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ION.X

 

ion.x is a code for calculating the relaxation of the d-doped, modulation-doped, ionized donors in the barrier region of a two dimensional electron gas (2DEG) heterostructure1. The code was written by M. Stopa and is available on the Harvard NNIN/C computation cluster for use under unix.

Physics
Silicon donors in AlxGa1-xAs have a complex structure2. Depending on the aluminum concentration Si donors can be either shallow, hydrogenic impurities or else so-called DX centers. DX centers2 are characterized by a coupling between a crystal deformation at the impurity site (a displacement of the Si atom from the exact lattice site) with the electronic state of the ion. Specifically, it is found that at low temperatures (<150K) and for Al concentration, x, in excess of 0.22, the ion becomes trapped in one of four nearly identical off-site locations and the electron, which at higher temperatures and lower x is in an effective mass hydrogenic state with radius the order of the effective Bohr radius (10 nm in GaAs), becomes tightly bound to the ion (orbital radius 2 Angstroms). In fact, the 2U model of the DX center indicates that two electrons can become trapped on the impurity site, presumably satisfying charge neutrality by having neighboring ions remain positively charged. The transition temperature is equivalent to the energy needed to move the ion out of its deformed state so that it spends most of its time near the lattice site where the electronic state is transiently (according to a Born-Oppenheimer approximation) weakly bound.

The detailed kinematics of the trapping-de-trapping physics is not fully understood. However, for 2DEG heterostructures another physical phenomenon is relevant. The density of the 2DEG is determined by the equivalence of the electrochemical potential between the 2DEG and in the donor layer3. At high temperature (when the ions do not dislocate and freeze their charge state) Si atoms ionize and contribute electrons to the 2DEG until its Fermi surface coincides with the highly degenerate bound state of the ions. In most cases the number of ionized impurities is less than and often substantially less than the total number of Si atoms. Thereby each electron has many Si atom sites to choose from and collectively the ionized sites can distribute among all Si atoms so as to minimize the total Coulomb energy. Since the position of the Si atoms themselves is fixed during crystal growth and essentially random (but constrained to a 2D plane in the d-doped case), the problem is that of a Coulomb gas on a disordered lattice. Defining F as the fraction of donors that are ionized (ignoring for simplicity the 2U model), the degree of possible Coulombic ion ordering clearly increases as F decreases. The program ion.x simulates the annealing process beginning with an initial distribution of N­ Si atoms and N+ ions (in a 2D layer) in a fixed area (containing typically about a thousand donors), with F=N­+/N, calculates the total Coulomb energy, and allows the system to evolve according to variable-range hopping4 events and a type of Metropolis, Monte-Carlo algorithm, to lower their energy. The 2DEG is included simply as an image plane for the ions, making the interaction a dipole-dipole interaction (i.e. ~1/R3). 2D is the upper critical dimension for a phase transition for interactions of this form. A crude form of    Etvos sum is performed by using a supercell arrangement. Hopping is restricted to a set of n nearest neighbors and proceeds sequentially. Since the system is essentially a glass1 the absolute energy minimum is difficult to find and presumably as the system cools it ends up in a metastable state at low temperature. This explains why the state of 2DEG heterostructure-based devices like quantum point contacts and quantum dots fluctuate significantly upon thermal cycling5.

Program
The code ion.x is a simple, single-processor code for calculating the annealing of a distribution of ions, chosen randomly from a distribution of donors with an ionization ratio of F and the system area fixed. The program can be limited by the number of hopping processes or the time of the simulation (here, time is a fictitious time associated with the Arrenhius thermal probability of hopping events, see Ref. 1). The executable is /group/nnin/ION/bin/ion.x and sample input decks can be found in /group/nnin/ION/example/ion_example.run, /group/nnin/ION/example/static.dat and /group/nnin/ION/example/ion_example.grid.  The .run file is the primary input file and is documented in the file itself. The static.dat file contains parameters that rarely change (contact NNIN for details). The name of the static.dat file is hardcoded. The .grid file is a template whereby the ions are placed into boxes (grid cells) at the end of a calculation for subsequent input into the SETE code for an electronic structure calculation with the specified ion distribution. Also included in this directory is a condor submit file, ion.submit. Copy the two input files and ion.submit to a directory under your home directory. Submit the job to condor using:

condor_submit ion.submit.

The ion.submit file contains the names of the two input decks. If you change these (by doing more simulations) you must change the corresponding names in this file.

Output files
The code ion.x outputs five principal output files. The naming convention is that if the input decks are, for example A1.run and A1.grid, then the output decks are A1.EvsT, A1.xysite, A1.ioni, A1.ionf and A1.sete. In the example case A1 is “ion_example.” These decks contain the following:

A1.EvsT – the total Coulomb energy as a function of simulation time and simulation step. This dependence typically shows the sudden decreases characteristic of glassy systems.

A1.xysite – the locations, in the xy-plane, of the Si donor atoms.

A1.ioni – the initial ion positions.

A1.ionf – the final ion positions.

A1.sete – the discretized ion positions for input to SETE.

The ion.x code can also compute the pair distribution function of the ionized impurities following the annealing process. This file is called pair.A1.

1. M. Stopa, Phys. Rev. B 53, 9595 (1996); M. Stopa, Y. Aoyagi and T. Sugano Surf. Sci. 305, 571 (1994).
2. P. M. Mooney, J. Appl. Phys. 67, R1 (1990).
3. T. Ando, A. B. Fowler and F. Stern, Rev. Mod. Phys. 54, 437 (1982).
4. B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors, (Springer-Verlag, Berlin 1984), chapter 9.
5. Some experimental evidence of these phenomena have been observed recently in C. Seigert et al., Nature Phys. 3, 315 (2007).

 
 

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