Radii of Rydberg states of isolated silicon donors

Juerong Li, Nguyen H. Le, K. L. Litvinenko, S. K. Clowes, H. Engelkamp, S. G. Pavlov, H.-W. Hübers, V. B. Shuman, L. М. Portsel, А. N. Lodygin, Yu. A. Astrov, N. V. Abrosimov, C. R. Pidgeon, A. Fisher, Zaiping Zeng, Y.-M. Niquet, and B. N. Murdin

Phys. Rev. B 98, 085423 – Published 15 August 2018

Abstract

We have performed high field magnetoabsorption spectroscopy on silicon doped with a variety of single and double donor species. The magnetic field provides access to an experimental magnetic length, and the quadratic Zeeman effect, in particular, may be used to extract the wave-function radius without reliance on previously determined effective mass parameters. We were, therefore, able to determine the limits of validity for the standard one-band anisotropic effective mass model. We also provide improved parameters and use them for an independent check on the accuracy of effective mass theory. Finally, we show that the optically accessible excited-state wave functions have the attractive property that interactions with neighbors are far more forgiving of position errors than (say) the ground state.

Received 21 March 2018

DOI:https://doi.org/10.1103/PhysRevB.98.085423

Physics Subject Headings (PhySH)

Atomic spectra Electronic structure Electronic transitions

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

Juerong Li¹, Nguyen H. Le¹, K. L. Litvinenko¹, S. K. Clowes¹, H. Engelkamp², S. G. Pavlov³, H.-W. Hübers^3,4, V. B. Shuman⁵, L. М. Portsel⁵, А. N. Lodygin⁵, Yu. A. Astrov⁵, N. V. Abrosimov⁶, C. R. Pidgeon⁷, A. Fisher⁸, Zaiping Zeng⁹, Y.-M. Niquet⁹, and B. N. Murdin^1,*

¹Advanced Technology Institute, University of Surrey, Guildford, GU2 7XH, United Kingdom
²High Field Magnet Laboratory (HFML-EMFL), Radboud University, Toernooiveld 7, 6525 ED Nijmegen, The Netherlands
³Institute of Optical Sensor Systems, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany
⁴Humboldt Universität zu Berlin, Institut für Physik, Newtonstrasse 15, 12489 Berlin, Germany
⁵Ioffe Institute, Russian Academy of Sciences, St. Petersburg, Russia
⁶Leibniz Institute for Crystal Growth, Max-Born-Straße 2, 12489 Berlin, Germany
⁷Institute of Physics and Quantum Science, SUPA, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom
⁸London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, London WC1H 0AH, United Kingdom
⁹Université Grenoble Alpes, CEA, INAC-MEM, L_Sim 38000 Grenoble, France

^*Corresponding author: b.murdin@surrey.ac.uk

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 98, Iss. 8 — 15 August 2018

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Linear Zeeman effect. (a) Lyman series of the Si:P,Sb co-doped sample. Labels indicate the excited state for three of the strongest Lyman series transitions. The color scale indicates the transmission ( $dark blue = high transmission$ ; $light yellow = high absorption$ ). (b) Splitting between the transitions for the same $n$ but a different $m$ , i.e., $h υ_{1 s \to n p_{+}} - h υ_{1 s \to n p_{-}} = E_{n p_{+}} - E_{n p_{-}}$ against $B$ . Data presented are for $n = 2$ and 3 for all species used in this paper ( $Si : X$ where $X = Li, P, Sb, Bi, Mg, Se, S e_{2}, S$ ) showing they all follow the same field dependence. Inset bottom: Expanded scale section from main panel showing 16 points at each field with a different color symbol for each species/ $n$ combination (and only three examples are labeled due to the small scatter). Inset top: Residuals from the linear fit (red) and from the nonparabolic model fit (blue) to the Si:P data from the main panel.
Reuse & Permissions
Figure 2
Quadratic Zeeman effect. (a) Transition energy, $h υ_{1 s \to n p_{\pm}}$ , for Si:P for $2 p_{\pm}$ (squares) and $3 p_{\pm}$ (circles), with $m = + 1$ transitions shown at positive field and $m = - 1$ transitions shown at negative field. The fits described in the text produced the residuals shown as an inset, along with the weighting function used. (b) The transverse radius. The effective transverse radius squared for the transition, i.e., ${\tilde{ρ}}_{n p_{\pm}}^{2} - {\tilde{ρ}}_{1 s}^{2}$ found by applying Eq. (3) to the experimental $h υ_{1 s \to n p_{\pm}} (B)$ shown in (a), (solid symbols) using the Savitzky-Golay method. Also shown is the theoretical effective transverse radius, i.e., Eq. (3) applied to $E_{n p_{\pm}} (B)$ from EMT with the parameters from Table 1 (not a fit), solid lines. The open symbols show $ρ^{2}$ the actual (as opposed to effective) transverse radius squared $〈 x^{2} + y^{2} 〉$ of the excited states from the EMT wave functions (with the same parameters, not a fit). The effective radius $\tilde{ρ}$ and actual radius $ρ$ are clearly the same at small field.
Reuse & Permissions
Figure 3
Experimentally determined values of the zero-field energy splittings between excited states for different donor centers in silicon. Centers are displayed in order of binding energy. Data from Ref. [46] are also included as open symbols. Error bars are from Gaussian fits (this paper, or in the case of Ref. [46] the instrumental resolution).
Reuse & Permissions
Figure 4
The wave functions. The top row shows the six ground-state envelope functions: The color scale shows the probability density increasing from black (zero density) to white, on a spherical surface around the donor averaged over a valley interference oscillation period. The bottom row shows some example wave functions including the valley interference term. The brightness shows the probability density on some illustrative surfaces around the donor, and the color scale shows the wave-function phase. The donor is at the origin at the central vertex of the image and the length scale is shown in units of $a_{0}^{*}$ . The $1 s (T_{2})$ state shown is the $z$ -valley component (last one on the top row). All illustrations take the lattice periodic part of the wavefunction $u (r) = 1$ simplicity.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics