Biased Structural Fluctuations due to Electron Wind Force
Microsoft Word - CurrentStressSSLb.doc
Biased Structural Fluctuations due to Electron Wind Force 
 O. Bondarchuk*, W.G. Cullen, M. Degawa and Ellen D. Williams 
Department of Physics 
University of Maryland at College Park 
College Park, MD 20742-4111 
T. Bole and P.J. Rous 
Department of Physics 
University of Maryland Baltimore County 
1000 Hilltop Circle 
Baltimore, MD 21250 
Direct correlation between temporal structural fluctuations and electron wind force is 
demonstrated, for the first time, by STM imaging and analysis of atomically-resolved motion on 
a thin film surface under large applied current (105A/cm2).  The magnitude of the momentum 
transfer between current carriers and atoms in the fluctuating structure is at least 5x to 15x (± one 
sigma range) larger than for freely diffusing adatoms. The corresponding changes in surface 
resistivity will contribute significant fluctuation signature to nanoscale electronic properties.    
PACS #s:  
73.63.-b  Electronic transport in nanoscale materials and structures  
73.25.+i  Surface conductivity and carrier phenomena 
73.50.Td  Noise processes and phenomena 
68.37.Ef  Scanning tunneling microscopy 
68.55.-a  Thin film structure and morphology   
*  Present address: Chemical Physics Department, Fritz-Haber-Insitut der Max- Planck -
Gesellschaft, Faradayweg 4-6, Berlin, 14195 Germany 
Introduction. 
 Due to the size-scaling of fluctuations, the effects of statistical mechanics will be very 
different at the nanoscale than for macroscopic systems. The effects of nanoscale thermal 
fluctuations will impact molecular electronic and nanoelectronic contacts 1-3, device stability 4, 5, 
electromigration 6-14 and noise 15-17.  In this work we quantify the relationship of thermal 
fluctuations with electrical transport by directly observing step fluctuations at the surface of a 
current-carrying metal film, as illustrated in Fig. 1.  Carrier scattering causes a force due to 
momentum transfer, known as the electromigration wind force, and corresponding changes in the 
surface resistivity.  By convention, this force is written in terms of an effective valence Z*, and 
the (macroscopic) applied electric field E: F=Z*eE 18-20.  The momentum transfer force felt by 
atoms at surfaces depends on the local environment: atoms at step edges, near defects, or freely 
diffusing at the terrace experience different forces 21-23.  These forces can cause substantial 
changes in surface morphology 24-27, due to mechanisms similar to those well known in 
electromigration-induced failure 6, 18-20.  Despite its substantial impact upon the morphological 
evolution of materials, the electromigration wind force is extremely weak, and detecting its 
effects have required following changes in structure after long periods of current stressing.  Here, 
we will describe direct observation of the effects of the electromigration force on a time scale of 
seconds by measuring the nanoscale fluctuations of atomic-layer steps 28 on the surface of a 
current-carrying metallic thin film.  
 The fluctuations of a surface step are observed via a direct measurement of the position 
of one element of the step as a function of time, x(t).  Near equilibrium, step fluctuations can be 
well-described using the continuum step model 28-31, which predicts that the time-correlation 
function grows as a power law for times less than the correlation time. For the system described 
here, steps on Ag(111), step motion is driven by step edge diffusion (SED) 30, 32, 33, for which the 
correlation function is: 
G t( ) " x(t) # x(0)[ ]
 .   (1)  
Here x(t) is the position of the step perpendicular to the average step-edge orientation and the 
average is taken by repeated observations, ax=0.25 nm and ay=0.29 nm are the lattice constants 
perpendicular and parallel to the step edge. The time characteristic of thermal fluctuations of the 
step edge, 
, is determined by the step stiffness, 
˜ " , and the hopping time 
 for atomic motion 
along the step edge: 
2$ 3 4( )
˜ + a
,    (2) 
where Γ is the Gamma function, and the value of τh has been measured to decrease from ~ 3 µs 
to ~3 ns between 300K and 460K 33.   
 Recently the step continuum model has been expanded to include the effect of an 
electromigration force acting perpendicular to a step that is fluctuating via SED 34.  The 
correlation function deviates from the equilibrium result as: 
G(t) = a
± t "
EM( )0
* 1+ e+2u
EM( )du , ax2
1± 0.3487
1/ 2- 
  (3) 
where the + and – signs correspond to a downhill and uphill direction of force respectively. The 
presence of the electromigration force gives rise to an additional time constant, 
, that 
represents the time when the time-correlation function begins to deviate significantly from its 
equilibrium behavior, 
T( ) ˜ # ax( )
) ) " h ,    (4) 
Since the electromigration force is weak, 
T  and 
<< ˜ " a
 we anticipate that 
.  The nature of the result indicated by Eq. 3 can be understood by analogy to the 
Bales-Zangwill `kinetic instability 35.  In both cases, a diffusional bias perpendicular to the step 
edge results in spontaneous increased deviations from the equilibrium position when the bias 
favors diffusion in the step-downhill direction.  For an up-hill bias, an anomalous straightening 
of the step edge occurs.   
In the remainder of this letter, we exploit this model.  Measurement of the correlations of 
thermal step-edge fluctuations in the presence of current stressing yield the electromigration time 
constant (Eq. 3) and, as a result, the electromigration force felt by atoms diffusing along the step 
edge (Eq. 4).  This represents a direct measurement of the effective valence of an atom at the 
surface of a current-carrying solid. 
The experimental methods for preparing atomically clean (111)-oriented silver films have 
been described previously 33, 36.  The films used here were 100 to 200 nm thick and 1-2 mm 
wide, with micron-scale areas of flat (111)-oriented surface separated by deep pits, which 
covered about 50% of the film area at the smallest film thickness (100 nm). Atomic cleanliness 
was confirmed by atomic-resolution STM imaging.  Imaging was performed using tunneling 
conditions of 0.6-0.8 nA and 1V, at a scan rate of ~9 pixels/ms, which are known not to perturb 
the measured step configurations 30, 32, 33.  After completing the STM measurements on each 
sample, the sample temperature was measured using a thermocouple brought into direct contact 
with the film surface 33.  The thermocouple values and the measured hopping time constant 33 
were used to determine the sample temperatures.   
The temporal evolution of the step fluctuations was observed by repeated STM scans 
across the step boundaries as shown in Figure 2.  The size of each image is 100 nm x 512 scans 
(56.6 ms/scan) and an electric current of 0.4 A (nominal current density 1x105 A/cm2) flows 
through the sample perpendicular to the step edges. The temperature of the current-stressed 
sample (due to Joule heating) was 370±30 K.  By following the motion of the edges of the steps 
shown in Fig. 2, we simultaneously determine the spatial variations x(t) for up-hill and down-hill 
steps fluctuating at the same sample temperature under the same current density.   
The time correlation functions obtained from the measured x(t) are shown in Fig. 3.  The 
magnitude of the correlation function grows more rapidly for the up-hill current than for the 
down-hill current, consistent with an electromigration force acting in the same direction as the 
electron flow in the sample (e.g. Z* < 0, an electron wind force).  The data were fit to Eq. 3, with 
the thermal and electromigration time constants (τ4 and τEM respectively) as the only adjustable 
parameters. The results of these fits are shown as solid curves in Fig. 3, and demonstrate 
excellent agreement between the measured correlation functions and those predicted by the 
Langevin theory (eqn. 4). As can be seen in fig. 4, the two fits yield clear χ2 minima for τEM of 
16s and 52s, respectively for the down-hill and up-hill steps.  Measurements on two additional 
steps subject to up-hill current stressing at a higher nominal current density (Jnom = 4x105A/cm2) 
gave fitting results similar to those of Fig. 3, with time constants of 98s (325K) and 32s (350K).  
The uncertainties in the fit parameters were 15-40%.  The correlation function for an unstressed 
sample measured at 325K is also shown in Fig. 3.  There is no minimum in the chi-squared value 
for the fit as a function of electromigration time constant, with virtually no change in the 
goodness of fit occurring for values larger than 9000s (see fig. 4).  This shows that data is well fit 
by a single parameter, τ4, as expected when no electromigration force is acting (i.e. 
#$).  
Similar results were consistently obtained for other steps measured without current stressing.   
The electromigration force is found using the measured values of τEM in Eqs. 1 and 4, 
given the step stiffness 
˜ " .  The stiffness is calculated 37, 38 using an effective kink energy of ε = 
0.117 eV 39, 40.  The four measurements of the electromigration time constant yield average 
values of the force per step-edge atom of -2.7x10-5 eV/nm for Jnom= 4x105A/cm2 (325-350K) and 
-9.7x10-6 eV/nm for Jnom= 1x105A/cm2 (370K).   Thus, within the cumulative uncertainties in the 
measured forces of ±50%, the force increases in direct proportion to the current density.  
The measured values of the force can be used to determine the effective valence Z* if the 
local surface current density and the resistivity in the surface region are known.   Since these 
local quantities cannot be measured directly, we first estimate them using the bulk current 
densities and the bulk resistivity of Ag, which is approximately 1.8x10-6Ω-cm at 325K and 
2.2x10-6Ω-cm at 370K 41, 42.  The resulting effective valence, obtained using the nominal current 
density, is Z* = -(4±2)x102.   The magnitude is substantially larger than the predicted effective 
valence of an isolated Ag adatom on Ag(111), which is is Z*=
"19  43. For atoms in a close-
packed site along a step edge, with a perpendicular current direction, the direct force per step 
atom may be as much as 2x higher than the force on an adatom 21, 23, which would yield a 
predicted valence of Z* ~ -38, still much smaller magnitude than the measured value.  A 
substantial systematic effect can be attributed to the film cross section, because as described 
earlier, at 100 nm film thickness there are vacancies in the film up to 50% of the surface area.  
Therefore the bulk current densities may be as much as 2x higher than the nominal values, and as 
a result the lower limit on the effective valence is Z* = -(2±1)x102.  The remainder of the 
difference compared with the perpendicular force on a close-packed step-edge atom may arise 
from the highly kinked environment suggested in Fig. 1.  Because diffusion is parallel to the step 
edge, only the component of the electromigration force tangential to the local step orientation 
will affect step-edge diffusion.  The largest impact of the electromigration force will thus occur 
for the most highly kinked step regions.  There have been no calculations of the electromigration 
force acting on low-symmetry kink sites.  However, such sites have enhanced valence charge 
density 44, and also present anomalous barriers to step-edge diffusion 45.  Such significant 
changes in local electronic structure may be reflected in significant changes in the scattering 
cross section (and thus Z* value).  In addition, the geometric effect of the kink configuration is 
likely to enhance scattering via blocking 21, or constriction-induced enhancement of local current 
density 46 analogous to current crowding 47, 48. 
The forces measured above are related to equal and opposite forces on the charge 
carriers, which translate into changes in the surface resistivity 22, 43.  This can be evaluated by 
treating the diffusing step-edge atoms as independent scattering sites of density nk = (LkLstep)-1, 
where Lk is the average distance between diffusing atoms along the step edge and Lstep is the 
average distance between steps.  Then the change in the surface resistivity ρs due to the diffusing 
step-edge atoms is 43: 
, , ,   (5) 
where Lf is the film thickness,  nAg = 58.5 nm-3 is the bulk carrier density for Ag, 
k is the wind 
force acting per atom (our measured value) and 
j  are additional changes in force on the 
carriers due to the perturbation of atomic structure in the immediate vicinity of the step-edge 
atoms.  Using the measured force per atom and the upper limit of the current density yields the 
component of the surface resistivity due to scattering at the diffusing step-edge atoms alone, 
~ (3±1.5nm
, where ρ0 is the bulk value of the Ag resistivity.   
The impact of diffusing step-edge atoms on surface resistivity will include the direct-
scattering term measured here as well as the perturbed lattice terms (
k in Eq. 5).  The latter 
term may contribute as much as 2/3 of the total resistivity change, thus it is reasonable to expect 
that scattering at the step-edge will contribute a total change in the surface resistivity 
$ (10 ± 5nm
.  As an example, for moderate step and kink densities (Lstep =10 nm and 
Lk =2 nm) and a very thin film (Lf = 10 nm), the change in surface resistivity due to scattering at 
the step-kink sites could be as large as 10% of the bulk resistivity.  This effect will be significant 
in nanoelectronic devices carrying large current densities 49, 50.  
The present observation of biased temporal fluctuations under current stress is the first 
direct correlation of nanoscale structural fluctuations with the electromigration wind force.  
Because of systematic uncertainty in the actual current density, analysis of the results yields a 
lower limit for the magnitude of the electron wind force on the diffusing step-edge atoms.   
Taking the one-sigma limits on the statistical uncertainties, we find that the value is five to 
fifteen times larger than that on individual adatoms 43.  Effects on resistivity, noise and 
electromigration susceptibility in metal nanoelectronic structures will thus be concomitantly 
higher than would have been expected 51. 
Acknowledgments 
This work was supported by the U.S. Department of Energy Award No. DOE-FG02-01ER45939.  
We also gratefully acknowledge support and SEF support from the NSF MRSEC under grant 
DMR 05-20471. 
Figure Captions 
Fig. 1:  Schematic illustration of current flow perpendicular to average orientation of steps on the 
surface.  Enhanced scattering from step sites at the surface is suggested by the arrows.  The inset 
illustrates adatoms on the terraces and the kinked (thermally roughened) step edge.   
Fig. 2: STM data for a current stressed sample. Applied current was 0.4 A and nominal sample 
cross section 200 nm thick x 2 mm wide.  Sample temperature = 380 K.  Upper panel – repeated 
scans across the edges of three steps (pseudo image) (from left to right one downhill step and 
two uphill steps) show fluctuations of step position in time x(t). Lower panel – height vs. 
position across the image is consistent with steps a single layer (0.236 nm) high. 
Fig. 3: Time correlation functions, G(t) in units Å2, for the step fluctuations (x(t) data) extracted 
from repeated measurements as shown in Fig. 1 and described in the text.  The data for the step-
down current (open circles) is the average of 10 separate measurements, and for the step-up 
current (open squares) is the average is over 9 data sets.  Fits to Eq. 4 are shown as the solid lines 
for each data set. Also shown is the measured correlation function (average of 19 separate 
measurements) for the unstressed sample (open triangles) fit to a single power law. 
Fig. 4  The reduced chi-squared plotted as a function of  τEM, while holding the value of τ4 at its 
optimum value.  Solid curve: step-down current. Dot-dashed curve: step-up current. Dashed 
curve: unstressed data.  Note that, for the unstressed data, there is no clear minimum value of 
τEM, i.e. a fit to a single parameter (τ4) suffices to describe the data.   
Fig. 1:  Schematic illustration of current flow perpendicular to average orientation of steps on the 
surface.  Enhanced scattering from step sites at the surface is suggested by the arrows.  The inset 
illustrates adatoms on the terraces and the kinked (thermally roughened) step edge. 
Fig.2: (Color on-line)  STM data for a current stressed sample. Applied current was 0.4 A and 
nominal sample cross section 200 nm thick x 2 mm wide.  Upper panel – repeated scans across 
the edges of three steps (pseudo image) (from left to right one downhill step and two uphill 
steps) show fluctuations of step position in time x(t). Lower panel – height vs. position across the 
steps of single layer height (0.236 nm). 
Fig. 3: (Color on-line)  Time correlation functions, G(t) for the step fluctuations (x(t) data) 
extracted from repeated measurements.  The data for the step-down current (open circles) is the 
average over 10 separate measurements, and for the step-up current (open squares) the average is 
over 9.  Fits to Eq. 3 are shown as the solid lines for each data set. Also shown is the measured 
correlation function (average of 19 separate measurements) for an unstressed sample (open 
triangles) fit to a single power law. 
0 20 40 60 80 100
Fig. 4  (color on-line)The reduced chi-squared plotted as a function of  τEM, while holding the 
value of τ4 at its optimum value.  Solid curve: step-down current. Dot-dashed curve: step-up 
current. Dashed curve: unstressed data.  Note that, for the unstressed data, there is no clear 
minimum value of τEM that is consistent with the data, i.e. a fit to a single parameter (τ4) suffices 
to describe the data.   
References  
1 V. V. Zhirnov and R. K. Cavin, Nature Materials 5, 11 (2006). 
2 H. Basch, R. Cohen, and M. A. Ratner, Nano Letters 5, 1668 (2005). 
3 C. Tao, , Nano Letters, in press (2007). 
4 A. Bid, A. Bora, and A. K. Raychaudhuri, Phys. Rev. B 72, 113415 (1 (2005). 
5 C.-H. Zhang, F. Kassubek, and C. A. Stafford, Phys. Rev. B 68, 165414 (1 (2003). 
6 H. Yasunaga and A. Natori, Surface Sci. Rep. 15, 205 (1992). 
7 P. J. Rous, T. L. Einstein, and E. D. Williams, Surface Sci. 315, L995 (1994). 
8 M. R. Gungor and D. Maroudas, Surface Sci. 415, L1055 (1998). 
9 O. Pierre-Louis and T. L. Einstein, Phys. Rev. B62, 13697 (2000). 
10 P. Kuhn, et al., Phys. Rev. Lett. 94 (2005). 
11 S. Heinze, N.-P. Wang, and J. Tersoff, Phys. Rev. Lett. 95, 186802 (2005). 
12 B. Stahlmecke, et al., Appl. Phys. Lett. 88, 053122 (2006). 
13 G. Essen and M. S. Fuhrer, Appl. Phys. Lett. 87, 263101 (2005). 
14 K.-F. Braun, et al., Appl. Phys. Lett. 90, 023118 (2007). 
15 J. Pelz and J. Clarke, Phys. Rev. B 55, 738 (1985). 
16 A. Bid, A. Bora, and A. K. Raychaudhuri, Nanotechnology 17, 152 (2006). 
17 M. Ishigami, et al., Appl. Phys. Lett. 88, 203116 (1 (2006). 
18 R. S. Sorbello and C. S. Chu, IBM J. Res. Develop. 32, 58 (1988). 
19 K. N. Tu, Phys. Rev. B45, 1409 (1992). 
20 P. S. Ho and T. Kwok, Rep. Prog. Phys. 52, 301 (1989). 
21 P. J. Rous, Phys. Rev. B59, 7719 (1999). 
22 H. Ishida, Phys. Rev. B 60, 4532 (1999). 
23 P. J. Rous, Phys. Rev. B 61, 8484 (2000). 
24 A. V. Latyshev, et al., Surface Sci. 227, 24 (1990). 
25 K. Yagi, Surface Sci. Rep. 17, 305 (1993). 
26 K. Thürmer, et al., Phys. Rev. Lett. 83, 5531 (1999). 
27 D.-J. Liu and J. D. Weeks, Phys. Rev. B57, 14891 (1998). 
28 H.-C. Jeong and E. D. Williams, Surface Sci. Rep. 34, 171 (1999). 
29 T. Ihle, C. Misbah, and O. Pierre-Louis, Phys. Rev. B 58, 2289 (1998). 
30 M. Giesen, Prog. Surface Sci. 68, 1 (2001). 
31 M. Ondrejcek, et al., Phys. Rev. B 73, 035418 (2006). 
32 F. Mugele, et al., Surface Sci. 400, 80 (1998). 
33 O. Bondarchuk, et al., Phys. Rev. B 71, 045426 (2005). 
34 P. J. Rous and T. Bole, submitted (2007). 
35 G. S. Bales and A. Zangwill, Phys. Rev. B41, 5500 (1990). 
36 A. Baski and H. Fuchs, Surface Sci. 313, 275 (1994). 
37 N. Akutsu and Y. Akustsu, J. Physics:  Condensed Matter 11, 6635 (1999). 
38 M. Nowicki, et al.l, Phys. Rev. B 67 (2003). 
39 C. Steimer, et al., Phys. Rev. B 64, 085416/1 (2001). 
40 T. J. Stasevich, et al., Phys. Rev. B 71, 245414(11) (2005). 
41 B. S. Verma and H. J. Juretschke, J. Appl. Phys. 41, 4732 (1970). 
42 R. A. Matula, J. Phys. Chem Ref. Data 8, 1260 (1979). 
43 M. F. G. Hedouin and P. J. Rous, Phys. Rev. B 62, 8473 (2000). 
44 F. Mehmood, A. Kara, and T. S. Rahman, Surface Sci. 600, 4501 (2006). 
45 J. Ikonomov, K. Starbova, and M. Giesen, Surface Sci. 601, 1403 (2007). 
46 N. Sai, et al., Cond-mat 07016341, 1 (2007). 
47 M. R. Gungor and D. Maroudas, J. Appl. Phys. 85, 2233 (1999). 
48 P. J. Rous, et al.s, J. Appl. Phys. 95, 2477 (2004). 
49 C. Durkan and M. E. Welland, Phys. Rev. B 61, 14215 (2000). 
50 W. Wu, et al., Appl. Phys. Lett. 84, 2838 (2004). 
51 D. Schumacher, Surface scattering experiments with conduction electrons (Springer, 
Dusseldorf, 1995).
