Supplementary Materials for “Suppression of Thermoelectric Thomson Effect in Silicon
Microwires under Large Electrical Bias and Implications for Phase-Change Memory
Devices”
The discussion of Thomson heat in the MacDonald’s book “Thermoelectricity: An
Introduction to the Principles” (Ref. 2 in the manuscript)
The nature of thermoelectric Thomson heat is discussed by D. K. C. MacDonald in comparison
to Joule heating. In this discussion, a simplified expression for the heat evolved in a current
carrying conductor subjected to a temperature gradient is considered to elucidate the relative
contributions of Joule and Thomson heat. Derivation of the simplified expression starting with
equation 1 in the manuscript text is below:
Eq. 1 in the manuscript text provides the generated power per volume as:
Qv  J 2   TJ  S
The conductor has a uniform medium, hence the gradient of Seebeck coefficient can be
expressed in terms of the temperature gradient using the Chain rule:
Qv  J 2   T
dS
J  T  J 2   J  T
dT
where µ is the Thomson coefficient. The volume of the conductor has to be incorporated to
evaluate power generated in the whole volume. Assuming a constant cross section (A) and length
(L), the total power becomes:
Q  LAJ 2   LAJ  T
Rewriting J as I/A and the temperature gradient as ∆T/L (assuming a linear gradient) reduces
above expression to:
Q  I 2 R  IT
The discussion in Ref. 2 assumes that a single charge is carried along the conductor in time t
creating the electric current (I=q/t). Hence, the heat generated on the conductor in time t
becomes:
Qt 
q2R
 qT
t
The final expression suggests that if the charge carrier moves slowly (large t), then Thomson
heat, independent of t, will dominate the total heat generated. For decreasing t, which can be
interpreted as increasing current (q/t), Joule heating is expected to increase and dominate the
total heat. Although this discussion sheds some light on the relative contributions of Joule and
Thomson heat, it assumes a constant temperature difference, independent of the generated heat,
which is not the case for our Si wires. Thomson heat, one of the factors determining the
temperature gradient, in turn depends on heat generation. Hence, for a clear understanding of
how Joule and Thomson heat depend on J, numerical modeling of the self-heating process is
performed.
Material parameters of the nc-Si wires, SiO2 layer and bottom Si substrate:
S (mV/K) CP (Jkg-1K-1) k (Wm-1K-1) (m.cm)
Figure 1 shows the electrothermal parameters of nc-Si films used in the simulations. Electrical
resistivity of nc-Si films is measured up to 620 K. The rest of the electrical resistivity and the
entire thermal conductivity is extracted on similar nc-Si wires as described in Ref. 1. The solid
part of the resistivity curve is from the measurements. Heat capacity of single crystalline silicon2
is used for the modeling of nc-Si wires. Latent heat of fusion3 is incorporated as a 10-K wide
peak. Seebeck coefficient of nc-Si films are measured up to 900 K (symbols), as well as
calculated using the doping level of the films and single crystallinity assumption (solid curve).
For the simulations, a Seebeck curve is extrapolated (dashed curve) using the measurement and
calculation results4. Liquid silicon values are used for nc-Si beyond 1700 K (Table 1). Constant
material parameters for SiO2 and single crystalline silicon substrate are used in the simulations
(Table 1).
30
20
10
0
1690 K
60
40
20
105
105
4
104
10
Latent heat:
1790 kJ/kg
103
3
1690
10
0.0
1700
-0.2
-0.4
300
600
900 1200 1500 1800
T (K)
Figure 1. Temperature dependent nc-Si parameters used in the modeling.
Table 1. Material parameters for nc-Si (at 300 K and beyond 1700 K), SiO2 and single crystal Si substrate (constant
for all T) used in the simulations. SiO2 and Si substrate values are obtained from COMSOL Multiphysics library.
ρ (mΩ.cm)
k (W m-1K-1)
CP (J kg-1K-1)
S (mV/K)
d (kg/m3)
nc-Si (300 K)
20.8
30
709
-0.14
2330
nc-Si (>1700 K)
0.07
60
890
-0.002
2330
(Ref. 5)
(Ref. 6)
(Ref. 7)
(Ref. 8)
SiO2
1021
1.4
700
-
2200
Si Substrate
200
130
700
-
2330
Calculating the heat diffusion time constant:
The 3-D geometry seen in Figure 2a is used to estimate the heat diffusion time constant. This is
the same geometry used for the simulations described in the main text with an additional small
structure on top of the wire to introduce an external heat source. For this study, the wires are
heated through keeping the top surface of the small structure (heat source) at a constant
temperature (T: 1600, 1200, 800 K) (Figure 2b-e). Initial temperature of the wire and underlying
SiO2 and Si substrate is 300 K. Average temperature of the wire is plotted as a function of time,
showing a 2nd order exponential decay, which allows for extraction of time constants (Figure 3).
The longer heat diffusion time constant is ~24 ns for all heat source temperatures studied. This
value is slightly smaller or larger for longer or shorter heat source length, respectively.
(a)
(b)
(c)
t: 0
(d)
t: 10 ns
(e)
t: 50 ns
t: 100 ns
Figure 2. (a) 3D geometry used for estimation of the heat diffusion time constant. Thermal properties of nc-Si are
used for the external heat source. (b-e) Heat diffusion process on the wire (external heat source dimensions: 100 x
500 nm2). Temperature scales in between 300 (blue) and 1600 K (red).
1.2
Tavg (x1000 K)
(a)
(b)
TS: 1600 K
0.9
TS: 1200 K
TS: 800 K
0.6
TS: 1600 K
1200 K
800 K
L: 100 nm
28 ns
-
-
L: 250 nm
24 ns
25 ns
23 ns
L: 500 nm
22 ns
-
-
Simulation
Fit
0.3
0
25
50
time (ns)
75
100
Figure 3. (a) Rise of the average wire temperature over time for 3 constant temperatures (TS) on the top surface of
the heat source and 2nd order exponential decay curves fit to simulation results. (b) Extracted heat diffusion time
constants (the longer exponential time constant) for different heat source lengths and surface temperatures.
Simulated PCM current
Figure 4. Current through the PCM cell during the applied pulses: 35 V / 0.1 ns, 29 V / 0.2 ns, 26.6 V / 0.3 ns, 25.1
V / 0.4 ns, 22.4 V / 1 ns, 20.05 V / 10 ns, 19.81 V / 100 ns. Fall/rise time is 1 ns for 100 ns pulse and 0.1 ns for the
rest.
Supplementary References
1
G. Bakan, L. Adnane, A. Gokirmak, and H. Silva, J. Appl. Phys. 112, 063527 (2012).
2
V.M. Glazov and A.S. Pashinkin, High Temp. 39, 413 (2001).
3
W.G. Hawkins, Appl. Phys. Lett. 42, 358 (1983).
4
G. Bakan, N. Khan, H. Silva, and A. Gokirmak, Sci. Rep. 3, 2724 (2013).
5
H. Sasaki, A. Ikari, K. Terashima, and S. Kimura, Jpn. J. Appl. Phys. 34, 3426 (1995).
6
H. Kobatake, H. Fukuyama, I. Minato, T. Tsukada, and S. Awaji, Appl. Phys. Lett. 90, 094102
(2007).
7
8
W.K. Rhim, S.K. Chung, A.J. Rulison, and R.E. Spjut, Int. J. Thermophys. 18, 459 (1997).
V.M. Glazov, S.N. Chizhevskaya, and N.N. Glagoleva, Liquid Semiconductors (Plenum, New
York, 1969), p. 364.