Pressure and Stress Effects on Atomic Transport in Semiconductors
last updated 1/19/2000
Pressure and Stress Effects on Atomic Transport in Semiconductors
last updated 1/19/2000
Contents:
The long-term objective of this research program is a fundamental understanding
of thin film crystallization and dopant diffusion processes in electronic
materials. A basic understanding of these processes becomes more essential
as semiconductor device dimensions become progressively smaller and fabrication
tolerances become more exacting. To design processes for the upcoming generation
of silicon integrated circuit devices, p/n junction depths will have to
be predicted to better than a few hundred Å - this will be exceedingly
difficult without a fundamental understanding of the concentrations and
mobilities of the point defects involved and their roles in diffusion.
Although electronic devices will never be fabricated under high pressure,
high
pressure can be used to perturb the rates of diffusion and crystallization,
thus providing direct information about the nature of the dominant point
defects involved.
Today all Si integrated circuits are manufactured using ion implantation
followed by high temperature annealing to restore the properties of the
perfect crystal. High implant doses are used to create transistor source
and drain regions for metal contact; low doses are used in the transistor
channels. Low doses introduce defects, the annealing of which involves
solid state diffusion. High doses create an amorphous surface layer, the
annealing of which involves solid phase epitaxial crystallization. Furthermore,
solid phase epitaxy is being investigated as a potential method for epitaxial
lateral overgrowth for single-crystal or large-grain SOI (silicon on insulator)
thin-film transistors.
In addition, nonhydrostatic stresses in crystalline strained-layer epitaxial semiconductors can be very large. An understanding of their effects on diffusion and crystallization is therefore an important part of the study of the stability of these materials. Although many are studying the effects of these stresses on the energetics of growth and other kinetic processes in strained epitaxial films, we could very well be the only group studying the stress effects on the atomic or interfacial mobilities.
The energetics and mechanisms of atomic diffusion control the kinetics
of such diverse phenomena as the fabrication of semiconductors and superconductors,
the tempering of steel, geological metamorphism, the precipitation hardening
of non-ferrous alloys, and corrosion of metals and alloys.
The nature of the defects that govern diffusion in semiconductors is
of considerable current interest. Self and dopant diffusivities in Si have
been very carefully and thoroughly tabulated. However, when one diffuses
two different species (say a donor and an acceptor) simultaneously into
the same sample, in general one cannot predict the results. (For this reason
there is a field of research called ÒcodiffusionÓ [1-3] The
fluxes are coupled because both species interact with the point defect
populations, and those populations are not understood. (For example, the
reported equilibrium vacancy concentrations in Si at 1300 K vary by five
orders of magnitude; the discrepancies tend to get still worse at lower
temperatures. ) This situation is complicated in Si because both vacancy
and interstitialcy mechanisms, as well as a direct interchange mechanism,
are quite plausible. The predicted activation energies for these three
mechanisms are very similar [4], so measurements of the temperature dependence
of the diffusivity are insufficient to distinguish the relative contributions
from these mechanisms. And there are far too few point defects to permit
a reliable direct measurement, such as a Simmons-Balluffi dilatometry and
diffraction experiment [5], of their concentrations.
Although a number of ideas about these defects have been proposed, the
picture is far from complete [6]. There is indirect evidence concerning
the dominant mechanisms from several sources. Non-Fickian profiles have
been predicted by models for hybrid mechanisms such as the kick-out (impurity
sits on a lattice site until it is kicked out by a Si self-interstitial,
whereupon it hops from interstitial site to interstitial site many times
before replacing a Si atom on a lattice site); fits of depth profiles and
their time-dependence have provided evidence about both the dominant impurity
transport mechanism and the self-diffusion mechanism. Additionally, measurements
have been performed under conditions of nonequilibrium point-defect concentrations,
or diffusion in samples with shifted Fermi levels due to the presence of
other dopants. For example, diffusivities change during oxidation or nitridation
of the surface (which are believed to inject self-interstitials and vacancies,
respectively, into the crystal based on observations of stacking fault
growth). In all of these cases, however, the interpretations are model-dependent,
and in most of them the results have significant uncertainties and are
sometimes contradictory [7]. The interpretation of our hydrostatic-pressure
results is far more direct.
Unlike the temperature-dependence, a measurement of the pressure-dependence
of diffusion is normally qualitatively different for the mechanisms
under consideration. The application of hydrostatic pressure will increase
the equilibrium concentration of interstitials and will decrease
that of vacancies through the action of the pDV
work term in the free energy of formation.
The effect of pressure on the diffusivity, D, is characterized by the activation volume, DV* or V*,
which is the pressure-analogue of the familiar activation energy DE* that characterizes the temperature-dependence of D. Unlike DE*, DV* can be either positive or negative, depending upon whether D decreases or increases with p, respectively. For a point defect mechanism, DV* is normally the sum of two components:
where DVf, the formation volume, is the volume change in the system upon formation of a defect in its standard state, and DVm, the migration volume, is the additional volume change when the defect reaches the saddle point in its migration path. The formation volume characterizes the pressure-dependence of the equilibrium point defect concentration while the migration volume characterizes the pressure-dependence of the defect mobility.DV* = DVf + DVm ,
Fig.
1. Basic atomic diffusion mechanisms. Top: vacancy mechanism;
Bottom: interstitialcy mechanism. Pressure effect is determined by volume
changes of sample upon defect formation (DVf)
and migration (DVm), which
are shown. To create a vacancy one moves an atom from the interior to a
surface site; to create an interstitial one moves an atom from a surface
site to the interior. Formation volume is ±W
+ DVr, where W
is the lattice site volume and the plus sign is for vacancy formation and
the minus sign is for interstitial formation.
DVr
is the amount of outward relaxation (if the relaxation is inward, DVr
is negative) around the newly-created point defect. The migration volume,
DVm,
is the additional volume change of the sample when the defect moves up
to the saddle-point of its migration path. In principle, DVm
and DVr can be calculated
reliably nowadays using ab-initio total-energy calculations. There is an
early calculation of DVr
for these defects in Si but none of DVm.
In practice, it is often assumed that they are both Ç W
in magnitude so that the effect of pressure on the diffusivity is dominated
by the ± W
term.
If the formation volume is ± W
and the migration volume is zero then we should expect large qualitative
differences between the pressure effects on a vacancy and an interstitialcy
mechanism. Under these circumstances even a large experimental uncertainty
is not a significant drawback - for example, a measurement of DV*
= (0.5 ± 0.5) times the atomic volume
(W) is still highly significant when we're trying
to distinguish between theoretical values in the vicinity of +1.0 W
and -1.0 W. In preliminary work using a NaCl
pressure-transmitting medium, we measured pressure-enhanced diffusion of
As in Si [8] and pressure-enhanced self-diffusion in Si [9]. If these results
hold up in planned "cleaner" experiments, the most obvious interpretation
would be in terms of a pressure-induced increase in the interstitial population.
Fig.
2 Pressure effect on diffusion rate of As in Ge in recent "clean"
experiment (pressure medium was clean argon). Simplest model (DVr=0=DVm)
of vacancy mechanism predicts slope of +1.0 W
and simplest model of interstitialcy mechanism predicts slope of -1.0 W;
neither fits data. Refining the predicted slopes requires atomistic calculations
of the relaxation around each defect in its ground-state and saddle-point
configurations; this has not yet been done for Ge.
Since that early work, we have developed much cleaner, hydrostatic pressure
environments for high-temperature annealing. We now use a high-temperature
Diamond Anvil Cell (DAC), cryogenically loaded with liquid argon, to provide
a clean and hydrostatic environment. It is subsequently heated by placing
the entire cell into a furnace, which permits us to know our temperature
really
well and to guarantee its uniformity and reproducibility -- this turns
out to be incredibly important, and temperature profiles from conventional
laser heating techniques just aren't uniform enough.
Fig.
3. Cross-section of high-temperature Diamond Anvil Cell, capable
of 1000 °C. Triangular pressure plates made of Ni-based superalloy
are squeezed together by three rhenium bolts. Pressure chamber is loaded
cryogenically with liquid argon. Entire cell is placed in a furnace for
temperature uniformity. Sm:YAG fluorescence can be used to measure pressure
at high temperature.
With this new tool, we discovered pressure-enhanced diffusion of arsenic
in germanium [10] (Fig. 2). This finding challenges "conventional wisdom"
that germanium is a vacancy diffuser under all circumstances; we may need
to think of diffusion in germanium in all the complexity with which we
now think of diffusion in silicon. Among the potential mechanisms is a
second-neighbor vacancy jump (which we call the "Tarzan mechanism" because
of the swinging path of the diffusing atom; see Fig. 4). It is plausible
that there may be a large, negative migration volume associated with this
trajectory, but a theoretical calculation of the migration volume must
be made in order to check this out.
Fig.
4. "Tarzan mechanism" of 2nd-neighbor vacancy jumps proposed
as possible unconventional diffusion mechanism for As in Ge. Large negative
migration volume is plausible.
We are currently studying the diffusion boron in silicon, which is the
most important dopant technologically. Transient enhanced diffusion after
boron implantation into silicon is making it very difficult to create shallow
p-n junctions in Si integrated circuits. There is mounting evidence that
boron diffuses by an interstitial-based mechanism. If this is the case
then we should expect to see a pressure-enhanced diffusivity.
We had to develop techniques to measure the pressure quickly and reliably
at high temperature in the DAC in order to perform these diffusion studies.
Ordinarily in a DAC one uses ruby fluorescence, the wavelength of which
shifts with pressure. However, above around 200 °C the ruby
signal becomes so faint and broad that it is virtually impossible to use.
Fluorescence from Sm-doped YAG is still going strong as high as 850 °C,
however, and we have now calibrated ten fluorescence peaks vs. T and P
up to 850 °C and 19 GPa. It turns out that we need all ten peaks
in order to get a reliable pressure measurement at high T, as described
in our publication [11] on this technique. Fitting ten fluorescence
peaks which are fairly broad at high T is non trivial and therefore we
are distributing the necessary tools via this website so that others can
readily use Sm:YAG. The calibration spectra and the fitting routines that
we have developed are available for free downloading under "free stuff".
Fig.
5. Furnace and pressure measurement system.
References
Nonhydrostatic Stress Effects1. R. Deaton, U. Gosele and P. Smith, J. Appl. Phys. 67, 1793 (1990).2. C. Gontrand, P. Ancey, H. Haddab and G. Chaussemy, Semiconductor Sci. and Tech. 7, 181 (1992).
3. M. Margesin, R. Canteri, S. Solmi, A. Armiglaato and F. Baruffaldi, J. Mater. Res. 6, 2353 (1991).
4. C.S. Nichols, C.G. VandeWalle and S.T. Pantelides, Phys. Rev. B 40, 5484 (1989).
5. R.O. Simmons and R.W. Balluffi, "Measurement of Equilibrium Vacancy Concentrations in Aluminum", Phys. Rev. 117, 52 (1960).
6. P.M. Fahey, P.B. Griffin and J.D. Plummer, "Point Defects and Dopant Diffusin in Silicon", Rev. Mod. Phys. 61, 289 (1989).
7. P.M. Fahey, G. Barbuscia, M. Moslehi and R.W. Dutton, Appl. Phys. Lett. 46, 784 (1985).
Nonhydrostatic stresses are common in lattice-mismatched heteroepitaxial
thin film growth; they arise even in single-phase materials from composition
gradients or from anisotropic thermal expansion or elasticity in polycrystals.
We recently discovered a set of thermodynamic relations for atomic diffusion
under nonhydrostatic stress. They relate the hydrostatic pressure effect,
the nonhydrostatic stress effect, and volume changes associated with formation
and migration of point defects. It has resolved a simmering controversy
among the people studying diffusion in coherently strained Si-Ge films
grown on Si. The derivation is available in [1], along with a successful
parameter-free prediction for the diffusion of Sb in biaxially-strained
Si. A more thorough explanation is available in [2], along with the application
of the thermodynamic relationships to boron diffusion in Si by an interstitial-based
(interstitialcy or kick-out) mechanism, and an explanation of what further
measurements and calculations are needed to "solve" the system.
Fig.
6. Nonhydrostatic stress effect on vacancy formation: anisotropic
formation volumes. Work done upon vacancy formation in crystal (red) free
of extended internal defects. Pressures, P1 and P3,
of incompressible fluids (blue) in contact with (100) and (001) faces,
respectively, maintained by pistons driven by masses m1
and m3. Crystal volume changes by Vf
upon vacancy formation at (100) or (001) surface but work done against
gravity differs (mgDh = pDV);
hence point defect concentrations in local equilibrium with (100) and (001)
surface differ.
Fig.
7. Nonhydrostatic stress effect on vacancy migration: anisotropic
migration volumes. Suppose you are concerned about diffusion in the [001]
direction in a coherently strained thin film on a thick substrate. The
change in the dimensions of the sample when a vacancy reaches its saddle
point in migration could be very different in the [001] and [100] directions
(as drawn, the migration volume in the [001] direction is large and negative;
the migration volume in the [100] direction is small). Hence different
components of the stress tensor couple differently to the different components
of the migration volume. It turns out that the migration volume is actually
a tensor, called the "migration strain tensor". Because a vacancy in the
diamond cubic lattice jumps along body diagonals (unlike in this cartoon),
crystal symmetry dictates that the migration strain tensor have only one
independent component and therefore no anisotropy; however, the migration
strain tensor for an interstitialcy mechanism has two independent components
and therefore can have a significant anisotropy.
References :
Crystallization of amorphous semiconductors1. M.J. Aziz, "Thermodynamics of Diffusion under Pressure and Stress: Relation to Point Defect Mechanisms", Applied Physics Letters 70, 2810 (1997).
Crystallization of amorphous Si is an ideal model system for studies
of growth of technologically useful semiconductors in particular and for
crystal growth in general. It is one of the few elemental systems in which
a planar interface between a crystal and an amorphous solid phase can be
produced and studied. The lack of a crystal structure in one of the phases
reduces the number of variables to a tractable number. The important thermodynamic
properties of amorphous Si are known, which eliminates guesswork from thermodynamic
modeling. The solid phase epitaxial growth (SPEG) rate had been carefully
measured at atmospheric pressure over a wide range of doping levels and
temperatures, and the resulting crystal is free of extended defects that
can complicate the growth kinetics. All these measurements still do not
permit a determination of the growth mechanism. Various models for growth
[1] relate the velocity to: a) the Fermi level, b) strain from impurity-size
mismatch, c) the concentration of vacancies in the crystal, d) the concentration
of interstitials in the crystal, e) the concentration of dangling bonds
at the interface, f) the concentration of dangling bonds in the amorphous
phase, g) the concentration of "floating bonds" (interstitial-like defects)
in the amorphous phase. Our experiments using pressure and nonhydrostatic
stress have permitted us to test these models and to eliminate most of
them, as described in [1]. Experiments to more severely test the remaining
model and others in its class are under way.
Fig.
8. Ion implantation causes amorphization of crystalline Si.
High-temperature annealing causes Solid Phase Epitaxial Growth (SPEG) which,
under some circumstances, proceeds by the steady motion of a planar interface
and leaves a defect-free single crystal behind. Under other circumstances
the interface does not remain so planar (read on...)
It was in the study of SPEG that we first discovered that nonhydrostatic compression can have the opposite effect of hydrostatic compression on the growth rate [2] as shown in Figs 9 and 10. We also developed a thermodynamic theory to explain this phenomenon [2], illustrated in Fig. 11. We have now discovered that nonhydrostatic stresses can lead to a kinetically-driven shape instability of an initially planar interface [3,4], as shown in Figs. 12-15, and we are developing a general model of the interaction of stress with the mobilities and the energetics of growth processes in general, as shown in Fig. 16.
Fig.
9. First measurement of nonhydrostatic stress effect on crystal
growth rate. (a) Amorphous layers on top and bottom surfaces of elastically
bent wafer grow at different rates: compressive side grows slower and tensile
side grows faster, as shown in (b).
Fig.
10. Newer in-situ time-resolved measurement of growth rate under
stress. As interface moves back towards free surface, alternating constructive
and destructive interference between reflected light waves from interface
and surface cause reflectivity oscillations in time.
Fig
11. . Rate of fluctuation from equiaxed state (a) to "short
and fat" transition state (a*) residing at amorphous/crystal interface
will be enhanced by compressive s33
(normal to interface) but retarded by compressive s11
(along interface); if total volume of system in transition state is smaller
than in starting state then hydrostatic pressure will enhance the rate.
This is what is observed in our experiments.
Fig.
12. Cross-section electron micrograph of Si (001) crystal-amorphous
interface that grew back in non-planar fashion under nonhydrostatic stress.
Corrugation wavelength is about 100 nm.
Fig.
13.. Controlled interface corrugation. This is how we make intentionally
rippled interfaces with a wavelength of our choice, so we can study the
rate of growth or decay of interface ripples vs. both stress and wavelength.
Fig.
14. . Dramatic demonstration that in-plane compressive stress
makes corrugations grow.
Fig.
15.. Interface instabilities under nonhydrostatic stress. (a)
Energetically-driven interface instability. It has long been known that
a corrugation will relieve elastic strain energy in the "tips" of the ripples
(the so-called Asaro-Tiller-Grinfeld instability). (b) We have proposed
a new, kinetically-driven interface instability. Under compressive in-plane
stress, stress concentrations in the grooves slow down the motion of the
grooves relative to the peaks, causing the corrugations to grow. Note that
in tension, (a) still leads to an instability whereas (b) leads to stability.
Fig.
16.. General model of interface motion based on "motion by weighted
mean curvature" [5] for capillarity effects; incorporating effects of stress
on both energetics and kinetics; also incorporating orientation-dependence
of interface speed. Model predicts that compressive stress makes corrugations
grow and stress-free conditions make corrugations shrink, as is observed
experimentally.
Noam Bernstein , a theory student who recently completed his Ph.D. and was jointly supervised by Efthimios Kaxiras and Michael Aziz, has been simulating SPEG using new classical potential called the "Environment Dependent Interatomic Potential" that has been optimized by Martin Bazant and Efthimios Kaxiras. The objective is to simulate the atomic structure and motion at the amorphous/crystal interface (which is too large for ab-initio calculations) to find realistic crystallization reaction pathways, and to compare the activation energy, volume, and strain for these pathways to each other and to the values we have measured in the laboratory.
For QuickTime movies of two of the crystallization pathways ("mechanisms")
that Noam has found and described in his
publication (see publication #114 in this list), click
here
for the "simple" mechanism;
here
for the "complex" mechanism.
Two "side" views of a bunch of atoms at and near the interface doing
interesting things are shown for each "mechanism". Among Noam's notable
achievements are his algorithms for finding interesting sequences of events
in close proximity in space and time and his methods for permitting us
to visualize what is going on in the interior of a condensed phase, in
the midst of lots of other uninteresting things going on at the same
time.
Three-fold coordinated Si atoms are color-coded
red, four-fold (the normal situation) are white, and five-fold are blue.
Note that migration sequences can be described by a "defect" alternating
between 3-fold and 5-fold configurations. This is especially apparent
in the "complex" mechanism. Other defect reactions that can occur:
a pair of neighboring four-fold atoms break the bond between them, thereby
creating a pair of three-fold "defects"; a pair of almost-neighboring four-fold
atoms push together to form a bond between them without breaking any others,
thereby creating a pair of five-fold "defects", etc.
The energy changes and shape changes in the bar
graphs in the QuickTime videos are plotted in his
publication (see publication #114 in this list), and agree remarkably
well with values of the activation energy, activation volume, and other
components of the activation strain tensor that we have measured in our
laboratory. (Remember, with a classical potential, we don't expect
to get perfect quantitative agreement, so we're pretty happy with the qualitative
agreement that we found.)
Noam's Ph.D. thesis is available in our online collection of theses, click here.
References :
Crystallization of quartz and the "free energy catastrophe"1. G.-Q. Lu, E. Nygren and M.J. Aziz, J. Appl. Phys. 70, 5323 (1991). (full paper, see publication #55 in this list)2. M.J. Aziz, P.C. Sabin and G.-Q. Lu, Phys. Rev. B 44, 9812 (1991). (full paper, see publication #49 in this list)
3. W. Barvosa-Carter and M.J. Aziz, "Effect of Non-Hydrostatic Stress on Kinetics and Interfacial Roughness during Solid Phase Epitaxial Growth in Si", Materials Research Society Symposia Proceedings 441, in press (1997) (abstract in HTML) (full paper, see publication #94 in this list).
4. W. Barvosa-Carter, M.J. Aziz, L.J. Gray and T. Kaplan, "Kinetically Driven Growth Instability in Stressed Solids", Physical Review Letters 81, 1445 (1998) (see publication #106 in this list).
5. J. E. Taylor, J.W. Cahn and C.A. Handwerker, "Geometric Models of Crystal Growth", Acta Metallurgica et Materialia 40, 1443 (1992).
Although not a semiconductor, amorphous SiO2 is
a covalent network that is topologically very similar to amorphous Si and
Ge. Earlier work had shown that pressure enhances the quartz growth rate
into amorphous SiO2 by reducing the free energy of
activation (i.e., a negative "activation volume"), just as it does for
Si or Ge. But if pressure reduces the barrier, how high can we crank the
pressure before the barrier goes all the way to zero? We call this a "free
energy catastrophe" because beyond this critical pressure (call it "PC"),
it would appear that the free energy of activation would be negative,
which is nonsense, of course. So what happens beyond PC?
If you keep raising the pressure on amorphous Si or Ge, a structural transition
to a high-pressure metallic phase gets in the way at roughly 10 GPa [1].
Sue Circone, Carl Agee, and Michael Aziz have studied crystal growth in
SiO2 under high pressure; this material has no high-pressure
metallic phase to get in the way. What we found was a steep pressure-enhancement
of the quartz growth rate up to 3 GPa, followed by a dramatic plunge in
the growth rate at yet higher pressures. This is the sought-after experimental
evidence for the pressure-induced vanishing of an atomic migration barrier,
in this case at PC = 3 GPa. What we believe
is going on is shown in Fig. 17. below. The result opens up the possibility
of studying chemical kinetics as barriers vanish under controllable conditions.
It also has important implications for subduction-zone magmatism of silica-rich
magmas, where it may imply that the viscosity decreases with depth until
a depth of 100 km, where the pressure is 3 GPa, and then suddenly starts
to increase. Such behavior may explain why lava reaching the earth's surface
never comes from deeper than 100 km. The results, plus additional information
that we had to cut out of the publication to save space, are available
online [2].
Fig.
17. Pressure-induced vanishing of atomic migration barrier. (a)
Measured quartz growth rate into amorphous SiO2 vs.
pressure at 1200 °C. Sharp peak at 3 GPa might be due to unusual
kinetics or unusual thermodynamics. The latter occurs because the amorphous
phase is more compressible than quartz and therefore the amorphous phase
is more stable than quartz below 0 and above 16 GPa. However, the thermodynamics
alone cannot account for the data as shown in the inset. Our explanation
for this unprecedented sharp peak in the growth rate is described in (b)-(d).
(b) Standard Gibbs free energy vs. configuration for 3 states: A: quartz
+ amorphous SiO2 + interface with no defect; B: same
system with defect (non-bridging oxygen (shown in red in (d)), after O-Si
bond has been broken at O-Si-OH site) in high-volume state; C: same system
with defect in low-volume state. The "reaction coordinate" is a path through
configuration space along which the reaction proceeds; it is proportional
to the number of atoms in the crystal. (c) Pressure dependence: by the
thermodynamic relation (¶G/¶P)T
= V, the slopes of curves A, B, and C are the volumes, VA,
VB, and VC,
of the system in these respective configurations. (d) Possible migration
mechanism, alternating between low-coordination state B (non-bridging O)
and high-coordination state C (overcoordinated Si). Si is at each vertex;
solid circles represent Si-O bonds normal to the page. The migration barrier
is | GC - GB | which vanishes at PC,
as shown in (c). Overall activation barrier in (c) is the greater of {GC,
GB} minus GA , which is smallest at PC.
References
1. G.-Q. Lu, E. Nygren and M.J. Aziz, J. Appl. Phys. 70, 5323 (1991).
Index
Mike Aziz
