Lower Critical Dimension Research Articles

Bicontinuous microemulsion in binary blends of complementary diblock copolymers.

The phase behavior of binary blends of AB diblock copolymers of compositions f and 1 - f is examined using field-theoretic simulations. Highly asymmetric compositions (i.e., f ≈ 0) behave like homopolymer blends macrophase separating into coexisting A- and B-rich phases as the segregation is increased, whereas more symmetric diblocks (i.e., f ≈ 0.5) microphase separate into an ordered lamellar phase. In self-consistent field theory, these behaviors are separated by a Lifshitz critical point at f = 0.2113. However, its lower critical dimension is believed to be four, which implies that the Lifshitz point should be destroyed by fluctuations. Consistent with this, it is found to transform into a tricritical point. Furthermore, the highly swollen lamellar phase near the mean-field Lifshitz point disorders into a bicontinuous microemulsion (BμE), consisting of large interpenetrating A- and B-rich microdomains. BμE has been previously reported in ternary blends of AB diblock copolymer with its parent A- and B-type homopolymers, but in that system the homopolymers have a tendency to macrophase separate. Our alternative system for creating BμE is free of this macrophase separation.

Gross-Neveu-Heisenberg criticality from 2+ε expansion

The Gross-Neveu-Heisenberg universality class describes a continuous quantum phase transition between a Dirac semimetal and an antiferromagnetic insulator. Such quantum critical points have originally been discussed in the context of Hubbard models on $\pi$-flux and honeycomb lattices, but more recently also in Bernal-stacked bilayer models, of potential relevance for bilayer graphene. Here, we demonstrate how the critical behavior of this fermionic universality class can be computed within an $\epsilon$ expansion around the lower critical space-time dimension of two. This approach is complementary to the previously studied expansion around the upper critical dimension of four. The crucial technical novelty near the lower critical dimension is the presence of different four-fermion interaction channels at the critical point, which we take into account in a Fierz-complete way. By interpolating between the lower and upper critical dimensions, we obtain improved estimates for the critical exponents in 2+1 space-time dimensions. For the situation relevant to single-layer graphene, we find an unusually small leading-correction-to-scaling exponent, arising from the competition between different interaction channels. This suggests that corrections to scaling may need to be taken into account when comparing analytical estimates with numerical data from finite-size extrapolations.

Lower critical dimension of the random-field XY model and the zero-temperature critical line.

The random-field XY model is studied in spatial dimensions d=3 and 4, and in between, as the limit q→∞ of the q-state clock models, by the exact renormalization-group solution of the hierarchical lattice or, equivalently, the Migdal-Kadanoff approximation to the hypercubic lattices. The lower critical dimension is determined between 3.81<d_{c}<4. When the random field is scaled with q, a line segment of zero-temperature criticality is found in d=3. When the random field is scaled with q^{2}, a universal phase diagram is found at intermediate temperatures in d=3.

Critical dynamics of the antiferromagnetic O(3) nonlinear sigma model with conserved magnetization.

We study the near-equilibrium critical dynamics of the O(3) nonlinear sigma model describing isotropic antiferromagnets with a nonconserved order parameter reversibly coupled to the conserved total magnetization. To calculate response and correlation functions, we set up a description in terms of Langevin stochastic equationsof motion, and their corresponding Janssen-De Dominicis response functional. We find that in equilibrium, the dynamics is well-separated from the statics, at least to one-loop order in a perturbative treatment with respect to the static and dynamical nonlinearities. Since the static nonlinear sigma model must be analyzed in a dimensional d=2+ɛ expansion about its lower critical dimension d_{lc}=2, whereas the dynamical mode-coupling terms are governed by the upper critical dimension d_{c}=4, a simultaneous perturbative dimensional expansion is not feasible, and the reversible critical dynamics for this model cannot be accessed at the static critical renormalization group fixed point. However, in the coexistence limit addressing the long-wavelength properties of the low-temperature ordered phase, we can perform an ε=4-d expansion near d_{c}. This yields anomalous scaling features induced by the massless Goldstone modes, namely subdiffusive relaxation for the conserved magnetization density with asymptotic scaling exponent z_{Γ}=d-2, which may be observable in neutron scattering experiments. Intriguingly, if initialized near the critical point, the renormalization group flow for the effective dynamical exponents recovers their universal critical values z_{c}=d/2 in an intermediate crossover region.

Extreme contact shrink for back end of line connectivity

For the past 50 years, Moore’s law has been well followed by the semiconductor industry. The scaling of transistors and interconnects has been enabled not only by various technological advancements but also by novel patterning approaches. However, in order to keep up with Moore’s law, further shrinking at all levels of the integrated circuit is needed. Among them is the back end of line (BEOL), where increasingly smaller metal pitches require tight specifications for vias connecting metal lines. In this paper, BEOL via shrink options is investigated, targeting the bottom critical dimension (CD) 10.5 nm in order to land on metal pitch 21 nm lines below, while maintaining low defectivity, as well as low global and local CD uniformity (CDU and LCDU, respectively). Approaches to this shrink consist of modifications to the etch chemistry at different levels of the mask etch and liner-assisted shrink, either organic or inorganic. Numerical analysis of CD-scanning electron microscopy (CDSEM) images quantitatively shows the efficiency of different approaches via shrink, together with associated CDU, LCDU, and defectivity values. CDSEM results are supplemented by large-area voltage contrast defectivity and transmission electron microscopy data sets.

Universal properties of the Kardar-Parisi-Zhang equation with quenched columnar disorders.

Inspired by the recent results on totally asymmetric simple exclusion processes on a periodic lattice with short-ranged quenched hopping rates [A. Haldar and A. Basu, Phys. Rev. Research 2, 043073 (2020)2643-156410.1103/PhysRevResearch.2.043073], we study the universal scaling properties of the Kardar-Parisi-Zhang (KPZ) equation with short-ranged quenched columnar disorder in general d dimensions. We show that there are generic propagating modes in the system that have their origin in the quenched disorder and make the system anisotropic. We argue that the presence of the propagating modes actually make the effects of the quenched disorder irrelevant, making the universal long wavelength scaling property belong to the well-known KPZ universality class. On the other hand, when these waves vanish in a special limit of the model, new universality class emerges with dimension d=4 as the lower critical dimension, above which the system is speculated to admit a disorder-induced roughening transition to a perturbatively inaccessible rough phase.

Disorder-Induced Long-Ranged Correlations in Scalar Active Matter.

We study the impact of quenched random potentials and torques on scalar active matter. Microscopic simulations reveal that motility-induced phase separation is replaced in two dimensions by an asymptotically homogeneous phase with anomalous long-ranged correlations and nonvanishing steady-state currents. Using a combination of phenomenological models and a field-theoretical treatment, we show the existence of a lower-critical dimension d_{c}=4, below which phase separation is only observed for systems smaller than an Imry-Ma length scale. We identify a weak-disorder regime in which the structure factor scales as S(q)∼1/q^{2}, which accounts for our numerics. In d=2, we predict that, at larger scales, the behavior should cross over to a strong-disorder regime. In d>2, these two regimes exist separately, depending on the strength of the potential.

Quasi-One-Dimensional Ising Models with Defects of the “Random Local Field” Type: the Imry–Ma Phase in Spaces with a Dimension Higher than the Lower Critical Dimensionality

The phase diagram in coordinates "temperature - concentration of defects" of quasi-one-dimensional Ising models with defects of the "random local field" type is investigated. The confrontation of the tendency to the emergence of the long-range order due to a weak interaction between one-dimensional spin chains and the tendency to the formation of the Imry-Ma phase in which the order parameter follows the fluctuations of the random field created by defects is studied. The possibility of the appearance of the Imry-Ma phase in a situation where the space dimension exceeds the lower critical dimension is shown. The question of the existence of the long-range order in the Ising model with random fields in space with the critical dimension dl = 2 is considered.

Dimensional crossovers and Casimir forces for the Bose gas in anisotropic optical lattices

We consider the Bose gas on a $d$-dimensional anisotropic lattice employing the imperfect (mean-field) gas as a prototype example. We study the dimensional crossover arising as a result of varying the dispersion relation at finite temperature $T$. We analyze in particular situations where one of the relevant effective dimensionalities is located at or below the lower critical dimension, so that the Bose-Einstein condensate becomes expelled from the system by anisotropically modifying the lattice parameters controlling the kinetic term in the Hamiltonian. We clarify the mechanism governing this phenomenon. Subsequently we study the thermodynamic Casimir effect occurring in this system. We compute the exact profile of the scaling function for the Casimir energy. As an effect of strongly anisotropic scale invariance, the Casimir force below or at the critical temperature $T_c$ may be repulsive even for periodic boundary conditions. The corresponding Casimir amplitude is universal only in a restricted sense, and the power law governing the decay of the Casimir interaction becomes modified. We also demonstrate that, under certain circumstances, the scaling function is constant for suffciently large values of the scaling variable, and in consequence is not an analytical function. At $T > T_c$ the Casimir-like interactions reflect the structure of the correlation function, and, for certain orientations of the confining walls, show exponentially damped oscillatory behavior so that the corresponding force is attractive or repulsive depending on the distance.

Critical Dimension Bimodality Both Within Wafer and Within Die

With reduction in the die sizes it is very critical to control the Critical Dimension (CD) within a very precise window. In this work we present a root cause analysis and solution to a bimodal CD distribution seen from the center of the wafer. The CD bimodality displays a sharp cliff at the center of the wafer, wherein one side of the wafer has the higher CD and the other side has lower CD with respect to the target value. The bimodality was also seen in the center column of the wafer.

One-loop functional renormalization group study for the dimensional reduction and its breakdown in the long-range random field O( N ) spin model near lower critical dimension

We consider the random-field O($N$) spin model with long-range exchange interactions which decay with distance $r$ between spins as $r^{-d-\sigma}$ and/or random fields which correlate with distance $r$ as $r^{-d+\rho}$, and reexamine the critical phenomena near the lower critical dimension by use of the perturbative functional renormalization group. We compute the analytic fixed points in the one-loop beta functions, and study their stability. We also calculate the critical exponents at the analytical fixed points. We show that the analytic fixed point which governs the phase transition in the system with the long-range correlations of random fields can be destabilized by the nonanalytic perturbation in both cases where the exchange interactions between spins are short ranged and long ranged. For the system with the long-range exchange interactions and uncorrelated random fields, we show that the $d\to d-\sigma$ dimensional reduction at the leading order of the $d-2\sigma$ expansion holds only for $N>2(4+3{\sqrt{3}})\simeq 18.3923\cdots$. Our investigation into the system with the long-range exchange interactions and uncorrelated random fields also gives the value of the boundary between critical behaviors in systems with long-range and short-range exchange interactions, which is identical to that predicted by Sak [Phys. Rev. B {\bf{8}}, 281 (1973)]. For the system with the long-range exchange interactions and the long-range correlated random fields, we show that the $d\to d-\sigma-\rho$ dimensional reduction does not hold within the present framework, as far as $N$ is finite.

Lower lower-critical spin-glass dimension from quenched mixed-spatial-dimensional spin glasses

By quenched-randomly mixing local units of different spatial dimensionalities, we have studied Ising spin-glass systems on hierarchical lattices continuously in dimensionalities 1 =< d =< 3. The global phase diagram in temperature, antiferromagnetic bond concentration, and spatial dimensionality is calculated. We find that, as dimension is lowered, the spin-glass phase disappears to zero temperature at the lower-critical dimension d_c=2.431. Our system being a physically realizable system, this sets an upper limit to the lower-critical dimension in general for the Ising spin-glass phase. As dimension is lowered towards d_c, the spin-glass critical temperature continuously goes to zero, but the spin-glass chaos fully sustains to the brink of the disappearance of the spin-glass phase. The Lyapunov exponent, measuring the strength of chaos, is thus largely unaffected by the approach to d_c and shows a discontinuity to zero at d_c.

Engineering Surface Critical Behavior of (2+1)-Dimensional O(3) Quantum Critical Points.

Surface critical behavior (SCB) refers to the singularities of physical quantities on the surface at the bulk phase transition. It is closely related to and even richer than the bulk critical behavior. In this work, we show that three types of SCB universality are realized in the dimerized Heisenberg models at the (2+1)-dimensional O(3) quantum critical points by engineering the surface configurations. The ordinary transition happens if the surface is gapped in the bulk disordered phase, while the gapless surface state generally leads to the multicritical special transition, even though the latter is precluded in classical phase transitions because the surface is in the lower critical dimension. An extraordinary transition is induced by the ferrimagnetic order on the surface of the staggered Heisenberg model, in which the surface critical exponents violate the results of the scaling theory and thus seriously challenge our current understanding of extraordinary transitions.

Dynamical implications of sample shape for avalanches in 2-dimensional random-field Ising model with saw-tooth domain wall

We study dynamics of a built-in domain wall (DW) in 2-dimensional disordered ferromagnets with different sample shapes using random-field Ising model on a square lattice rotated by 45 degrees. The saw-tooth DW of the length Lx is created along one side and swept through the sample by slow ramping of the external field until the complete magnetisation reversal and the wall annihilation at the open top boundary at a distance Ly. By fixing the number of spins N=Lx×Ly=106 and the random-field distribution at a value above the critical disorder, we vary the ratio of the DW length to the annihilation distance in the range Lx∕Ly∈[1∕16,16]. The periodic boundary conditions are applied in the y-direction so that these ratios comprise different samples, i.e., surfaces of cylinders with the changing perimeter Lx and height Ly. We analyse the avalanches of the DW slips between following field updates, and the multifractal structure of the magnetisation fluctuation time series. Our main findings are that the domain-wall lengths materialised in different sample shapes have an impact on the dynamics at all scales. Moreover, the domain-wall motion at the beginning of the hysteresis loop (HLB) probes the disorder effects resulting in the fluctuations that are significantly different from the large avalanches in the central part of the loop (HLC), where the strong fields dominate. Specifically, the fluctuations in HLB exhibit a wide multi-fractal spectrum, which shifts towards higher values of the exponents when the DW length is reduced. The distributions of the avalanches in this segments of the loops obey power-law decay and the exponential cutoffs with the exponents firmly in the mean-field universality class for long DW. In contrast, the avalanches in the HLC obey Tsallis density distribution with the power-law tails which indicate the new categories of the scale invariant behaviour for different ratios Lx∕Ly. The large fluctuations in the HLC, on the other hand, have a rather narrow spectrum which is less sensitive to the length of the wall. These findings shed light to the dynamical criticality of the random-field Ising model at its lower critical dimension; they can be relevant to applications of the dynamics of injected domain walls in two-dimensional nanowires and ferromagnetic films.

Relaxational processes in the one-dimensional diluted Ising model with long-range interactions

ABSTRACTRelaxational processes in the one-dimensional diluted Ising model with long-range interactions are numerically investigated. When the dilution is relevant, the power-law decay of autocorrelation functions is observed as the droplet theory predicts. The power-law decay is expected to disappear in the high dimensional mean-field ordered phase and in the low dimensional Kosterlitz-Thouless phase. Numerical results, however, show that the power-law decay survives in the mean-field ordered phase, and nontrivial stretched exponential decay is observed at the lower critical dimension.

Chiral Potts spin glass ind=2and 3 dimensions

The chiral spin-glass Potts system with q=3 states is studied in d=2 and 3 spatial dimensions by renormalization-group theory and the global phase diagrams are calculated in temperature, chirality concentration p, and chirality-breaking concentration c, with determination of phase chaos and phase-boundary chaos. In d=3, the system has ferromagnetic, left-chiral, right-chiral, chiral spin-glass, and disordered phases. The phase boundaries to the ferromagnetic, left- and right-chiral phases show, differently, an unusual, fibrous patchwork (microreentrances) of all four (ferromagnetic, left-chiral, right-chiral, chiral spin-glass) ordered phases, especially in the multicritical region. The chaotic behavior of the interactions, under scale change, are determined in the chiral spin-glass phase and on the boundary between the chiral spin-glass and disordered phases, showing Lyapunov exponents in magnitudes reversed from the usual ferromagnetic-antiferromagnetic spin-glass systems. At low temperatures, the boundaries of the left- and right-chiral phases become thresholded in p and c. In d=2, the chiral spin-glass Potts system does not have a spin-glass phase, consistently with the lower-critical dimension of ferromagnetic-antiferromagnetic spin glasses. The left- and right-chirally ordered phases show reentrance in chirality concentration p.

Lower-critical spin-glass dimension from 23 sequenced hierarchical models.

The lower-critical dimension for the existence of the Ising spin-glass phase is calculated, numerically exactly, as dL=2.520 for a family of hierarchical lattices, from an essentially exact (correlation coefficent R2=0.999999) near-linear fit to 23 different diminishing fractional dimensions. To obtain this result, the phase transition temperature between the disordered and spin-glass phases, the corresponding critical exponent yT, and the runaway exponent yR of the spin-glass phase are calculated for consecutive hierarchical lattices as dimension is lowered.

Infinite volume extrapolation in the one-dimensional bond diluted Levy spin-glass model near its lower critical dimension

We revisited, by means of numerical simulations, the one-dimensional bond diluted Levy Ising spin glasses outside the limit of validity of mean-field theories. In these models the probability that two spins at distance $r$ interact (via disordered interactions, ${J}_{ij}=\ifmmode\pm\else\textpm\fi{}1)$ decays as ${r}^{\ensuremath{-}\ensuremath{\rho}}$. We have estimated, using finite size scaling techniques, the infinite volume correlation length and spin glass susceptibility for $\ensuremath{\rho}=5/3$ and $\ensuremath{\rho}=9/5$. We have obtained strong evidence for divergences of the previous observables at a nonzero critical temperature. We discuss the behavior of the critical exponents, especially when approaching the value $\ensuremath{\rho}=2$, corresponding to a critical threshold beyond which the model has no phase transition. Finally, we numerically study the model right at the threshold value $\ensuremath{\rho}=2$.

Diffusive quantum criticality in three-dimensional disordered Dirac semimetals

Three dimensional Dirac semimetals are stable against weak potential disorder, but not against strong disorder. In the language of renormalization group, such stability stems from the irrelevance of weak disorder in the vicinity of the noninteracting Gaussian fixed point. However, beyond a threshold, potential disorder can take Dirac semimetals into a compressible diffusive metallic phase through a quantum phase transition (QPT), where density of states at zero energy, quasiparticle lifetime and metallic conductivity at $T=0$, are finite. Universal behavior of such unconventional QPT is described within the framework of an $\epsilon(=d-2)$-expansion near the lower critical dimension. Various exponents near this QCP are obtained after performing a two loop perturbative expansion in the vanishing replica limit and we demonstrate that the theory is renormalizable at least to two loop order.We argue that such QPT is always continuous in nature, share same university class with a similar transition driven by odd-parity disorder, critical exponents are independent of flavor number of Dirac femrions and thus can be germane to disordered Cd$_3$As$_2$, Na$_3$Bi. Scaling behaviors of various measurable quantities, such as specific heat, density of states across such QPT are proposed.

Topological mott insulator in three-dimensional systems with quadratic band touching.

We argue that a three-dimensional electronic system with the Fermi level at the quadratic band touching point such as HgTe could be unstable with respect to the spontaneous formation of the (topological) Mott insulator at arbitrary weak long-range Coulomb interaction. The mechanism of the instability can be understood as the collision of Abrikosov's non-Fermi liquid fixed point with another, quantum critical, fixed point, which approaches it in the coupling space as the system's dimensionality d→dlow+, with the "lower critical dimension" 2<dlow<4. Arguments for the existence of the quantum critical point based on considerations in the large-N limit in d=3, as well as close to d=2, are given. In the one-loop calculation we find that dlow=3.26, and thus above, but not far from three dimensions. This translates into a temperature or energy window (Tc, T*) over which the non-Fermi liquid scaling should still be observable, before the Mott transition finally takes place at the critical temperature Tc∼T*exp[-zC/(dlow-d)1/2]. We estimate C=π/1.1, dynamical critical exponent z≈1.8, and the temperature scale kBT*≈(4m/melϵ^2)13.6 eV, with m as the band mass and ϵ as the dielectric constant.