a maximumabsolute deviation of less than 3K.

We have carried out additional simulations (data notshown) which have demonstrated that the maximum sub-strate temperature line (dotted line in Fig. 4(d)) is shifteddown by a few degree Kelvin if the notch is removedfrom the geometry. The You model temperature andthe maximum substrate temperature curves then coin-

cide for 2 . t . 10 ns. If, furthermore, we increase thewire length from 5µm to 30µm, the two curves coincidefor 2 . t . 300 ns.Both the notch and the relative shortness of the wire

decrease the accuracy of the prediction of the You model:the notch roughly shifts all temperature curves up by afew degrees whereas the length of the wire determinesthe time when regime (ii) is entered.

In contrast to the previous examples the nanowire witha notch in Fig. 3 is attached to a relatively thin siliconnitride membrane of thickness 100 nm (the silicon anddiamond substrates for the discussion above are of thesize order of 500µm). The You model should not beused in the regime of membranes as the model expectsan infinite substrate.However, one could argue that the You model should

be applicable for very small t until the heat front emerg-ing from the wire has propagated through the 100 nmthick substrate. Additional simulations (data not shown)reveal that this is the case after t ≈ 0.1 ns. The Youmodel data shown in Fig. 3(d) (inset) for t . 0.1 ns over-

10

estimates the substrate maximum temperature becausethe finite heat capacity of the wire is important at thistime scale.

We note that the You model cannot be expected toprovide accurate temperature predictions for thin sub-strates. The deviation of the You curve in Fig. 3(d) orig-inates in the inappropriate application of the model toa system with a thin, effectively two-dimensional, sub-strate.

In summary, we find that the You model provides anaccurate description of the maximum substrate temper-ature if used within its bounds of applicability.

In the You model, the nanowire has no heat capacityand this results in the model slightly overestimating thetemperature for very small t (visible in Fig. 5(d) for t .1 ns).

For a ‘thick’, effectively three-dimensional, substratewe find that the applicability of the model is limited bythe finite length of the wire. The maximum time tc upto which the You model is appropriate, can be estimatedby calculating the characteristic time scale of the heatconduction equation (1). For a wire of length L, thischaracteristic time tc is

tc ∼

(

L

2

)2ρC

k(5)

where k, ρ, and C are the thermal conductivity, densityand specific heat capacity of the substrate material. Thegreater the nanowire length, the longer it takes for theheat front to assume spherical shape around the nanowireheating source. The greater the heat capacity (ρC) andthe smaller the thermal conductivity, the slower is thepropagation of the heat front in the substrate. The cur-rent density does not enter the equation as it only affectsthe temperature and not the time or length scale.

Substituting the corresponding parameters for eachcase study, we compute the characteristic time tc for thenanowire with a notch from Sec. IIID to be 70 ns, for thezigzag nanowire from Sec. III E to be 9µs, and for thenanowire on diamond from Sec. III F to be 0.3µs. Thesevalues are in good agreement with the corresponding fi-nite element results from Fig. 4(d), 5(d), and 6(d).

The You model cannot be applied for thin, effectivelytwo-dimensional, substrates such as the membrane sub-strate case study in Sec. III C and Fig. 3.

The temperature within the nanowire can show a gra-dient (hotter at the top, cooler at the interface to thesubstrate), and the You model computes the smaller tem-perature in the wire. For the studies carried out here wefind this temperature difference to be less than 10 K inall cases although this difference depends on material pa-rameters and wire thickness.

B. Analytic expression for nanowire on amembrane

Equation (4) is valid for pulse durations up to the crit-ical duration tc (Eq. (5)) for effectively three-dimensionalsubstrates, i.e. substrates whose thickness is sufficientlylarge so that the heat front in the substrate does notreach the substrate boundary for t < tc. This conditionis fulfilled for the case studies in Sec. IIID, III E, andIII F.If the substrate is effectively two-dimensional (such as

in Sec. III C), then Eq. (4) cannot be applied. In thissection, we show how Eq. (4) can be adapted to the 2dcase.The assumptions made in the derivation of Eq. (4) re-

quire a system of nanowire and substrate that is trans-lationally invariant in one direction. In Sec. IIID–III Fthis direction was along the long axis of the wire, and werefer to this axis as x, and assume that the height h ofthe wire extends along the z axis.For the case of a nanowire on a thin membrane sub-

strate, we can regard the system as two-dimensional byassuming invariance in the perpendicular direction. Inother words, to apply a modified version of Eq. (4), weassume that the temperature distribution in the mem-brane system is invariant along the z axis. We model thenanowire as embedded in the substrate (not grown on topof the substrate as in the real system) and imagine an in-crease of thickness of both the wire and the substratesuch that they expand from −∞ to +∞ in z-direction.The cooling and heating in each slice (in the x-y plane)of the nanowire and substrate system is not affected bycooling and heating from slices above and below; it doesnot matter whether we consider only one isolated slice (asin the real system) or imagine an infinite stack of slicesclosely packed on top of each other.In more detail, we first convert the system of the

nanowire of height h and substrate of thickness d to a2d system of equal height. We increase the height of thenanowire by a factor c = d/h so that the wire and thesubstrate are now both of height d (assuming that gener-ally d > h but the derivation also holds for d < h). Thisincreases the volume of the wire by a factor of c, and thuswe will have to correct down the heat emerging from thewire by the same factor at a later point.Second, to obtain translational invariance in the z-

direction we imagine a stack of such identical 2d systemson top of each other. Using the substitutions w → L andh → w, we obtain

T 2d(t) =Lwj2

πkσ

1

c

1

2arcsinh

(

2√

tk/(ρC)

0.5L

)

(6)

=whLj2

2dπkσarcsinh

(

2√

tk/(ρC)

0.5L

)

(7)

The first fraction in Eq. (6) is based on Eq. (4) and in-cludes the substitutions w → L and h → w, and we also

11

substitute w → L in the denominator of the arcsinh argu-ment. The second fraction (1/c) reduces the temperatureincrease by c to compensate for the increase of heatingby the factor c above when we increased the thicknessof the nanowire to the thickness of the substrate. Thethird fraction (1/2) in Eq. (6) is a correction becausethe nanowire is now surrounded by substrate in all di-rections, and not only in one half-space as in Eq. (4),thus the cooling is twice as efficient and the temperatureincrease is halved.Equation (7) can be used to compute the maximum

temperature increase T 2d(t) for a nanowire of length L,width w, and height h on a two-dimensional substrate ofthickness d.In contrast to T 3d(t) there is no upper time limit tc

for the validity of Eq. (7) as the emerging heat-front willalways stay translationally invariant.The comparison of T 2d(t) with the finite element sim-

ulation results from Sec. III C is shown in Fig. 7. Theoverall agreement with the simulation results is good forall times t.

FIG. 7. Comparison of simulated wire and substrate temper-atures with the analytical expression T 2d(t) from Eq. (7), asa function of pulse length.

V. SUMMARY

We have carried out detailed numerical simulations ofthe current distribution, Joule heating, and dissipationof temperature and heat through the nanowire and thesubstrate for a number of experiments and three differentsubstrate types. We find that the silicon nitride mem-brane (thickness 100 nm) is the least efficient in coolinga ferromagnetic nanowire that experiences Joule heat-ing. Due to the quasi-two-dimensional nature of themembrane, the temperature in the nanowire will keep

increasing proportionally to the logarithm of time forlonger current pulses while the heat front (forming a cir-cle in the membrane substrate) propagates away from thenanowire, which is located in the center of the heat frontcircle.Using a (effectively three-dimensional) silicon wafer

substrate instead of the (effectively two-dimensional) sil-icon nitride membrane, there is a qualitative change:once the steady state is entered, the heat propagatesin three dimensions and keeps the temperature in theheated nanowire virtually constant. In addition to thethree-dimensionality of a silicon wafer, silicon has a ther-mal conductivity that is about one order of magnitudegreater than silicon nitride’s (see Tab. I), which helpswith removing Joule heating energy from the nanowire.If we replace silicon in the 3d substrate with diamond,

the cooling is improved again significantly: diamond’sthermal conductivity is about an order of magnitudegreater than that of silicon (see Tab. I).In addition to these generic insights, we have worked

out the temperature increase quantitatively for a series ofrecent experimental publications.4,12,13,23,24 The modelsimulations presented here show for all of them that thetemperature increase due to the Joule heating did notresult in the temperature exceeding the Curie tempera-ture.We compare these results to the approximating but

analytical model expression provided by You, Sung, andJoe1 and investigate the limits of its applicability. Weprovide an estimate for the characteristic time tc overwhich the You model is valid for three-dimensional sub-strates.Finally, we provide a new analytical expression that

allows to compute the temperature for a nanowire on atwo-dimensional substrate in the presence of Joule heat-ing. This expression should be of significant value inthe design and realization of spin-torque transfer studieson membranes where experimenters are often operatingvery close to the Curie temperature or even the meltingtemperature, and where no estimate of the wire’s tem-perature has been possible so far.

The research leading to these results has received fund-ing from the European Community’s Seventh Frame-work Programme (FP7/2007-2013) under Grant Agree-ment n233552 (DYNAMAG), and from the EPSRC(EP/E039944/1,EP/G03690X/1). Financial support bythe Deutsche Forschungsgemeinschaft via SFB 668 aswell as GK 1286 and from the Cluster of Excellence”Nanospintronics” funded by the Forschungs- und Wis-senschaftsstiftung Hamburg is gratefully acknowledged.We thank Sebastian Hankemeier for useful discussions.

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