Comment on "Shallow donor states near a semiconductor-insulator-metal interface"
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Comment on ”Shallow donor states near a semiconductor-insulator-metal interface”
L.F. Makarenko ∗ Faculty of Applied Mathematics and Computer Science,Belarusian State University, Independence Ave. 4, 220030 Minsk, Belarus
O.A. Lavrova
Faculty of Mechanics and Mathematics, Belarusian State University, Independence Ave. 4, 220030 Minsk, Belarus (Dated: December 6, 2018)In a recent paper Hao et al. [Phys. Rev. B 80, 035329 (2009)] reported variational calculationsof energy spectrum for shallow hydrogenic donor in the structure of semiconductor/insulator/metalwith a new type of trial wave function. They also performed calculations for semiconductor/insulatorsystem and found that their method gives energy values lower than those obtained by MacMillen andLandman [Phys. Rev. B 29, 4524 (1984)]. As follows from these results MacMillen and Landmanhave got much larger errors in energy values than they expected. However we confirm that thetheoretical approach suggested by MacMillen and Landman gives rather accurate energy values forthe system of hydrogenic donor near the interface between semiconductor and insulator.
In a recent paper , Hao et al. (HDAP) reportedvariational calculations of energy spectrum for shal-low hydrogenic donor in the structure of semiconduc-tor/insulator/metal with a new type of trial wave func-tion. To evaluate the quality of the suggested trial func-tion HDAP compared the calculated ground state en-ergies at different donor positions with the calculationsobtained by MacMillen and Landman . The authors ofRef. [2] ensured that their calculated energy values areaccurate to four significant figures. However, in Ref. [1]it was claimed that the errors in the result of Ref. [2] maybe even grater than 9%. The aim of this comment is toresolve this contradiction. We argue that the comparisonin Ref. [1] is not adequate.To calculate the electronic spectrum under consider-ation an eigenvalue problem for the following equation(written in dimensionless form) should be solved : (cid:18) −∇ − r D − Q ∗ r ¯ D + Q ∗ z (cid:19) F ( ~r ) = EF ( ~r ) . (1)All distances are scaled to units of effective Bohr radius( a ∗ B ) and the energy E is given in effective Rydbergs(Ry ∗ ). The parameter Q ∗ is given by: Q ∗ = ǫ s − ǫ ins ǫ s + ǫ ins , where ǫ s and ǫ ins are the dielectric constants of the semi-conductor and the insulator respectively. The distance ofthe donor from the interface is R and r D and r ¯ D are thedistances of an impurity electron from the donor ( D ) andits image ( ¯ D ), respectively.HDAP compared their results calculated for Q ∗ = 1 inEq. (1) (see Table I in Ref. [1]) to the values calculatedin Ref. [2] for Q ∗ ≈ . ǫ s = 11 . ǫ ins = 1). Thelatter one corresponds to the interface between siliconand vacuum (see Table II in Ref. [2]). It seems that thisincorrectness has been arisen due to the same mistakemade in Ref. [3]. In order to test this assertion we have performed cal-culations for Q ∗ ≈ . trial = P c ik φ ik with the basis wave functions: φ ik = 2 α / Rπ − / exp [ − αR ( ξ − η )] L i ( ξ ) P k ( η ) , (2)where ξ = r D + r ¯ D R and η = r ¯ D − r D R are prolate spheroidalcoordinates, α is a variational parameter, and L i ( ξ ) and P k ( η ) are polynomials of i -th and k -th degrees, respec-tively. As P k ( η ) we choose the Legendre polynomials ofodd degree which allow to satisfy the boundary condition F ( ~r ) = 0 at the interface ( z = 0). Choosing this basiswe avoid the necessity of numerical integration. Anotheradvantage of this basis set is a lower number of basisfunctions which is necessary to obtain the same accuracyfor the ground state energy E as in Ref. [2].To carry out numerical computations using finite ele-ment method (FEM), the system is assumed to be rota-tionally symmetric around z -axis and the problem is re-formulated in cylindrical coordinates ( r, z ). The equationis posed in a bounded domain Ω = (0 , δ ) × ( − δ − R, δ = 10 chosen for an approximation of the semi-infinite region (0 , + ∞ ) × ( −∞ ,
0) with a donor positionat (0 , − R ). Using MATLAB we discretize this problemby linear finite elements on a triangular mesh with anumber of unknowns ≈ Q ∗ ≈ . TABLE I. The values of the ground state energy E of theimpurity electron near the semiconductor-insulator interfacefor different values of R/a ∗ B calculated on the basis of thevariational wave functions of Eq. (2) with Q ∗ = 0 . E found in Ref. [2] for the same value of Q ∗ . R/a ∗ B α E / Ry ∗ E / Ry ∗ E / Ry ∗ variational, FEM, variational,present present Ref. [2]0.2 0.854 -0.6064 -0.6062 -0.60640.4 0.908 -0.6508 -0.6507 -0.65070.6 0.961 -0.7223 -0.7222 -0.72210.8 0.988 -0.8099 -0.8098 -0.80981.0 0.908 -0.8946 -0.8946 -0.89451.2 0.827 -0.9640 -0.9643 -0.96401.4 0.773 -1.0158 -1.0164 -1.01581.6 0.639 -1.0522 -1.0530 -1.05211.8 0.666 -1.0767 -1.0771 -1.07672.0 0.666 -1.0925 -1.0932 -1.09253.0 0.666 -1.1085 -1.1089 -1.10864.0 0.666 -1.0943 -1.0952 -1.09445.0 0.720 -1.0794 -1.0804 -1.07946.0 0.720 -1.0676 -1.0679 -1.0676TABLE II. The values of the ground state energy E of theimpurity electron near the semiconductor-insulator interfacefor different values of R/a ∗ B calculated on the basis of thevariational wave functions of Eq. (2) with Q ∗ = 1, and E found in Ref. [1] for the same value of Q ∗ . R/a ∗ B E / Ry ∗ E / Ry ∗ E / Ry ∗ Realtivevariational, FEM, variational, errorpresent present Ref. [1] in %0.4 -0.7208 -0.7208 -0.716 0.61.0 -0.9491 -0.9491 -0.927 2.31.6 -1.0917 -1.0925 -1.077 1.42.0 -1.1256 -1.1264 -1.116 0.93.0 -1.1323 -1.1327 -1.128 0.44.0 -1.1131 -1.1139 -1.111 0.36.0 -1.0806 -1.0810 -1.080 < . variational and FEM calculations for Q ∗ = 1. The ob-tained results are presented in Table II. As seen from thistable the trial function suggested by HDAP gives rathergood bound energy values. However, they are less thanthe values obtained using the trial wave function consist-ing of a sum over a basis set of wave functions (as in Ref.[2] or ours) or by FEM.HDAP also examined the quality of their trial func-tions with two and three parameters by comparison withresults of their FEM calculations. As it was found in Ref.[1] the use of three parameter functions leads to a lowererror. It was estimated as about 1% when the distancefrom semiconductor/insulator interface is equal to a ∗ B .However, as it was shown in Ref. [1], the errors inenergy values calculated even with the three parameterfunction become much greater in the case Q ∗ = − R = a ∗ B as equal to E / Ry ∗ =0 . , . , . E / Ry ∗ = 0 . trial ) which is ratherclose to our FEM value E / Ry ∗ = 0 . .
4% for any donor distances from the interface. Howeverall these trial functions do not provide the ’spectroscopicaccuracy’ of calculated energies.In conclusion, we confirm that the theoretical approachsuggested by MacMillen and Landman gives rather ac-curate energy values for the system of hydrogenic donornear the interface between semiconductor and insulator.One can use their results as a reference in searching forother types of trial functions considering electronic prop-erties of such systems. ∗ [email protected] Y. L. Hao, A. P. Djotyan, A. A. Avetisyan, and F. M.Peeters, Phys. Rev. B , 035329 (2009). D. B. MacMillen and U. Landman, Phys. Rev. B , 4524(1984). M. J. Calder´on, B. Koiller, and S. D. Sarma, Phys. Rev. B , 125311 (2007), cond-mat/0612093. Z. J. Shen, X. Z. Yuan, B. C. Yang, and Y. Shen, Phys.Rev. B , 1977 (1993). L. F. Makarenko, Proc. of the Natl. Academy of Sciences ofBelarus, Ser. Phys.-Math. Sci.1