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110
RLE Progress Report Number 133
Chapter 1. Statistical Mechanics of Surface Systems
Chapter 1. Statistical Mechanics of Surface Systems
and Quantum-Correlated Systems
Academic and Research Staff
Professor A. Nihat Berker
Graduate Students
Daniel P. Aalberts, Alexis Falicov, William C. Hoston, Jr., Roland R. Netz
Technical and Support Staff
Imadiel Ariel
1 .1 Introduction
Sponsor
Joint Services Electronics Program
Contract DAAL03-89-C-0001
Our objectives are (1) to produce predictive quantitative properties from first principles for surface
systems and systems in which quantum correlations are important, (2) to deduce, using the
renormalization-group
method
of
statistical
mechanics, broadly relevant properties of condensed matter, and to explore their application to
systems of coupled electronic, structural, and magnetic degrees of freedom. In this research, fluctuation dominated - due to finite temperatures,
impurities, and/or constrained environments properties are of paramount interest.
Our recent results, detailed in the following
sections, show that both objectives can be
achieved. We are in the uniquely fortunate position of having integrated the finite-temperature
renormalization-group expertise of our group with
the electronic energy calculations of Professor
Joannopoulos' group. Thus, microscopic theories
can be produced that start with Schroedinger's
equation and end with predictions directly observable in the laboratory.
consequences for the growth of GaAs on Si(100),
DL steps not disturbing the epitaxy conditions,
while SL steps leading to undesirable antiphase
domains. We have predicted, in the variables of
temperature and crystal cut angle, a phase
boundary between these two regimes, which has
been quantitatively confirmed by experiments
(figure 1).
-
SL
a
0
no
0
DL
100
200
I
300
I ,- -f
400
500 600
TEMPERATURE
1.2 Finite-Temperature
Properties of Vicinal Si(100)
Surfaces
In collaboration with Professor J.D. Joannopoulos,
we have combined electronic energy calculations
and finite-temperature statistical mechanics to
study, for the first time, the equilibrium properties
of the Si(100) surface. The occurrence of singlelayer (SL) or double-layer (DL) steps on these purposefully - misoriented surfaces has important
a
(K)
Figure 1. Our calculated phase diagram of vicinal
Si(100) in the variables of crystal cut angle and temperature. The solid curve is our theoretically predicted
line of first-order phase transitions between the singlelayer (SL) and double-layer (DL) stepped surface
phases. Open and solid bars represent experimental
observations of SL and DL stepped surfaces. The bar
at -2.5' in fact represents observation of a mixed phase
of mostly DL steps!
To conduct this work, we were able to formulate a
new Hamiltonian for the temperature-roughened
111
Chapter 1. Statistical Mechanics of Surface Systems
steps that embodied the accurate electronic calculations of the energies and that was amenable to
statistical mechanics calculations. In addition to
the phase diagram, which is consistent with new
experimental data otherwise unexplained and
which brings together into a coherent picture all
the existing data on the domain structure of
stepped Si(100), we have obtained the free
energy, the entropy, as well as the step profiles
(figure 2) which are in good agreement with
experimental observations. For annealed surfaces,
we find that the critical angle at which the transition between SL and DL stepped surfaces occurs
is .2', also in agreement with experiment. Before
we obtained these results, it was erroneously
believed that vicinal Si(100) surfaces have only
one equilibrium phase with only DL steps present!
Our work has directly and immediately motivated a
new set of (confirming) experiments.
T= 400 K
X SA 20
Figure 2. Our calculated step profile on Si(100) at 1
In our theory, straight steps
degree misorientation.
occur at the horizontal boundaries of the figure. This
picture is in remarkable agreement with subsequent
observations using scanning tunneling microscopy.
1.3 Impurity-Induced Critical
Behavior
Another recent theoretical prediction that we made
using the renormalization-group method appears
to have general and far-reaching consequences:
We discovered that even an infinitesimal amount
of randomness in interactions (e.g., distribution of
defects), in surface systems, converts first-order
phase transitions, characterized by discontinuities,
to second-order phase transitions, characterized by
infinite response functions. In bulk systems, a
(calculable) threshold randomness is needed for
this conversion to occur. This general prediction
appears to be supported by experiments on doped
KMnF 3 and by most recent computer simulations.
112
RLE Progress Report Number 133
More specifically, temperature-driven first-order
phase transitions that involve a symmetry breaking
are converted to second-order phase transitions by
the introduction of infinitesimal bond randomness,
2 or d < 4 respectively
in spatial dimensions d
for systems composed of discrete or continuous
Even strongly
microscopic degrees of freedom.
first-order transitions undergo this conversion to
second order! Above these dimensions, this phenomenon still occurs, but requires a threshold
amount of bond randomness. For example, under
bond randomness, the phase transitions of q-state
Potts models, widely encountered in the context of
structural and magnetic transitions, are second
order for all q in d < 2. If no symmetry breaking is
involved, temperature-driven first-order phase transitions are eliminated under the above conditions.
Another consequence of this phenomenon is that
bond randomness drastically alters multicritical
phase diagrams. For example, tricritical points and
2) or
critical endpoints are entirely eliminated (d
depressed in temperature (d > 2 for both discrete
and continuous degrees of freedom). These predictions have been confirmed by a renormalization-group calculation. Similarly, bicritical phase
diagrams are converted (d < 2) reentrant-disorder-line or decoupled-tetracritical phase diaThese quenched-fluctuation- induced
grams.
second-order transitions constitute a diametric
opposite to the previously known annealedfluctuation-induced first-order transitions, and
point to a multitude of new universality classes of
criticality, including many experimentally accessible cases.
This general result should have applications to
crystals used as probing devices. These probing
crystals are plagued by first-order phase transitions
with non-equilibrium hysteresis loops. It would be
useful to replace these transitions, via the controlled introduction of randomness (which, we
think, could be achieved by crossed laser beams
reflected from rough surfaces), by second-order
phase transitions with large response functions.
1.4 Monte Carlo Mean-Field
Theory and Frustrated Systems
in Two and Three Dimensions
We have recently developed a new method of statistical mechanics, merging the effective-field and
Monte Carlo approaches. This method brings for
the first time to effective-field theory the hard-spin
condition, essential to (frustrated) spin systems
with competing interactions, and uses much less
This
sampling then Monte Carlo simulation.
method was successfully tested on frustrated Ising
magnets in d=2 and 3, in zero and non-zero
uniform fields. The phase diagram of the d=2 tri-
Chapter 1.
angular antiferromagnet was easily obtained with
remarkable global quantitative accuracy.
The
phase diagram of the d=3 stacked triangular
antiferromagnet was found to show three ordered
phases, in a new finite-field multicritical topology
of lines of XY, Ising, and 3-state Potts transitions,
accessible to experiments with layered magnets.
This result also explains for the first time critical
exponents measured at zero field, via crossover
phenomena.
Our new method, thus applicable to frustrated
system, will be developed towards frustrated
quantum spins, which is relevant to high-temperature superconducting systems.
1.5 Quantum Systems
One aim of our research program is to effect the
statistical mechanics of quantum mechanical
systems. We now report encouraging preliminary
results.
Progress is achieved by systematically
mapping d-dimensional quantum mechanical
systems onto (d+1)-dimensional classical systems,
but with complicated many-body interactions.
Figures 3 and 4 show our calculated results for
chains of s=1/2 spins exchange-coupled via their
x and y components (known as the XY magnet).
Figure 3 shows that our successive approximations
to the internal energy systematically and quickly
converge to the exact result. Moreover, exact
information available to-date on quantum systems
is very limited and piecemeal. By contrast, the statistical mechanical solution of the (d+1) dimensional system in our procedure is an entire
solution, providing every equilibrium property of
the original quantum system.
Thus, figure 4
shows the correlation function of the XY chain, for
which no information had been available.
Statistical Mechanics of Surface Systems
I.
A
x
9
bx
V
0.8
0.6
o04
0.2
I0.
30.
20.
40.
50.
r
Figure 4. Our calculated correlation functions of the
quantum XY spin chain as a function of spin separation. The curves are for temperatures T = J/ks, 2J/kB,
3J/kB, and 5J/kB, where J is the exchange coupling
constant. No exact result exists for the correlation
function.
We plan to extend our calculations to twodimensional quantum systems.
One system of
interest is the d=2 XY magnet, in which the occurrence of a distinctive algebraically ordered phase is
controversial. Another system of interest is the
d=2 triangular Heisenberg antiferromagnet, which
is relevant to high-temperature superconductors.
In the latter systems, the interplay of frustration
and thermal vacancies is crucial. We believe that
our previous work on frustrated triangular systems
(Section 4 above) and on thermal vacancies will
be helpful.
By applying renormalization-group
statistical mechanics, we should be able to include
even the weak interplanar coupling of the real
materials. The subsequent aim of our studies is to
include both particle and spin degrees of freedom
in considering quantum-mechanical electronic
systems.
Publications
k8 T/J
0.5
I.
1.5
2.
-0.5.
-0.75- .--
-1.25
-1.5-1.75.
-2.
Figure 3. Internal energy of the quantum XY spin
chain as a function of temperature. The upper curve is
the exact result. The lower curves are our calculations
of systematically improved approximations.
Alerhand, O.L., A.N. Berker, J.D. Joannopoulos, D.
Vanderbilt, R.J. Hamers, and J.E. Demuth.
"Finite-Temperature Phase Diagram of Vicinal
Si(100) Surfaces." Phys. Rev. Lett. 64 (20):
2406-2409 (1990).
Alerhand, O.L., A.N. Berker, J.D. Joannopoulos,
and D. Vanderbilt. "Phase Transitions on Misoriented Si(100) Surfaces." In 20th International
Conference
on
the
Physics
of
Semiconductors. Eds. E.M. Anastassakis and
J.D. Joannopoulos. Singapore: World Scientific, 1990.
Alerhand, O.L. "Equilibrium Properties of Steps on
Si(100) Surfaces." Paper to be presented at the
General Meeting of the American Physical
113
Chapter 1. Statistical Mechanics of Surface Systems
Society, Cincinnati, Ohio, March 18-22, 1991.
Bull. Am. Phys. 36(3): 587 (1991).
Berker, A.N. "Harris Criterion for Direct and
Orthogonal Quenched Randomness." Phys.
Rev. B 42(13): 8640-8642 (1990).
Alerhand, O.L., A.N. Berker, J.D. Joannopoulos, D.
Vanderbilt, R.J. Hamers, and J.E. Demuth.
"Alerhand et al. Reply." Phys. Rev. Lett. 66(7):
962 (1991).
Hui, K., and A.N. Berker. "Random-Field Mechanism in Random-Bond Multicritical Systems."
J. Appi. Phys. 67(9): 5991 (1990).
of
K. Hui. "Absence
and
Berker, A.N.,
Temperature-Driven First-Order Phase Transitions in Systems with Random Bonds." In
Science and Technology of Nanostructured
Magnetic Materials. Eds. G.C. Hadjipanayis, G.
Prinz, and L. Paretti. New York: Plenum, 1990.
McKay, S.R., and A.N. Berker. "Magnetization of
the d-Dimensional Random-Field Ising Model:
An Intermediate Critical Dimension." In New
Trends in Magnetism. Eds. M.D. CoutinhoRezende. Singapore: World
Filho and S.M.
Scientific, 1990.
Berker, A.N. "Quenched Fluctuation Induced
Second-Order Phase Transitions." Paper to be
presented at the General Meeting of the American Physical Society, Cincinnati, Ohio, March
18-22, 1991. Bull. Am. Phys. 36(3): 439
(1991).
Netz, R.R., and A.N. Berker. "Monte Carlo MeanField Theory and Frustrated Systems in Two
and Three Dimensions." Phys. Rev. Lett. 66(3):
377-380 (1991).
114
RLE Progress Report Number 133
Netz, R.R. Frustration in Magnetic, Liquid Crystal,
and Surface Systems: Monte Carlo Mean-Field
Theory. S.M. thesis, Dept. of Physics, MIT,
1991.