Doping Simulations

(Dated: December 16, 2021)

A computer model of an electron confined to a one-dimensional crystal lattice with doped wells demonstrates that a small number of wells with slightly lower potential energies leads to the electron being spatially isolated in just one of the wells. The computer model is used to determine how slight the doping can be and still lead to such effects.

I. INTRODUCTION

A. Motivation

Matrix methods make the solution of Schrodinger's equation for the unbound states of a potential composed of side-by-side finite square wells of different depths quite tractable [1]. It is possible to solve the problem of bound states of such a potential analytically [2], but the calculations required to match continuity conditions at the boundaries quickly become unwieldy for more than a handful of wells. Computer simulations are ideal for this type of problem.

B. Description of method

We developed a simple model of doping within a one dimensional lattice of limited extent. In the model, we employ atomic units, with the Planck reduced constant and the rest mass of an electron equal to unity, so energy is in units of hartrees, equal to approximately 22.7 eV [3]. The entire region is 600 units long, using the Bohr radius as the unit of length as specified in atomic units, divided into 2048 discrete points. Each well is 10 points across, about 2.93 of these units of length, with the same distance between wells. The interstitial space between atoms is at potential V = 0, and periodic square potentials wells are at potential V = −1.0, on the order of the potential energy of the ground state of the hydrogen atom [4]. We then solved the time-independent Schrodinger equation numerically using the finite differences approximation [5].

C. Preliminary results

Using the finite differences method to solve the time independent Schrodinger equation leads to the expected results when we plot the normalized square of the wavefunction versus position with the lattice, resembling the bound states of a deep square well [6]. The number of nodes is dependent on the excitation state (ignoring the dense intermediate peaks and valleys that are results of the discrete approximation), and the expected symmetries are apparent. Figures 1 and 2 illustrate this for the two lowest energy states.

The computer program produced 32 states that have negative energy for all permutations of minor variation of well depth that we examined; inspection of the plots of probability density of these states confirms these wavefunctions are bound within the well. Finding 32 bound states in a potential composed of 16 symmetric or nearly symmetric wells might imply some interesting physics, but we did not place too much stock in that number for reasons discussed near the end of this paper. Higher energy states, with energy greater than zero, appear unbound, with more-or-less sinusoidal wavefunctions that reach the edge of the plot. Particularly in the lower positive energy states, the wavefunctions predicted by the program occasionally exist on just one side of the potential wells.

II. VARYING THE DEPTH OF THE WELLS

A. Ground state

We then adjusted the potential of a single arbitrary well to be 10% more negative. The model predicts the electron in its ground state will be almost completely confined to the well with the lower potential energy, as

FIG. 2. Plot of the square of the wavefunction for first excited state with all wells at the same negative potential energy.

shown in Figures 3 and 4. The model further predicts that at this level of doping, two doped wells will cause the electron to be confined to only one of the wells, but no discernible pattern is present as to which well it will be. This inconsistency may be an artifact of the algorithm we employ. Figure 5 demonstrates this seemingly random behavior.

FIG. 3. Ground state of electron with a single well with reduced potential energy. A scaled representation of the potential energy is superimposed to indicate where the spikes occur.

B. Excited states

When subjected to the same regimen, the first excited state of the electron demonstrates only partial isolation when just one doped well is present, but the same sharp peak is present when multiple wells are doped, within limits discussed later.

FIG. 4. Ground state of electron with a different well with reduced potential energy.

FIG. 5. Ground state of electron with two wells with reduced potential energy. 

Single doping frequently causes a paradoxical repulsion. The magnitude of the square of the wavefunction is noticeably reduced in the region of the lower potential well. This might be explained by the electron's greater kinetic energy in those regions- the electron spends less time there because it is traveling faster. Higher excited states demonstrate the same pattern. The n-th state (with n = 1 the ground state) requires n wells to have the lower potential to effectively isolate the electron. If the doping is insufficient to completely isolate the electron, each doped well still leads to a loss of a maximum of the square of the wavefunction in all instances we observed (e.g. the second excited state resembles the first excited state with one lowered well, while it resembles the ground state with two lowered wells). Figures 6 and 7 demonstrate the process of reducing the wavefunction and then isolating the electron for the first excited state.

 FIG. 6. Plot of the first excited state of the electron with one doped well. The potential energy scaled by 1/5 is indicated below the squared wavefunction.

FIG. 7. Plot of the first excited state of the electron with two doped wells.

The limits of this process are discussed at the end of the paper. The effect is delicate, however-if too many wells have the lower potential, there is a chance of measuring the electron in any of several wells. Figures 8 and 9 provide an example. When more than half of the wells have the more negative potential, it is equivalent to lowering the potential energy of non-doped well and selectively doping wells to have a less negative potential energy. Beyond noting the suppressive effect on the likelihood of the electron being found in a well doped to have a less negative potential, we did not explore this direction further.

FIG. 8. Tuning the wells to isolate the electron. Five doped wells is too many to completely confine the ground state...

FIG. 9. ...but five is just right for isolating the fourth excited state.

 III. ADJUSTING THE DEPTH OF THE WELLS

A. Further refinements

The question we ultimately wanted to explore was how deep the wells with altered potential energy must be to demonstrate this phenomenon of isolating the electron. Reducing the depth of the doped wells by half, to 5% deeper than the unchanged wells, had no noticeable effect. Further reducing the depth, to 1% deeper than the other wells, allows some small offshoots to appear- the wavefunction overlaps the undoped wells slightly, as shown in Figure 10. More central wells have the offshoots on both sides, and the sharpness of the peak is reduced because the electron can be measured on either side of the principal peak.

FIG. 10. Small ripples of the square of the wavefunction appear when doping is reduced. The relative depth of the potential wells is superimposed for reference.

B. Reducing doping until the isolation effect disappears

Doped wells that are 1% deeper demonstrate a similar isolation effect to the 10% discussed above, but with minor rippling shown in Figure 10, while a well that is only 0.1% deeper than the other wells results in a square of the wavefunction that appears very similar to that produced by a row of wells all of the same depth, as in Figure 11.

FIG. 11. A very slightly reduced well depth produces a probability density comparable to no doping at all. The location of the potential wells is again superimposed for reference.

Intermediate steps between a well that is 1% deeper than the rest and one that is one tenth of that show the wavefunction progressively spreading as the relative depth of the doped well decreases. The process is apparent in Figures 12 through 15. More central wells show a similar spreading, as do excited states with the appropriate number of doped wells to demonstrate the isolation effect.

FIG. 12. Successive reductions in the relative depth of a well with lower relative depth cause the wavefunction to spread across the array of wells.

FIG. 13. Wavefunction spreads further as relative depth of doped well decreases.

 We did not extensively explore combs of wells of greater depth, but cursory examination indicates the range of energy where the effect is found is proportional to the depth of the majority of the undoped wells.

IV. EFFECT OF WELLS OF REDUCED POTENTIAL ON HIGHER ENERGY STATES

Figures 8 and 9 demonstrated that doping five wells was able to confine an electron up to the fourth excited state in a single well, though it also allowed the wavefunction of the ground state to leak into other doped wells. For the particular choice of potentials we made, the fourth excited state is where the total confinement stops.

FIG. 14. Wavefunction is in an intermediate state as further reduction in relative potential is applied.

FIG. 15. Wavefunction is just slightly biased towards doped well as relative depth decreases further.

More doped wells will isolate the electron to some extent, but it has a probability of being any of several wells. In the cases we examined, it still has the highest likelihood of being in a particular well, with much less probability of its position being measured in the concomitant wells. In a computer simulation, we determine which wells have the lower potential, so we can control where the extraneous peaks occur to some degree (Figures 16 and 17). The presence of more doped wells necessitates that the doped wells will be closer to each other, so this might be explained by tunneling effects to some extent. Notably, increasing the number of total wells so the doped wells are farther apart restores the single sharp peak. We did not go too far in testing this, but Figure 18 shows an example, using 40 wells.

FIG. 16. Confinement breaks down for higher states.

FIG. 17. Manipulating which wells have the lower potential changes where the spikes occur.

V. WRAPPING UP

A. Discussion of computer code and graphs

The program we used was written in MATLAB, and adapted from Fortran code written by Kristian Dolghier that used the discrete variable representation approximation to solve Schrodinger's equation [7]. The original program predicted more bound states, with the lowest energies about 10% more negative than those found using the finite differences approximation, but the shapes of the squares of the wavefunctions were reassuringly similar. The built-in diagonalization methods of MATLAB produces more jagged-looking graphs, however. More data points would also have improved this jaggedness. The graphs produced are a little unconventional. We opted not to follow the standard practice of labeling the axes, while still leaving the tick marks on them, because the 6

FIG. 18. In a wider lattice, the confinement returns.

y-axis denotes more than one piece of information, and the shapes of the squares of the wavefunctions are more significant than their dimensionality. The software automatically resized the graphs, leading to an inconsistent look, but this is a failing more of the lead author's mastery of graphic techniques within the software than of MATLAB or the program itself.

B. Limitations of the simulation

The obvious departures from physical reality by the model we employ are that the model is one-dimensional and assumes the electron is already in a specific energy level, so time-dependent interactions between different levels can be ignored. The model treats the wells as simple square wells instead of the approximately parabolic wells that would actually be found [8], but the relative depth of the wells compared to their widths make this discrepancy a correction to be handled rather than a fatal flaw. The values we chose for potentials and distances are more or less physically plausible [9, 10], but it might be challenging to develop dopants that alter the potential of a well by a fraction of a percent. Finally, the wells in the model are static, so phononic phenomena that are so important in contemporary condensed matter physics [11] do not appear.

C. Conclusion

In the computer simulation we conducted, the effect of confining an electron to a single well in a small array of wells of depth of 1 hartree is not observed when that well is only 0.1% deeper than the others, but is almost complete when it is 1% deeper than the rest. Intermediate effects occur in doped potential levels between the two extremes. As the number of doped wells increases, the effect appears in higher energy levels, but eventually tunneling-like phenomena occur and ripples of the wavefunction appear in nearby doped wells. It appears this can be mitigated by performing the doping in a larger lattice, so the doped cells are farther apart. A surprising result is that higher energy wavefunctions can be caused to have a shape similar to lower levels when doping is not sufficient to confine the electron to a single well. We did not investigate superpositions of states of the electron or solve for time-dependent phenomena, but our results point to intriguing effects in that arena.

 [1] G. Grosso and G. Parravicini, in Solid State Physics (Academic Press, Cambridge, 2014) Chap. 1, pp. 12-25, 2nd ed.

[2] D. McIntyre, in Quantum Mechanics (Pearson, San Francisco, 2012) Chap. 5, pp. 128-132.

[3] National Institute for Standards and Technology, https://physics.nist.gov/cgi-bin/cuu/value?hr (2021).

[4] D. McIntyre, in Quantum Mechanics (Pearson, San Francisco, 2012) Chap. 8, p. 272.

 [5] R. Becerril, F. S. Guzm´an, A. Rend´on-Romero, and S. Valdez-Alvarado, Solving the time-dependent Schr¨odinger equation using finite difference methods, Revista Mexicana de F´ısica E 54, 120 (2008).

[6] D. McIntyre, in Quantum Mechanics (Pearson, San Francisco, 2012) Chap. 5, pp. 125-128.

[7] K. Dolghier, https://kdolghier.webnode.com/effects-ofdoping-on-finite-wells/ (2020).

[8] S. Simon, in The Oxford Solid State Basics (Oxford, Oxford, 2013) Chap. 6, p. 57.

[9] National Institute for Standards and Technology, https://www.ctcms.nist.gov/potentials/ (2021).

[10] C. Kittel, in Introduction to Solid State Physics (John Wiley & Sons, Inc, New York, 1986) Chap. 1, p. 17, 6th ed.

[11] S. Simon, in The Oxford Solid State Basics (Oxford, Oxford, 2013) Chap. 9,10, pp. 78-96.