Physica B 302–303 (2001) 1–6 Electric field broadening of arsenic donor states in strongly compensated n-type Ge:(As, Ga) Jiro Katoa, Kohei M. Itoha,*, Eugene E. Hallerb a Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Yokohama 223-8522, Japan b UC Berkeley and Lawrence Berkeley National Labs, Berkeley, CA 94720, USA Abstract We discuss on the broadening of ground-state to bound excited-state transitions of shallow donors in strongly compensated n-type Ge in the presence of electric fields and their gradients arising from randomly distributed ionized impurities. Low-temperature (T ¼ 3:2 K) far-infrared absorption spectra of strongly compensated n-type Ge:(As,Ga) 13 14 À3 have been obtained for samples having ionized impurity concentration NI ¼ 2:2  10 22:6  10 cm . Absorption peaks corresponding to 1s–2p Æ transition of arsenic impurities are observed, and broadened linearly with the ionized impurity concentration due to interactions between electrons and the quadrupole moments of ionized donors. The slope 13 À3 of the peak line width plotted against NI changes at NI 8  10 cm due to the transition of ionized impurity 13 À3 13 À3 distribution from random (NI58  10 cm ) to correlated (NI > 8  10 cm ). # 2001 Elsevier Science B.V. All rights reserved. PACS: 78.40; 02.50.N; 78.55.A Keywords: Impurity absorption; Compensated semiconductors; Impurity distribution 1. Introduction the theory based on the correlated distribution of ionized impurities [2] than the theory assuming a In a recent publication [1], Itoh et al. have random distribution [3,4]. They also have found reported on the broadening of the 1s–2p-like Ga that the quadrupole interaction is a dominant acceptor absorption peak in heavily compensated broadening mechanism. Ge samples having ionized impurity concentration In this work, we have repeated a similar 13 15 À3 NI ¼ 9:0  10 26:8  10 cm . Their line investigation on donors because the relevant widths as a function of NI have been compared theories [2–4] have been developed assuming with theories for the hydrogenic donor 1s–2p hydrogenic donors, not acceptors, in semiconduc- transitions [2–4]. For the concentration range they tors. Moreover, we have probed the lower investigated, they found very good agreement with concentration region for which the random distribution of ionized impurities is predicted. As it will be shown below, the ionized impurity *Corresponding author. Tel.: +81-45-566-1594; fax: +81- 45-566-1587. distribution is predicted to be random when NI5 13 À3 E-mail address: [email protected] (K.M. Itoh). 1  10 cm [3,4] and correlated when 0921-4526/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0921-4526(01)00398-2 2 J. Kato et al. / Physica B 302–303 (2001) 1–6 14 À3 NI > 3  10 cm [2] for our Ge:(As,Ga), i.e., The second term in Eq. (1), which is proportional there may be a transition of the ionized impurity to E, represents the linear Stark effect, while the 2 distribution as a function of NI from random to third term, which is proportional to E , represents 13 14 À3 correlated between NI ¼ 10 and 10 cm . The the quadratic Stark effect. Apart from Eq. (1), present work reports on the experimental observa- there exists a quadrupole interaction between tion of such a transition for the first time in doped donor-bound electrons and the quadrupole mo- semiconductors. The experimentally determined ments of ions leading to an energy shift propor- critical ionized-impurity concentration for the tional to the gradients of the electric field. The line 13 À3 transition is NI ¼ 8  10 cm for Ge:(As,Ga) width of the absorption peaks broadened by the at T ¼ 3:2 K. It will be also shown that the electric field is proportional to [3,4] quadrupole interaction is a dominant broadening 2=3 mechanism of the 1s–2p Æ absorption peaks of As (1) ðNI=N0Þ when the linear Stark effect is in Ge for the concentration range we investigated. dominant, 4=3 In the past, four major broadening mechanisms (2) ðNI=N0Þ when the quadratic Stark effect is of the impurity absorption lines have been treated dominant, and theoretically: (3) NI=N0 when the quadrupole interaction is dominant, (1) phonon lifetime broadening due to interaction of bound electrons (holes) with phonons [5,6]; where N0 is the concentration of the neutral (2) concentration broadening due to overlap of absorption centers. In this study we probe 1s– the extended hydrogenic impurity wave func- 2p Æ transition of electrons bound to As donors in tions [7]; Ge. Such a transition corresponds to n ¼ 2; n1 ¼ (3) strain broadening due to strains arising from 0; n2 ¼ 0 and m ¼Æ1 in Eq. (1), i.e., the linear the presence of dislocations, precipitates, and Stark term (/ E) vanishes. Moreover, the quad- impurities [8]; and rupole interaction is expected to be dominant in (4) Stark and quadrupole broadening due to the limit of small NI and for the transition having n random electric fields arising from ionized even integers (n ¼ 2; 4; 6; etc:). Therefore, we impurities [2–4]. expect our 1s–2p Æ transition peaks to be broa- dened dominantly by the quadrupole effect. We focus in this work on the fourth broadening For the quantitative calculation of absorption mechanism. For weak electric fields, the energy e line widths broadened by the electric field, it is of the bound state of a hydrogenic impurity with important to assume an appropriate distribution quantum numbers n, m, n1,andn2 in the unit of of ionized impurities. Up to now, two kinds of ion effective Rydberg of the hydrogenic impurity is distributions have been proposed for compensated given by semiconductors at low temperatures: random distribution [3,4] and correlated distribution [2]. 1 3 ¼À þ ðÞÀ Fig. 1 depicts examples of the two distributions for e 2 nn1 n2 eEa n 2 Ge:(As,Ga). There are more As donors than Ga À½17n2 À 3ðn À n Þ2 À 9m2 þ 19 acceptors, so all Ga lose their holes at low 1 2 E2a2e2n4 temperatures and are ionized negatively. The  þÁÁÁ Ry; ð1Þ negatively ionized Ga acceptors cannot move, so 32 that their positions are fixed. However, electrons where n is the principal quantum number, m is the bound to As donors are free to choose As sites to magnetic quantum number, a is the effective Bohr which they bind. In the example shown in Fig. 1, radius, and n1 and n2 are integers greater than or four electrons can hop between eight As sites. The equal to zero which obey the condition ionized impurity distribution is random [3,4] when these four electrons distribute themselves ran- n ¼ n1 þ n2 þjmjþ1: ð2Þ domly among eight As sites that are available to J. Kato et al. / Physica B 302–303 (2001) 1–6 3 Fig. 1. Two kinds of distributions of ionized impurities in compensated Ge(As,Ga): random distribution (left half) and correlated distribution (right half). them. On the other hand, the ionized impurity distribution is correlated [2] when electrons choose specific sites to lower the total Coulombic energy of the system. As atoms that are close to negatively charged Ga ions may prefer to remain positive in order to lower the Coulombic energy. The average electric field arising from the random distribution is larger than that from correlated distribution, i.e., the absorption line widths become broader for the random distribution than for the correlated distribution. Whether the distribution becomes random or correlated depends on the concentra- tions of ionized impurities and temperature. The random distribution is preferred for the lower N I Fig. 2. Experimentally determined free carrier concentration since the larger distance between ions leads to (filled circles) vs. inverse temperature. The dotted curve is the weaker correlation. According to the estimation best fit to the experimental data using Eq. (5) with ½As¼ due to Larsen, the random distribution is domi- 2:19  1013 cmÀ3 and ½Ga¼1:85  1013 cmÀ3. nant when À5 À3 tion energy is larger than the thermal energy, i.e., NI40:7Â10 a ; ð3Þ 2 1=3 e ND =k > kBT. Assuming NI ¼ 2KND where K is the compensation ratio defined by the minority where a is the Bohr radius of absorption centers. impurity concentration divided by the majority Inserting a ¼ 87:6 A( for As in Ge, we expect the impurity concentration, the correlated distribution random distribution to be dominant when becomes dominant when [3,4] N  13 À3 I51 10 cm . The correlation energy, on the other hand, is of the same order of magnitude k Tk 3 N 42K B : ð4Þ as the Coulomb interaction energy between I e2 2 1=3 impurities, i.e., e ND =k, where k is the dielectric 14 À3 constant and ND is the donor concentration in an Eq. (4) becomes NI > 3  10 cm for our mea- n-type semiconductor. The correlated distribution surements performed at T ¼ 3:2 K. The purpose of ionized impurities is expected when the correla- of this study is to obtain evidences for both kinds 4 J. Kato et al. / Physica B 302–303 (2001) 1–6 of ion distributions via precise absorption mea- It has been recorded at T ¼ 3:2 K with a resolu- surements using a series of Ge:(As,Ga) samples tion of 0.05 cmÀ1 in the wave number range À1 having appropriate concentrations of NI.