Abstract
The retarded Green’s function (E−H + iε)^{−1}is given for a dimensionally hybrid Hamiltonian which interpolates between one and two dimensions. This is used as a model for dimensional competition in propagation effects in the presence of onedimensional subsystems on a surface. The presence of a quantum wire generates additional exponential terms in the Green’s function. The result shows how the location of the onedimensional subsystem affects propagation of particles.
Keywords:
Fermions in reduced dimensions; Nanowires; Quantum wiresIntroduction
Onedimensional field theory is frequently used for quantum wires or nanowires [13] or nanotubes [47]. Twodimensional field theory has become a universally accepted tool for the theoretical modeling of particles and quasiparticles on surfaces, interfaces, and thin films. The success of lowdimensional field theory in applications to the quantum hall effects [810], impurity scattering in lowdimensional system (see e.g. [1116]), and the confirmation of lowdimensional critical exponents in experimental samples [1722] confirm that lowdimensional field theories are useful tools for the description of lowdimensional condensed matter systems.
The properties of a physical system have a strong dependence on the number of dimensionsd. A straightforward example is provided by the zero energy Green’s functionG(r)_{E=0}, which is proportional torind = 1 and proportional to lnrind = 2, and decays liker^{2 − d}in higher dimensions. Green’s functions determine correlation functions, twoparticle interaction potentials, propagation of initial conditions, scattering off perturbations, susceptibilities, and densities of states in quantum physics. It is therefore of interest to study systems of mixed dimensionality, where competition of dimensions can manifest itself in the properties of particle propagators.
To address questions of dimensional competition analytically in the framework of interfaces in a bulk material, dimensionally hybrid Hamiltonians of the form
were introduced in [23]. The corresponding first quantized Hamiltonian is
Here the convention is to use vector notation for directions parallel to an interface, while z is orthogonal to the interface. From a practical side, Hamiltonians of the form (2) may be used to investigate propagation effects of weakly coupled particles in the presence of an interface. From a theoretical side, the Hamiltonians (1,2) are of interest for the analytic study of competition between twodimensional and threedimensional motion.
The twodimensional mass parameter μ is a mass per length. In simple models it is given by
where depending on the model, is either a bulk penetration depth of states bound to the interface at z = z_{0} or a thickness of the interface, see [24].
The zero energy Green’s function for the Hamiltonians (1,2) for perturbations in the interface (z′ = z_{0} = 0, ) satisfies
and was found in [23] (),
The Green’s function in the interface is given in terms of a Struve function and a Bessel function,
and interpolates between twodimensional and threedimensional distance laws (see Fig. 1),
Figure 1. The upper dotted line is the threedimensional Green's function (4πr) ^{−1} in units of ℓ^{−1}, the continuous line is the Green's function (4), and the lower dotted line is the twodimensional logarithmic Green’s function. x = r/ℓ
The corresponding energydependent Green’s function was also recently reported [24]. However, another system of great practical and theoretical interest concerns quantum wires or nanowires on surfaces. Preparation techniques for onedimensional nanostructures were recently reviewed in reference [25]. We will examine the corresponding dimensionally hybrid Hamiltonian and its Green’s function in this paper.
The Hamiltonian
We wish to discuss effects of dimensionality of nanostructures on the propagation of weakly coupled particles in the framework of a simple model system. We assume large de Broglie wavelengths h/p compared to lateral dimensions of nanostructures, and for our model system we also neglect electromagnetic effects or interactions, bearing in mind that these effects are highly relevant in realistic nanostructures [26,27].
The model system which we have in mind consists of nonrelativistic particles or quasiparticles tied to a surface. The surface carries a onedimensional wire. The x direction is along the wire and the y direction is orthogonal to the wire. The wire is located at y = y_{0}. The particles can move with a mass m on the surface, but motion along the wire may be described by a different effective mass m_{*}. In case of a weak attraction to substructures, kinetic operators can be split between bulk motion and motion along substructures [24]. Alternatively, for large lateral de Broglie wavelength relative to lateral extension of a substructure, one can also argue that the lateral integral of the kinetic energy density along a substructure only yields a factor in the kinetic energy for motion along the substructure. In either case we end up with an approximation for the kinetic energy operator of the particles of the form
where the mass parameter is a mass per lateral attenuation length of bound states, or a mass per lateral extension of the substructure. The operator (7) is the second quantized kinetic Hamiltonian for the particles. The corresponding first quantized Hamiltonian is
The wire corresponds to a channel in which propagation of a particle comes with a different cost in terms of kinetic energy. It is intuitively clear that existence of this channel will affect propagation of the particles on the surface, and we will discuss this in terms of a resulting Green’s function for the Hamiltonians (7,8).
The Green’s Function inkSpace
The Hamiltonians (7,8) yield the Schrödinger equation
and the corresponding equation for the Green’s function in the energy representation,
The last equation reads in (x,y) representation and with the convention
Substitution of the Fourier transform
yields
This yields with
the condition
This result implies that must have the form
with the yet to be determined function satisfying
Substitution of
yields
We finally find
where the definition
was used. ) is the Green’s function which we would use inkspace Feynman rules. It is also instructive to switch toyrepresentation for the transverse direction to see the impact of the wire on particle propagation.
The Green’s Function in Mixed Representations and Impurity Scattering
It is well known in surface science that Green’s functions can also be given in closed form in mixed representations, where momentum coordinates are used along the surface and configuration space coordinates are used for the transverse directions. The same observation applies here. In particular, the Green’s function with one transverse momentum replaced by a transverse coordinate is given by
The Green’s function with both arguments for the transverse direction given in terms of configuration space coordinates is
The first order perturbation of a state ψ_{0}(x,y) due to scattering off an impurity potential V(x,y) corresponds to
The result (16) shows peculiar distance effects between the location of the wire and the perturbation or impurity on the one hand, and between the location of the wire and theycoordinate of the wave function on the other hand. In both cases, the wavelength (for ) or attenuation length (for ) are the same as in the terms from the unperturbed surface propagator. In the evanescent case, the impact of the wire on impurity scattering is exponentially suppressed if either the impurity is located far from the wire or if the wave function is considered far from the wire. In the nonevanescent case the perturbation of the propagator due to the wire becomes a strongly oscillating function of far from the wire. Therefore the impact of the wire will also be small if we consider wave packets far from the wire.
For a simple application of (17) consider a wire at y_{0} = 0 and an impurity potential
The plane wave with is a solution of the Schrödinger Eq. 9 which satisfies the conditions for the approximation (7). In this case we get a scattering amplitude
The equation for ℓ = 0 is just the standard result for scattering from a pointlike impurity in mixed representation. The presence of the wire reduces the scattering cross section of the impurity for orthogonal infall.
Equations 15 and 16 also show that the effects of the additional terms should be most noticeable ifk_{x}ℓ ≫ 1. Since promising samples should have an effective mass m_{*} for motion along a quantum or nanowire which is much smaller than the effective mass m for motion along the surface. What comes to mind is an InSb nanowire on a Si surface. Scattering of surface particles off impurities in the presence of the wire should exhibit the additional propagator terms.
Conclusion
A simple model system for dimensional competition in nanostructures has been proposed. The system assumes that motion along a wire on a surface comes with a different cost in terms of kinetic energy, e.g. due to effective mass effects. The dimensionally hybrid retarded Green’s function for the propagation of free particles in the system was found in closed analytic form both inkspace and in mixed (k_{x},y) representations. The wire generates extra exponential terms in the propagator of the particles. The attenuation lengths or wavelengths in the evanescent or oscillating case, respectively, are the same as for the unperturbed propagator, but the extra terms exhibit distance effects between the particles and the wire.
Acknowledgments
This work was supported in part by NSERC Canada. I also gratefully acknowledge the generous hospitality of the Perimeter Institute for Theoretical Physics while this work was completed.
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