Supporting Online Materials for
“Nanometer-Scale Temperature Imaging for Independent Observation
of Joule and Peltier Effects in Phase Change Memory Devices”
By: Kyle L. Grosse, Eric Pop, and William P. King
University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA
I. Frequency Domain Thermoelectric Equations
The frequency domain thermoelectric transport equations are derived as follows.
Equations S1 and S2 show the modified Poisson and heat diffusion equations to account for
thermoelectric transport.1,2
 ( S T  V )  0
 d cP
T
 (k   S 2T )T  ( ST )V   ( S T V  [V ]2 )
t
(S1)
(S2)
The density, heat capacity, thermal conductivity, electrical conductivity, thermopower,
temperature, and voltage are given by ρd, cP, k, σ, S, T, and V. Equations S1 and S2 are coupled
and must be solved simultaneously. Equation S3 shows the applied sinusoidal bias which
generates the temperature oscillations shown in Eq. S4.
V  V0  V1 cos(2t )
(S3)
T  T0  T1 cos(2t )  T2 cos(2 2t )
(S4)
The subscripts 0ω, 1ω, and 2ω denote the amplitude of V and T at the zero, first, and second
harmonics. The device is biased at a frequency ω = 43 kHz. The first and second harmonic
components are complex numbers, and the first and second harmonic peak-to-peak temperature
rises are defined ΔT1ω = 2|T1ω| and ΔT2ω = 2|T2ω|. Driving the device with a bias at 0ω (DC) and
1ω bias causes a 0ω, 1ω, and 2ω temperature rise as Joule heating scales with V2. The Fourier
transform of Eqs. S1-S4 yields Eq. S5 the frequency domain thermoelectric transport equation.
1
 4(k   S 2T0 )
 T0 
2 S 2T1
2 S 2T2
4 ST0
2 ST1

  
2
2
2
2k   S (2T0  T2 )
 S T1
2 ST1  S (2T0  T2 )   T1 
 2 S T1
  T2  
  2 S 2T2
 S 2T1
2(k   S 2T0 ) 2 ST2
 ST1

  
S
0
0

0

 V0 

 V1 
0
S
0
0



(S5)
2
2


2 S (2T0 V0  T1 V1 )  2(V0 )  (V1 ) 


 S (2T0 V1  2T1 V0  T2 V1 )  4V1 V0   i 4 d cPT1 

...  
 S (T1 V1  2T2 V0 )  (V1 ) 2   i 4 d cPT2 2



0




0
The terms on the left hand side of Eq. S5 are the transport terms. The terms on the right hand
side without frequency dependence are generation terms, and the terms with frequency
dependence describe periodic heat diffusion. The model is coupled with a thermo-mechanical
model for comparison with scanning Joule expansion microscopy (SJEM) measurements.
II. Finite Element Analysis Model
Equation S5 was implemented in a finite element analysis (FEA) model of a device,
shown in Figure 1(a), to predict Joule and thermoelectric device behavior. The model is similar
to 2D phase change memory (PCM) device models used in previous SJEM measurements.3,4 The
model is bounded by the electrode bias as shown, a heat sink of T0ω = 300 K along the bottom
surface, and free surfaces along the top. All other surfaces are insulated and constrained. Joule
and Peltier effects are generated in the Ge2Sb2Te5 (GST) device from its high resistivity ρGST =
2.5×10-4 Ω m and large thermopower SGST = 250 μV K-1. The device was modeled as a 22 nm
thin GST device with a 4 μm channel length. Joule heating also occurs at the contacts due to
finite interface resistivity (ρC = 2×10-9 Ω m2) between the GST and metal.
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III. The Seebeck Effect
Figures 2(e) and (f) show a slight change in ΔT2ω at the channel center with changing
thermopower due to the Seebeck effect.5 The model predicts a thermoelectric voltage is induced
across the device due to the asymmetric temperature gradient at the edges of the device and its
contacts. The induced voltage slightly changes the power dissipation of the device. The changes
are typically too small to be observed experimentally. Equations S1 and S2 describe the Seebeck
effect, but the PE and JH equations in Figs. 1(b) and (c) do not describe the Seebeck effect.
IV. Scanning Joule Expansion Microscopy
Scanning Joule expansion microscopy (SJEM) was used to observe Joule and Peltier
effects of the PCM devices. The SJEM technique utilizes a sinusoidal waveform with voltage
VDS to drive the device. An atomic force microscopy (AFM) cantilever in contact with the
surface measures the surface expansion, and two lock-in amplifiers at 1ω and 2ω simultaneously
record the peak-to-peak sample surface expansion Δh1ω and Δh2ω. The measured Δh is
proportional to the device temperature rise ΔT, and the two are related by the FEA model.
V. Measurement and Prediction Uncertainty
Observing Peltier effects using a unipolar bias requires two measurements which
increases the measured uncertainty. A single SJEM measurement had an average uncertainty of
~2-3 pm. Using a unipolar bias requires two measurements to observe Peltier effects which
doubles the variance (the square of the uncertainty) from a single measurement and the drift
between the two measurements further increases the average uncertainty. The uncertainty
therefore increased from ~2-3 pm for a single SJEM measurement to ~7-8 pm when observing
3
Peltier effects using a unipolar bias. Using a bipolar bias requires a single measurement to
observe Peltier effects and the measured uncertainty is the same as a single SJEM measurement.
The predicted uncertainty of SGST increases with increasing GST thermopower for the
following reason. The measured percent uncertainty of Δh1ω, the measured uncertainty of Δh1ω
divided by the measured Δh1ω, was similar for each device. The observed Δh1ω is due to local
Peltier heating and cooling, and the PE is proportional to the GST thermopower. Therefore, the
measured percent uncertainty of Δh1ω causes a proportional percent uncertainty in the predicted
GST thermopower.
Supplemental References
1
E. M. Lifshitz, L. D. Landau, and L. P. Pitaevskii, Electrodynamics of Continuous Media.
(Butterworth-Heinemann, 1984).
2
J. Martin, in the COMSOL Conference, Hannover, Germany, 4-6 November, 2008, pp. 1-7.
3
K. L. Grosse, E. Pop, and W. P. King, Submitted to J. Appl. Phys. (2014).
4
K. L. Grosse, F. Xiong, S. Hong, W. P. King, and E. Pop, Appl. Phys. Lett. 102, 193503 (2013).
5
D. K. C. MacDonald, Thermoelectricity: An Introduction to The Principles. (John Wiley and
Sons, 1962).
4