- Email: [email protected]
On the theory of classical cyclotron resonance line broadening in two- and three-dimensional semiconductors
Physica B 406 (2011) 4283–4288
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
On the theory of classical cyclotron resonance line broadening in two- and three-dimensional semiconductors S.P. Andreev a, Yu. A. Gurvich b, V.A. Nebogatov a, T.V. Pavlova c,n a
National Research Nuclear University MEPHI, Kashirskoe Sh. 31, 115409 Moscow, Russia Moscow State Pedagogical University, M. Pirogovskaya Str. 1, 119991 Moscow, Russia c Prokhorov General Physics Institute, Russian Academy of Science, Vavilov Str. 38, 119991 Moscow, Russia b
a r t i c l e i n f o
abstract
Article history: Received 22 December 2010 Accepted 22 August 2011 Available online 25 August 2011
A broadening of the absorption line of the classical cyclotron resonance (CR) in two- and three-dimensional semiconductors with neutral impurities of arbitrary depth is investigated. A dependence of the half-width of the classical CR line on the impurity characteristics (depth and range of the potential) is obtained in the wide range of the parameters. The broadening is studied within the Born approximation, and also at resonant and non-resonant scattering. Fundamental differences between the line broadening by neutral impurities in two- and three-dimensional semiconductors are revealed. An alternative explanation of classical experiments on electron scattering by neutral impurities (donors) is proposed. & 2011 Published by Elsevier B.V.
Keywords: Neutral impurities Two-dimensional gas Cyclotron resonance Resonant scattering
1. Introduction The cyclotron resonance (CR) in semiconductors is a powerful tool for getting versatile information about the interaction of carriers with lattice vibrations and defects, types and concentration of the defects, etc. (see Refs. [1–4] and references therein). The measurement of an absorption line’s half-width do gives an opportunity to identify various mechanisms of charge carriers scattering, since do is proportional to inverse relaxation time of current and, roughly, represents the sum of probabilities of the scattering mechanisms. At essentially low temperatures, scattering by phonons is suppressed. A specific role of CR at a study of impurities arises from its selectivity, since each scattering mechanism, generally speaking, gives rise to own specific shape of the absorption line. Broadening of the classical CR line by ionized impurities in bulk semiconductors has been satisfactory investigated. The situation with neutral impurities was inspected worse. To describe the broadening of the classical CR line by shallow neutral donors, the modified Erginsoy formula is used [1]. The modification originates from an account of the polarization effects on the scattering cross-section by the hydrogen atom. At this the effective Bohr radius and the dielectric constant of a semiconductor are substituted to the cross-section, noted above. The approach [1] allows one to get a satisfactory agreement between the theory and experiments. However, it can not be accepted as comprehensive, because it needs the use of a modulating function,
n
Corresponding author. Tel.: þ7 916 9109810. E-mail address: [email protected] (T.V. Pavlova).
0921-4526/$ - see front matter & 2011 Published by Elsevier B.V. doi:10.1016/j.physb.2011.08.051
introduced ‘‘manually’’, which provides a correct energy dependence of the scattering cross-section of electrons by shallow neutral donors in the desired range of temperatures. At changing depth of the neutral impurity potential, the character of carriers scattering is also changed. The scattering becomes resonant ones a real or virtual electron state appears in the impurity potential. Except this, in the presence of a magnetic field there can appear specific magnetic-impurity states in the impurity potential, which radically affect the kinetics of low-temperature phenomena [5,6]. Account of carriers resonant scattering by these states has yielded a few important results [5]. In particular, this allowed one to eliminate discrepancy between the theory and experiments on broadening of a quantum CR line by neutral impurities in bulk semiconductors [7,8], which took place for more than 30 years.1 In the case of classical CR, the role of resonant states in the potential of neutral centers on absorption line broadening was not discussed at all. To date, the following issues remain open: (1) Dependence of classical CR line half-width on depth of the potential of a separate impurity, particularly, in the conditions of resonant electron scattering in bulk semiconductors at various temperatures.2 1 All efforts to achieve agreement between the dependences obtained in experiments and theory were made on the base of Erginsoy’s formula, leading to discrepancies of more than an order of magnitude [4]. 2 In Ref. [9], it was obtained the dependence of relaxation time on temperature, which was explained by using the scattering cross-section of slow electrons by the hydrogen atom with all phase shifts accounted. At the same time, the
4284
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
(2) Derivation of the classical CR line broadening by shallow neutral donors in the three-dimensional semiconductors in the frames of a quantum-mechanical consideration, like it is done in the case of the quantum CR [7,8]. (3) Analysis of the classical CR line broadening in two-dimensional semiconductors because of the presence of neutral impurities with the potential of arbitrary depth. (4) Comparison of the features of the classical CR line broadening in two- and three-dimensional structures. This point has fundamental importance, since the corresponding dependences of the scattering cross-sections on depth of the impurity potential considerably differ. The present paper is dedicated to a thorough analysis of the issues listed above.
2. Half-width of the absorption line Consider a two-(three-)dimensional semiconductor placed in ~ and weak resonant electric fields the crossed uniform magnetic H iot ~ ~ EðtÞ ¼ E 0 e ðooH 5 oÞ (in two-dimensional case the magnetic field has the direction perpendicular to free surface of semiconductor). Let non-degenerate gas of noninteractive electrons in the semiconductor scatters on chaotically distributed neutral impurities under conditions of the classical CR ðT b ‘ oH Þ. Concentration of impurities is assumed to be low, i.e. simultaneous carriers scattering on two and more centers is not accounted for, implying gas approximation holds [10]. The magnetic field is assumed to be that weak to neglect its influence upon the interaction between electrons and impurities. We use the kinetic Boltzmann equation for electrons momentum ð‘ ~ kÞ distribution function f in crossed electric and magnetic fields in the presence of scatterers: ! ~ @f @f e~ E e½~ k, H þ ð1Þ ¼ Ist , @t mn c @~ ‘ k where Ist ¼ f =tNd , tNd the electron momentum relaxation time in a N -dimensional semiconductor, mn the effective mass of charge carriers assumed to be isotropic. Assuming weakness of the pffiffiffiffiffiffiffiffiffi ffi electric field ðE0 5 oH mn T =e, i.e. neglecting electron heating through the cyclotron period), we reduce the distribution function at the first approximation to the form: f ¼ f ð0Þ þf ð1Þ , where f ð0Þ is an equilibrium distribution function and f ð1Þ is a linear summand. In the weak fields while degeneracy of the electrons is absent, f ð0Þ is the Boltzmann distribution function. Electric current expressed in terms of the distribution function is nNd e‘ ~ J¼ e n m
Z
~ kf ð1Þ d~ k
ð2Þ
(nNd e is the electrons concentration in N-dimensional volume, with 2d N ¼ 3n3d e being the volumetric electrons concentration and N ¼ 2ne the surface electrons concentration). An absorption line in such a semiconductor is characterized by a ratio of average energy absorbed by carriers through the cyclotron period to average electromagnetic energy density: ~ J~ E QNd ¼
2 ~ E =4p
2
¼
2 2pnNd e e ‘ Im mn2 T
Z
Assume further that the condition oH tNd b 1 holds, i.e. that broadening of levels due to carriers scattering on neutral impurities is small in comparison with a distance between the levels. The relaxation time is defined by the scattering cross-section: Z 1 ‘k dsNd ð1cos WÞ, ¼ nNd ð4Þ i mn tNd where dsNd is the differential scattering cross-section of charge carriers on neutral impurities in two-dimensional (N¼2) or bulk (N¼ 3) semiconductors, W the scattering angle, nNd the amount of i scatterers per unit of N-dimensional volume (n3d the volumetric i concentration of impurities and n2d i surface concentration of impurities). Let us consider a model potential of a separate impurity to be central-symmetric two- or three-dimensional square-well with depth U and radius rc for the two- and three-dimensional semiconductors, respectively. Stress here that the following formulas are expressed in terms of two- and three-dimensional scattering lengths, not depending on potential’s shape, viz., the present results are model-independent. We will consider further the dependence of an absorption 2 line’s half-width on carries energy e ðe ¼ ‘ k2 =2mn Þ at various depths of impurity potential U, where the absorption line halfwidth is estimated as doNd ðeÞ ¼ 1=tNd ðkÞ.
3. Absorption line broadening due to scattering of fast electrons on neutral impurities Consider broadening of absorption line while presence of scattering of fast electrons (kr c b 1) on neutral impurities. In Born approx2 imation, the impurity potential depth is limited by U 5 ‘ k=mn rc . Two-dimensional ds2d and three-dimensional ds3d differential scattering cross-sections3 obtained in Born approximation are !2 2p Ur 2c mn J1 ðqr c Þ ds2d ¼ d W, ð5Þ qr c k ‘2 8prc2 ds3d ¼ k2
!2 Ur 2c mn sinðqr c Þqr c cosðqr c Þ
‘2
ðqr c Þ3
q dq,
ð6Þ
where q ¼ 2k sinðW=2Þ and J is the Bessel function. Substituting Eqs. (5) and (6) in Eq. (4), we obtain the inverse relaxation time of fast electrons on neutral impurities in two-dimensional ð1=t2d Þ and bulk ð1=t3d Þ semiconductors as: 4pmn rc2 U 2 n2d 1 i ¼ t2d ðkÞ ‘ 3 k2
Z p=2
8pmn rc2 U 2 n3d 1 i ¼ t3d ðkÞ ‘ 3 k3
Z
0
2krc
J12 ð2qr c sin wÞdw,
ðsin ww cos wÞ2
w3
0
ð7Þ
dw:
ð8Þ
Under the integrals signs in Eqs. (7) and (8) we factorize functions 2 sin w and cos w, using condition kr c b Umn rc2 =‘ . Leaving in Eqs. (7) and (8) only the main terms, we get: 4mn rc4 U 2 n2d 1 i ln kr c , t2d ðkÞ ðkr c Þ3 ‘3
ð9Þ
2
k3 expð‘ k2 =2mn TÞ dk: ooH þ i=tNd ðkÞ
ð3Þ
(footnote continued) question about dependence of relaxation time on depth of the impurity potential remained open.
1
t3d ðkÞ
4pmn rc5 U 2 n3d i ln kr c
‘3
ðkr c Þ3
:
ð10Þ
3 Note that dimension of scattering cross-section in two-dimensional case is length.
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
For the absorption line’s half-width at carriers scattering on neutral 2 impurities at large energies (e b 2U 2 mn rc2 =‘ ), we find: r U 2 n2d ln ð2rc2 mn e=‘ 2 Þ
c do2d pffiffiffiffiffiffiffiffiffiinffi
2m
e3=2
,
2 prc2 U 2 n3d ln ð2rc2 mn e=‘ Þ pffiffiffiffiffiffiffiffiffiffii e3=2 2mn
ð11Þ
4285
assumed to be center-symmetrical; hence electron states are classified through projection m of momentum in the two-dimensional and orbital momentum l in the three-dimensional case, accordingly.
ð12Þ
5.1. Scattering of electrons with m ¼0 in two-dimensional and l¼0 in three-dimensional cases
in the two-dimensional and bulk semiconductors, respectively. In accordance with the obtained formulas, the energy dependences of the absorption line’s half-width in two- (11) and three-dimensional (Eq. (12)) cases are similar apart from the coefficients (impurity 2d concentration): do3d =do2d ¼ prc n3d i =ni . In both two- and threedimensional cases, scattering of fast particles occurs basically on small angles that’s why the dimension of problem has no impact on absorption line’s broadening. While electrons energy is increasing, half-widths do2d and do3d are decreasing, i.e. the absorption curves become narrower. This is caused by a decrease of scattering cross-section of fast electrons on neutral impurities with energy raising both in two- and three-dimensional cases. However a further energy raise activates carriers scattering on acoustic phonons and ionized impurities so an account must be taken of other scattering mechanisms at calculating do2d and do3d for higher energies 2 (and for higher temperatures T b2U 2 mn rc2 =‘ , correspondingly).
To find the amplitude of two- and three-dimensional scattering, we solve by a standard method [11] the two- and threedimensional Schrodinger equations for electron with m ¼0 and l¼0 in potential U. Inside the square-well potential ðr t rc Þ we neglect energy of particle; in the intermediate area ðrc t r 51=kÞ we neglect both particle energy and potential U; at the largest distances ðr \ 1=kÞ it is neglected only the potential. Matching the solutions on the areas’ boundaries one can find the wave function of particle. The scattering amplitude is determined by a coefficient in the wave function, which stands before a diverging spherical (cylindrical) wave in the three- (two-) dimensional case. Thus the scattering amplitudes for slow electrons in two- and three-dimensional case are found as4: rffiffiffiffiffiffi p eip=4 m¼0 , ð17Þ ¼ f2d 2k lnðig ka2d =2Þ
do3d
l¼0 ¼ f3d
4. Absorption line broadening due to scattering of slow electrons on shallow neutral impurities To calculate the broadening of absorption line given by slow 2 particles scattering (kr c 51) on shallow impurities U 5 ‘ =mn rc2 , we may again use Born approximation. As for inverse relaxation times Eqs. (7) and (8) are correct, we can factorize them at small values krc: ! p2 mn rc4 U 2 n2d 1 3ðkr c Þ2 i 1 , ð13Þ t2d ðkÞ 4 ‘3 8pmn rc5 U 2 n3d 1 i kr c : 3 t3d ðkÞ 9‘
ð14Þ
Half-widths of absorption lines at scattering of carriers with low 2 energy ðe 5 ‘ =mn rc2 Þ on shallow neutral impurities in two- and three-dimensional semiconductors become: p2 mn rc4 U 2 n2d 3rc2 mn i , ð15Þ do2d 1 e 2 ‘3 2‘ pffiffiffi pffiffiffi 8 2pðmn Þ3=2 rc6 U 2 n3d i ð16Þ do3d e: 4 9‘ In contrast to the three-dimensional case, in the two-dimensional one half-width of the absorption line at e-0 is finite and does not depend on energy. The different forms of energy dependences do2d (Eq. (15)) and do3d (Eq. (16)) appear due to a principal difference between low-energy scattering in the two- and threedimensional cases. The amplitude of two-dimensional scattering pffiffiffi (in Born approximation) is proportional to 1= k, whereas the amplitude of three-dimensional scattering in first approximation does not depend on energy.
5. Absorption line broadening due to scattering of slow electrons on neutral impurities with arbitrary depth Consider scattering of slow electron (kr c 5 1) on a center with finite radius and arbitrary depth U. A potential of the center is
1 , ik1=a3d
ð18Þ
where g ¼ eC 0:577 (C is the Euler constant), a2d and a3d are the lengths of two- and three-dimensional electron scattering with m¼0 and l ¼0, accordingly. The scattering lengths a2d and a3d for square-well potential U are as follows: J0 ðKrc Þ , ð19Þ a2d ¼ rc exp 0 Krc J0 ðKrc Þ J3=2 ðKrc Þ tan Krc ¼ rc 1 , ð20Þ J1=2 ðKrc Þ Krc pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 where K ¼ 2mn U =‘ ¼ U=U0 =rc and U0 ¼ ‘ =ð2mn rc2 Þ. Scattering length a2d (Eq. (19)) coincides with the length of two-dimensional scattering [12]. Cross-sections obtained from Eqs. (17) and (18) are a3d ¼ rc
s2d ¼
p2
1 , k ln2 f2=gka2d g þ p2 =4
s3d ¼ 4pa23d :
ð21Þ
ð22Þ
Substituting the scattering cross-sections (Eqs. (21) and (22)) in Eq. (4), we get inverse relaxation times of slow electrons on impurity centers in the two- and three-dimensional cases: 2
1 p2 ‘ n2d 1 p 2 ka2d g i ¼ þ ln , ð23Þ n t2d ðkÞ m 4 2 2 2p‘ n3d 1 i a3d k ¼ : n m t3d ðkÞ
ð24Þ
From Eqs. (23) and (24), we find half-width of the absorption lines at slow electrons scattering with m¼0, l ¼0 on impurities for twoand three-dimensional semiconductors: ¼0 dom ¼ 2d
4‘ n2d p2 i , n 2 m p2 þln fg2 mn a2 e=2‘ 2 g 2d
ð25Þ
4 Results are correct for any potential of impurity which is decreasing with the distance faster than 1=r 3 .
4286
do
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
l¼0 3d
pffiffiffi 2 pffiffiffi 2 2pn3d i a3d pffiffiffiffiffiffi ffi ¼ e: mn
ð26Þ
Energy dependences of CR line half-widths in two-dimensional (Eq. (25)) and bulk (Eq. (26)) semiconductors seen to be essen2 tially different. If U 5 ‘ =mn rc2 , Eq. (26) for three-dimensional case coincides with the result within Born approximation (16). In the two-dimensional case the expression for absorption line halfwidth (Eq. (25)) at the limit of shallow impurities also coincides with the calculations in Born approximation (15), but the usabil2 ity condition for Born approximation is stronger (U 5 ‘ =½mn 2 rc ln fkr c g=2g) than in three-dimensional one. Consider separately the absorption line broadening due to slow particles scattering while weakly bounded s-state in the potential well (the energy of this state is small as compared with U). Let Ures is the depth of potential well where s-state with zero energy exists and DUðDU 5 Ures Þ is the detuning of potential U from Ures. Resonant depth of the potential well is determined by the conditions of levels escape from the two- and three-dimensional potential wells: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J00 ð Ures =U0 Þ ¼ 0, ð27Þ J1=2 ð
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ures =U0 Þ ¼ 0
ð28Þ
e DU 2 =4U0 (Fig. 1). In the two-dimensional case, at the potential res well deepening the scattering length is small ares 2d 5 rc ða2d -0 at
DU-0Þ so at small energies the logarithm in Eq. (29) is large and 2
res
s2d appears to be small as well: 1=ln fka2d g. Therefore the m¼0 absorption line half-width do2d (Eq. (33)) continuously rises with growth of e and does not have a maximum see (the lower curve in Fig. 2). In the three-dimensional case, dol3d¼ 0 (Eq. (34)) does not depend on sign of the scattering length, and, hence, on sign of res DU (while DU o09ares 3d 9 b rc ,a3d o0). In the two-dimensional one the scattering length depends on sign of detuning DU, however always a2d Z 0. At depth of the potential less than its resonant value (at DU o 0), scattering occurs on a virtual level. So, scatterres ing length ares 2d brc ða2d -1 while DU-0Þ and the logarithm’s value in Eq. (33) is small. As a result, the absorption line halfwidth in the two-dimensional case has a peak at energy of carriers, corresponding to energy of the virtual level e 4U0 expf4U0 =9DU9g=g2 (see the upper curves in Fig. 2). Energies of carriers, at which do2d reaches a maximum, are significantly less than energies, at which resonant broadening of the absorption line occurs in three-dimensional semiconductors. Half-width of the line (33) is limited by a maximum value 2d n domax 2d ¼ 4‘ ni =m , which does not depend neither on the energy nor on the impurity parameters.
(the values of resonant depths for the two- and three-dimensional potential differ one from another, however we label them equally Ures). To find cross-section of resonant scattering, we factorize the amplitudes of two- (Eq. (17)) and three-dimensional (Eq. (18)) cases at DU 5Ures and use the conditions of levels escape from potential well (Eqs. (27) and (28)). The resonant electron scattering cross-sections with m¼0, l¼0 in the two- and three-dimensional cases are as follows:
sres 2d ¼ sres 3d ¼
p2
1 , k ln2 f2=gkares g þ p2 =4 2d 4p
k2 þ1=ðares Þ2 3d
ð29Þ
,
ð30Þ
and the length of resonant scattering of particles on the potential with depth U ¼ Ures þ DU becomes: 2U0 , ð31Þ ares 2d ¼ rc exp DU ares 3d ¼ rc
2U0 : DU
ð32Þ
Fig. 1. The dependence of the absorption line half-width on energy of carriers in bulk semiconductors for various detunings of a separate impurity depth.
Using Eqs. (29) and (30), we find half-width of absorption line at the resonant scattering of slow electrons with m¼ 0, l¼0 on neutral impurities in plane and bulk semiconductors to be: ¼0 dom ¼ 2d
4‘ n2d p2 i , mn p2 þ ln2 fg2 mn ðares Þ2 e=2‘ 2 g
ð33Þ
2d
dol3d¼ 0 ¼
pffiffiffi 2 2 2p‘ n3d i ðmn Þ3=2
pffiffiffi
e : e þ U0 ðrc =ares Þ2 3d
ð34Þ
Let’s analyze broadening of absorption line in the case of resonant electron scattering on neutral impurities in two-dimensional (Eq. (33)) and bulk (Eq. (34)) semiconductors. There is a level with zero energy in the well with resonant depth, and this level becomes deeper at depth increasing. In the tree-dimensional res case, the scattering length is large and positive (ares 3d b rc ,a3d 40) while deepening the well ðU ¼ Ures þ DU, DU 40Þ. Scattering crosssection is resonant if a scattered electron has energy close to energy of a weak-bounded state: s3d 1=k2 b rc2 . Hence halfpffiffiffi width of absorption line increases resonantly dol3d¼ 0 1= e at energies comparable with energy of the weak-bounded state
Fig. 2. The dependence of the absorption line half-width on the energy of carriers in plane semiconductors for the various detunings of a separate impurity depth.
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
5.2. The electron scattering with m a0 in two- and l a 0 in threedimensional cases Consider broadening of the absorption line under scattering of slow electrons with nonzero projection of angular momentum and nonzero angular momentum on the impurity center in twoand three-dimensional semiconductors, correspondingly. Solving the Schrodinger equation for the electron with arbitrary m a0ðl a0Þ scattering on the two-(three)-dimensional rectangular potential well, we can find partial scattering amplitudes. Calculating partial scattering cross-sections and substituting them in Eq. (4), we can obtain relaxation time of electrons with m a0 ðl a 0Þ and get half-widths of the absorption lines in twoand three-dimensional semiconductors. At scattering of electrons with 9m9 ¼ 1 in two-dimensional and l ¼1 in bulk semiconductors on neutral impurities, the absorption line half-widths become:
n 4p2 n2d J2 ðKrc Þ 2 2 9m9 ¼ 1 i m do2d ¼ e , ð35Þ J0 ðKrc Þ ‘3
dol3d¼ 1 ¼
pffiffiffi
6 n 3=2 J5=2 ðKrc Þ 2 5=2 2 2pn3d i rc ðm Þ e : J1=2 ðKrc Þ ‘4
ð36Þ
In the three-dimensional case, the partial contribution of electrons with nonzero angular momentum in the total cross section is small compared with the contribution of electrons with l ¼0. The resonant scattering cross-section with lþ1 gives the same contribution of the order of magnitude as the non-resonant scattering cross-section with l [11] therefore on the background of dol3d¼ 0 (Eq. (26)) we are interested in additional broadening of the absorption line only due to the resonant scattering of electrons with l ¼1. Resonant electron scattering cross-section with l¼1 is calculated by analogy with resonant electron scattering cross-section with l ¼0. The levels with l a 0 escapes the three-dimensional rectangular potential well at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jl1=2 ð Ures =U0 Þ ¼ 0, ð37Þ so this equality defines the resonant depth of the impurity potential Ures for arbitrary l. Then we obtain half-width of the absorption curve in the case of resonant scattering of electrons with l ¼1 on neutral impurities in bulk semiconductors: pffiffiffi pffiffiffi 2 2p2 rc2 n3d e pffiffiffiffiffiffiffi i : ð38Þ dol3d¼ 1 ¼ mn ð1þ DU=eÞ2 þ5=4
4287
(Eq. (33)) at scattering of electrons with m¼ 0 on the real level. Broadening of the classical CR line due to scattering of electrons with 9m9 ¼ 1 on the real level (at DU 40) is less than to scattering of electrons on the virtual level (while DU o 0). Partial amplitudes of the scattering of electron with 9m9 Z 2 ðl Z2Þ are small compared to the amplitude of s-scattering, even if the weakly bound level with 9m9 Z 2 ðl Z2Þ exists in the potential well. We do not provide here the formulas for halfwidth of the absorption line at scattering of electrons with arbitrary 9m9 and l. We only give (see below) the main dependences of half-widths on energy and detuning of the potential at small DU for two-dimensional and bulk semiconductors: 9m9 Z 2
do2d
dol3dZ 2
e29m92 ðe=U0 Þ
29m92
þð1 þ DU=eÞ2
e29m93=2 ðe=U0 Þ
29m93=2
þ ð1 þ DU=eÞ2
,
ð41Þ
:
ð42Þ
6. Comparison with experiment and calculations by modified Erginsoy formula The experiment described in Ref. [1] was carried out at temperatures ð1:625:5Þ K in n-Ge samples doped with Sb at concentrations ð1:523:4Þ 1014 cm3 . Fig. 3 shows a comparison between inverse relaxation time obtained in the present work and one from Ref. [1]. The experimental data are given for a sample with concentration of neutral donors 2:6 1014 cm3 , effective ˚ and effective mass of carriers Bohr radius of impurities 44:6 A, mn ¼ 0:22me , where me is the mass of free electron. Temperature dependence of inverse relaxation time was obtained by averaging Eq. (34) with the use of Boltzmann distribution function. The temperature dependence of inverse relaxation time, obtained in the present work, with a good accuracy matches the experimental data [1]. We choose the value of detuning of the impurity potential DU equal to 6.6 meV. This allowed us to make a conclusion that scattering of electrons by neutral impurities, which takes place in the experimental conditions [1], is not strictly resonant. In the meantime, if the resonance character of scattering is not taken into account, it seems to be impossible to describe the experimental values of inverse relaxation time with good accuracy. According to our calculations, the resonant
In a bulk semiconductor under resonant scattering of electrons with l ¼1 the absorption line half-width is a monotonically rising pffiffiffi function of carrier energy dol3d¼ 1 e (this is true for all l a0), in contrast to half-width of the classical CR line dol3d¼ 0 at resonant electron scattering with l¼0, which has a sharp maximum. Consider scattering of electrons on a two-dimensional potential well with weakly bounded state with 9m9 ¼ 1 inside. For m a0, resonant depth Ures is defined by the condition: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð39Þ J9m91 ð Ures =U0 Þ ¼ 0: Using Eq. (39), we obtain the absorption line half-width at resonant scattering of electrons with 9m9 ¼ 1 on neutral impurities in two-dimensional semiconductors: 9m9 ¼ 1
do2d ¼ 2d 4‘ ni
e=U0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi : mn e=U0 þ ½DU=ðp Ures U0 Þð1 þ e=DUÞ þ e=U0 lnð4U0 =g2 eÞ2 ð40Þ
At resonant scattering of electrons with 9m9 ¼ 1 on the virtual or real level, half-width of the absorption line increases monotoni2 ¼0 cally with increasing energy 1=ln feg as it does dom ðeÞ 2d
Fig. 3. Dependence of inverse relaxation time on temperature at electrons scattering by neutral donors in bulk semiconductors. The experimental data of Ref. [1] are shown by crosses; solid curve is built using Eq. (34), averaged over energy; dashed curve represents the theoretical results [1].
4288
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
scattering length is 117 A˚ (see Eq. (34)), while the energy value of the bounded level in the impurity potential is 1.3 meV. Thus, if one can find experimentally the temperature dependence of relaxation time by neutral impurities, a value of the potential detuning and energy of weakly bounded state in the impurity potential can be obtained with good accuracy by Eq. (34).
[3] [4] [5] [6] [7]
References
[9]
[1] L.E. Blagosklonskaya, E.M. Gershenzon, Yu.P. Ladizhinskiy, A.P. Popova, Fiz. Tverd. Tela 11 (1969) 2967. [2] L.E. Blagosklonskaya, E.M. Gershenzon, Yu.A. Gurvich, N.G. Ptitzina, N.A. Serebryakova, Fiz. Tverd. Tela 8 (1966) 332; E.M. Gershenzon, Yu.A. Gurvich, S. L Orlova, N.G. Ptitzina, Zh. Eksp. Theor. Fiz. 67 (1974) 627.
[8]
[10] [11] [12]
E.J. Otsuka, J. Appl. Phys. Jpn. 25 (1986) 303. H. Kobori, T. Ohyama, E. Otsuka, J. Phys. Soc. Jpn. 59 (1990) 2141. S.P. Andreev, Usp. Fiz. Nauk. 143 (1984) 213. S.P. Andreev, T.V. Pavlova, Physica E 40 (2008) 1377. A.S. Andreev, S.P. Andreev, Yu.A. Gurvich, I.F. Podlivaev, Laser Phys. 6 (1996) 1214; A.S. Andreev, S.P. Andreev, Yu.A. Gurvich, Laser Phys. 7 (1997) 488. S.P. Andreev, T.V. Pavlova, Laser Phys. 12 (2002) 1381; S.P. Andreev, T.V. Pavlova, Laser Phys. 13 (2003) 897. K.M. Itoh, W. Walukiewicz, H.D. Fuchs, J.W. Beeman, E.E. Haller, J.W. Farmer, V.I. Ozhogin, Phys. Rev. B 50 (1994) 16995. P.S. Zyrianov, M.I. Klinger, Quantum Theory of Electron Kinetic Phenomena in Crystallite Semiconductors, Nauka, Moscow, 1976. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Pergamon, Oxford, 1977. B.J. Verhaar, L.P.H. de Goey, J.P.H.W. van den Eijnde, E.J.D. Vredenbregt, Phys. Rev. A 32 (1985) 1424.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
On the theory of classical cyclotron resonance line broadening in two- and three-dimensional semiconductors S.P. Andreev a, Yu. A. Gurvich b, V.A. Nebogatov a, T.V. Pavlova c,n a
National Research Nuclear University MEPHI, Kashirskoe Sh. 31, 115409 Moscow, Russia Moscow State Pedagogical University, M. Pirogovskaya Str. 1, 119991 Moscow, Russia c Prokhorov General Physics Institute, Russian Academy of Science, Vavilov Str. 38, 119991 Moscow, Russia b
a r t i c l e i n f o
abstract
Article history: Received 22 December 2010 Accepted 22 August 2011 Available online 25 August 2011
A broadening of the absorption line of the classical cyclotron resonance (CR) in two- and three-dimensional semiconductors with neutral impurities of arbitrary depth is investigated. A dependence of the half-width of the classical CR line on the impurity characteristics (depth and range of the potential) is obtained in the wide range of the parameters. The broadening is studied within the Born approximation, and also at resonant and non-resonant scattering. Fundamental differences between the line broadening by neutral impurities in two- and three-dimensional semiconductors are revealed. An alternative explanation of classical experiments on electron scattering by neutral impurities (donors) is proposed. & 2011 Published by Elsevier B.V.
Keywords: Neutral impurities Two-dimensional gas Cyclotron resonance Resonant scattering
1. Introduction The cyclotron resonance (CR) in semiconductors is a powerful tool for getting versatile information about the interaction of carriers with lattice vibrations and defects, types and concentration of the defects, etc. (see Refs. [1–4] and references therein). The measurement of an absorption line’s half-width do gives an opportunity to identify various mechanisms of charge carriers scattering, since do is proportional to inverse relaxation time of current and, roughly, represents the sum of probabilities of the scattering mechanisms. At essentially low temperatures, scattering by phonons is suppressed. A specific role of CR at a study of impurities arises from its selectivity, since each scattering mechanism, generally speaking, gives rise to own specific shape of the absorption line. Broadening of the classical CR line by ionized impurities in bulk semiconductors has been satisfactory investigated. The situation with neutral impurities was inspected worse. To describe the broadening of the classical CR line by shallow neutral donors, the modified Erginsoy formula is used [1]. The modification originates from an account of the polarization effects on the scattering cross-section by the hydrogen atom. At this the effective Bohr radius and the dielectric constant of a semiconductor are substituted to the cross-section, noted above. The approach [1] allows one to get a satisfactory agreement between the theory and experiments. However, it can not be accepted as comprehensive, because it needs the use of a modulating function,
n
Corresponding author. Tel.: þ7 916 9109810. E-mail address: [email protected] (T.V. Pavlova).
0921-4526/$ - see front matter & 2011 Published by Elsevier B.V. doi:10.1016/j.physb.2011.08.051
introduced ‘‘manually’’, which provides a correct energy dependence of the scattering cross-section of electrons by shallow neutral donors in the desired range of temperatures. At changing depth of the neutral impurity potential, the character of carriers scattering is also changed. The scattering becomes resonant ones a real or virtual electron state appears in the impurity potential. Except this, in the presence of a magnetic field there can appear specific magnetic-impurity states in the impurity potential, which radically affect the kinetics of low-temperature phenomena [5,6]. Account of carriers resonant scattering by these states has yielded a few important results [5]. In particular, this allowed one to eliminate discrepancy between the theory and experiments on broadening of a quantum CR line by neutral impurities in bulk semiconductors [7,8], which took place for more than 30 years.1 In the case of classical CR, the role of resonant states in the potential of neutral centers on absorption line broadening was not discussed at all. To date, the following issues remain open: (1) Dependence of classical CR line half-width on depth of the potential of a separate impurity, particularly, in the conditions of resonant electron scattering in bulk semiconductors at various temperatures.2 1 All efforts to achieve agreement between the dependences obtained in experiments and theory were made on the base of Erginsoy’s formula, leading to discrepancies of more than an order of magnitude [4]. 2 In Ref. [9], it was obtained the dependence of relaxation time on temperature, which was explained by using the scattering cross-section of slow electrons by the hydrogen atom with all phase shifts accounted. At the same time, the
4284
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
(2) Derivation of the classical CR line broadening by shallow neutral donors in the three-dimensional semiconductors in the frames of a quantum-mechanical consideration, like it is done in the case of the quantum CR [7,8]. (3) Analysis of the classical CR line broadening in two-dimensional semiconductors because of the presence of neutral impurities with the potential of arbitrary depth. (4) Comparison of the features of the classical CR line broadening in two- and three-dimensional structures. This point has fundamental importance, since the corresponding dependences of the scattering cross-sections on depth of the impurity potential considerably differ. The present paper is dedicated to a thorough analysis of the issues listed above.
2. Half-width of the absorption line Consider a two-(three-)dimensional semiconductor placed in ~ and weak resonant electric fields the crossed uniform magnetic H iot ~ ~ EðtÞ ¼ E 0 e ðooH 5 oÞ (in two-dimensional case the magnetic field has the direction perpendicular to free surface of semiconductor). Let non-degenerate gas of noninteractive electrons in the semiconductor scatters on chaotically distributed neutral impurities under conditions of the classical CR ðT b ‘ oH Þ. Concentration of impurities is assumed to be low, i.e. simultaneous carriers scattering on two and more centers is not accounted for, implying gas approximation holds [10]. The magnetic field is assumed to be that weak to neglect its influence upon the interaction between electrons and impurities. We use the kinetic Boltzmann equation for electrons momentum ð‘ ~ kÞ distribution function f in crossed electric and magnetic fields in the presence of scatterers: ! ~ @f @f e~ E e½~ k, H þ ð1Þ ¼ Ist , @t mn c @~ ‘ k where Ist ¼ f =tNd , tNd the electron momentum relaxation time in a N -dimensional semiconductor, mn the effective mass of charge carriers assumed to be isotropic. Assuming weakness of the pffiffiffiffiffiffiffiffiffi ffi electric field ðE0 5 oH mn T =e, i.e. neglecting electron heating through the cyclotron period), we reduce the distribution function at the first approximation to the form: f ¼ f ð0Þ þf ð1Þ , where f ð0Þ is an equilibrium distribution function and f ð1Þ is a linear summand. In the weak fields while degeneracy of the electrons is absent, f ð0Þ is the Boltzmann distribution function. Electric current expressed in terms of the distribution function is nNd e‘ ~ J¼ e n m
Z
~ kf ð1Þ d~ k
ð2Þ
(nNd e is the electrons concentration in N-dimensional volume, with 2d N ¼ 3n3d e being the volumetric electrons concentration and N ¼ 2ne the surface electrons concentration). An absorption line in such a semiconductor is characterized by a ratio of average energy absorbed by carriers through the cyclotron period to average electromagnetic energy density: ~ J~ E QNd ¼
2 ~ E =4p
2
¼
2 2pnNd e e ‘ Im mn2 T
Z
Assume further that the condition oH tNd b 1 holds, i.e. that broadening of levels due to carriers scattering on neutral impurities is small in comparison with a distance between the levels. The relaxation time is defined by the scattering cross-section: Z 1 ‘k dsNd ð1cos WÞ, ¼ nNd ð4Þ i mn tNd where dsNd is the differential scattering cross-section of charge carriers on neutral impurities in two-dimensional (N¼2) or bulk (N¼ 3) semiconductors, W the scattering angle, nNd the amount of i scatterers per unit of N-dimensional volume (n3d the volumetric i concentration of impurities and n2d i surface concentration of impurities). Let us consider a model potential of a separate impurity to be central-symmetric two- or three-dimensional square-well with depth U and radius rc for the two- and three-dimensional semiconductors, respectively. Stress here that the following formulas are expressed in terms of two- and three-dimensional scattering lengths, not depending on potential’s shape, viz., the present results are model-independent. We will consider further the dependence of an absorption 2 line’s half-width on carries energy e ðe ¼ ‘ k2 =2mn Þ at various depths of impurity potential U, where the absorption line halfwidth is estimated as doNd ðeÞ ¼ 1=tNd ðkÞ.
3. Absorption line broadening due to scattering of fast electrons on neutral impurities Consider broadening of absorption line while presence of scattering of fast electrons (kr c b 1) on neutral impurities. In Born approx2 imation, the impurity potential depth is limited by U 5 ‘ k=mn rc . Two-dimensional ds2d and three-dimensional ds3d differential scattering cross-sections3 obtained in Born approximation are !2 2p Ur 2c mn J1 ðqr c Þ ds2d ¼ d W, ð5Þ qr c k ‘2 8prc2 ds3d ¼ k2
!2 Ur 2c mn sinðqr c Þqr c cosðqr c Þ
‘2
ðqr c Þ3
q dq,
ð6Þ
where q ¼ 2k sinðW=2Þ and J is the Bessel function. Substituting Eqs. (5) and (6) in Eq. (4), we obtain the inverse relaxation time of fast electrons on neutral impurities in two-dimensional ð1=t2d Þ and bulk ð1=t3d Þ semiconductors as: 4pmn rc2 U 2 n2d 1 i ¼ t2d ðkÞ ‘ 3 k2
Z p=2
8pmn rc2 U 2 n3d 1 i ¼ t3d ðkÞ ‘ 3 k3
Z
0
2krc
J12 ð2qr c sin wÞdw,
ðsin ww cos wÞ2
w3
0
ð7Þ
dw:
ð8Þ
Under the integrals signs in Eqs. (7) and (8) we factorize functions 2 sin w and cos w, using condition kr c b Umn rc2 =‘ . Leaving in Eqs. (7) and (8) only the main terms, we get: 4mn rc4 U 2 n2d 1 i ln kr c , t2d ðkÞ ðkr c Þ3 ‘3
ð9Þ
2
k3 expð‘ k2 =2mn TÞ dk: ooH þ i=tNd ðkÞ
ð3Þ
(footnote continued) question about dependence of relaxation time on depth of the impurity potential remained open.
1
t3d ðkÞ
4pmn rc5 U 2 n3d i ln kr c
‘3
ðkr c Þ3
:
ð10Þ
3 Note that dimension of scattering cross-section in two-dimensional case is length.
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
For the absorption line’s half-width at carriers scattering on neutral 2 impurities at large energies (e b 2U 2 mn rc2 =‘ ), we find: r U 2 n2d ln ð2rc2 mn e=‘ 2 Þ
c do2d pffiffiffiffiffiffiffiffiffiinffi
2m
e3=2
,
2 prc2 U 2 n3d ln ð2rc2 mn e=‘ Þ pffiffiffiffiffiffiffiffiffiffii e3=2 2mn
ð11Þ
4285
assumed to be center-symmetrical; hence electron states are classified through projection m of momentum in the two-dimensional and orbital momentum l in the three-dimensional case, accordingly.
ð12Þ
5.1. Scattering of electrons with m ¼0 in two-dimensional and l¼0 in three-dimensional cases
in the two-dimensional and bulk semiconductors, respectively. In accordance with the obtained formulas, the energy dependences of the absorption line’s half-width in two- (11) and three-dimensional (Eq. (12)) cases are similar apart from the coefficients (impurity 2d concentration): do3d =do2d ¼ prc n3d i =ni . In both two- and threedimensional cases, scattering of fast particles occurs basically on small angles that’s why the dimension of problem has no impact on absorption line’s broadening. While electrons energy is increasing, half-widths do2d and do3d are decreasing, i.e. the absorption curves become narrower. This is caused by a decrease of scattering cross-section of fast electrons on neutral impurities with energy raising both in two- and three-dimensional cases. However a further energy raise activates carriers scattering on acoustic phonons and ionized impurities so an account must be taken of other scattering mechanisms at calculating do2d and do3d for higher energies 2 (and for higher temperatures T b2U 2 mn rc2 =‘ , correspondingly).
To find the amplitude of two- and three-dimensional scattering, we solve by a standard method [11] the two- and threedimensional Schrodinger equations for electron with m ¼0 and l¼0 in potential U. Inside the square-well potential ðr t rc Þ we neglect energy of particle; in the intermediate area ðrc t r 51=kÞ we neglect both particle energy and potential U; at the largest distances ðr \ 1=kÞ it is neglected only the potential. Matching the solutions on the areas’ boundaries one can find the wave function of particle. The scattering amplitude is determined by a coefficient in the wave function, which stands before a diverging spherical (cylindrical) wave in the three- (two-) dimensional case. Thus the scattering amplitudes for slow electrons in two- and three-dimensional case are found as4: rffiffiffiffiffiffi p eip=4 m¼0 , ð17Þ ¼ f2d 2k lnðig ka2d =2Þ
do3d
l¼0 ¼ f3d
4. Absorption line broadening due to scattering of slow electrons on shallow neutral impurities To calculate the broadening of absorption line given by slow 2 particles scattering (kr c 51) on shallow impurities U 5 ‘ =mn rc2 , we may again use Born approximation. As for inverse relaxation times Eqs. (7) and (8) are correct, we can factorize them at small values krc: ! p2 mn rc4 U 2 n2d 1 3ðkr c Þ2 i 1 , ð13Þ t2d ðkÞ 4 ‘3 8pmn rc5 U 2 n3d 1 i kr c : 3 t3d ðkÞ 9‘
ð14Þ
Half-widths of absorption lines at scattering of carriers with low 2 energy ðe 5 ‘ =mn rc2 Þ on shallow neutral impurities in two- and three-dimensional semiconductors become: p2 mn rc4 U 2 n2d 3rc2 mn i , ð15Þ do2d 1 e 2 ‘3 2‘ pffiffiffi pffiffiffi 8 2pðmn Þ3=2 rc6 U 2 n3d i ð16Þ do3d e: 4 9‘ In contrast to the three-dimensional case, in the two-dimensional one half-width of the absorption line at e-0 is finite and does not depend on energy. The different forms of energy dependences do2d (Eq. (15)) and do3d (Eq. (16)) appear due to a principal difference between low-energy scattering in the two- and threedimensional cases. The amplitude of two-dimensional scattering pffiffiffi (in Born approximation) is proportional to 1= k, whereas the amplitude of three-dimensional scattering in first approximation does not depend on energy.
5. Absorption line broadening due to scattering of slow electrons on neutral impurities with arbitrary depth Consider scattering of slow electron (kr c 5 1) on a center with finite radius and arbitrary depth U. A potential of the center is
1 , ik1=a3d
ð18Þ
where g ¼ eC 0:577 (C is the Euler constant), a2d and a3d are the lengths of two- and three-dimensional electron scattering with m¼0 and l ¼0, accordingly. The scattering lengths a2d and a3d for square-well potential U are as follows: J0 ðKrc Þ , ð19Þ a2d ¼ rc exp 0 Krc J0 ðKrc Þ J3=2 ðKrc Þ tan Krc ¼ rc 1 , ð20Þ J1=2 ðKrc Þ Krc pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi 2 where K ¼ 2mn U =‘ ¼ U=U0 =rc and U0 ¼ ‘ =ð2mn rc2 Þ. Scattering length a2d (Eq. (19)) coincides with the length of two-dimensional scattering [12]. Cross-sections obtained from Eqs. (17) and (18) are a3d ¼ rc
s2d ¼
p2
1 , k ln2 f2=gka2d g þ p2 =4
s3d ¼ 4pa23d :
ð21Þ
ð22Þ
Substituting the scattering cross-sections (Eqs. (21) and (22)) in Eq. (4), we get inverse relaxation times of slow electrons on impurity centers in the two- and three-dimensional cases: 2
1 p2 ‘ n2d 1 p 2 ka2d g i ¼ þ ln , ð23Þ n t2d ðkÞ m 4 2 2 2p‘ n3d 1 i a3d k ¼ : n m t3d ðkÞ
ð24Þ
From Eqs. (23) and (24), we find half-width of the absorption lines at slow electrons scattering with m¼0, l ¼0 on impurities for twoand three-dimensional semiconductors: ¼0 dom ¼ 2d
4‘ n2d p2 i , n 2 m p2 þln fg2 mn a2 e=2‘ 2 g 2d
ð25Þ
4 Results are correct for any potential of impurity which is decreasing with the distance faster than 1=r 3 .
4286
do
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
l¼0 3d
pffiffiffi 2 pffiffiffi 2 2pn3d i a3d pffiffiffiffiffiffi ffi ¼ e: mn
ð26Þ
Energy dependences of CR line half-widths in two-dimensional (Eq. (25)) and bulk (Eq. (26)) semiconductors seen to be essen2 tially different. If U 5 ‘ =mn rc2 , Eq. (26) for three-dimensional case coincides with the result within Born approximation (16). In the two-dimensional case the expression for absorption line halfwidth (Eq. (25)) at the limit of shallow impurities also coincides with the calculations in Born approximation (15), but the usabil2 ity condition for Born approximation is stronger (U 5 ‘ =½mn 2 rc ln fkr c g=2g) than in three-dimensional one. Consider separately the absorption line broadening due to slow particles scattering while weakly bounded s-state in the potential well (the energy of this state is small as compared with U). Let Ures is the depth of potential well where s-state with zero energy exists and DUðDU 5 Ures Þ is the detuning of potential U from Ures. Resonant depth of the potential well is determined by the conditions of levels escape from the two- and three-dimensional potential wells: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J00 ð Ures =U0 Þ ¼ 0, ð27Þ J1=2 ð
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ures =U0 Þ ¼ 0
ð28Þ
e DU 2 =4U0 (Fig. 1). In the two-dimensional case, at the potential res well deepening the scattering length is small ares 2d 5 rc ða2d -0 at
DU-0Þ so at small energies the logarithm in Eq. (29) is large and 2
res
s2d appears to be small as well: 1=ln fka2d g. Therefore the m¼0 absorption line half-width do2d (Eq. (33)) continuously rises with growth of e and does not have a maximum see (the lower curve in Fig. 2). In the three-dimensional case, dol3d¼ 0 (Eq. (34)) does not depend on sign of the scattering length, and, hence, on sign of res DU (while DU o09ares 3d 9 b rc ,a3d o0). In the two-dimensional one the scattering length depends on sign of detuning DU, however always a2d Z 0. At depth of the potential less than its resonant value (at DU o 0), scattering occurs on a virtual level. So, scatterres ing length ares 2d brc ða2d -1 while DU-0Þ and the logarithm’s value in Eq. (33) is small. As a result, the absorption line halfwidth in the two-dimensional case has a peak at energy of carriers, corresponding to energy of the virtual level e 4U0 expf4U0 =9DU9g=g2 (see the upper curves in Fig. 2). Energies of carriers, at which do2d reaches a maximum, are significantly less than energies, at which resonant broadening of the absorption line occurs in three-dimensional semiconductors. Half-width of the line (33) is limited by a maximum value 2d n domax 2d ¼ 4‘ ni =m , which does not depend neither on the energy nor on the impurity parameters.
(the values of resonant depths for the two- and three-dimensional potential differ one from another, however we label them equally Ures). To find cross-section of resonant scattering, we factorize the amplitudes of two- (Eq. (17)) and three-dimensional (Eq. (18)) cases at DU 5Ures and use the conditions of levels escape from potential well (Eqs. (27) and (28)). The resonant electron scattering cross-sections with m¼0, l¼0 in the two- and three-dimensional cases are as follows:
sres 2d ¼ sres 3d ¼
p2
1 , k ln2 f2=gkares g þ p2 =4 2d 4p
k2 þ1=ðares Þ2 3d
ð29Þ
,
ð30Þ
and the length of resonant scattering of particles on the potential with depth U ¼ Ures þ DU becomes: 2U0 , ð31Þ ares 2d ¼ rc exp DU ares 3d ¼ rc
2U0 : DU
ð32Þ
Fig. 1. The dependence of the absorption line half-width on energy of carriers in bulk semiconductors for various detunings of a separate impurity depth.
Using Eqs. (29) and (30), we find half-width of absorption line at the resonant scattering of slow electrons with m¼ 0, l¼0 on neutral impurities in plane and bulk semiconductors to be: ¼0 dom ¼ 2d
4‘ n2d p2 i , mn p2 þ ln2 fg2 mn ðares Þ2 e=2‘ 2 g
ð33Þ
2d
dol3d¼ 0 ¼
pffiffiffi 2 2 2p‘ n3d i ðmn Þ3=2
pffiffiffi
e : e þ U0 ðrc =ares Þ2 3d
ð34Þ
Let’s analyze broadening of absorption line in the case of resonant electron scattering on neutral impurities in two-dimensional (Eq. (33)) and bulk (Eq. (34)) semiconductors. There is a level with zero energy in the well with resonant depth, and this level becomes deeper at depth increasing. In the tree-dimensional res case, the scattering length is large and positive (ares 3d b rc ,a3d 40) while deepening the well ðU ¼ Ures þ DU, DU 40Þ. Scattering crosssection is resonant if a scattered electron has energy close to energy of a weak-bounded state: s3d 1=k2 b rc2 . Hence halfpffiffiffi width of absorption line increases resonantly dol3d¼ 0 1= e at energies comparable with energy of the weak-bounded state
Fig. 2. The dependence of the absorption line half-width on the energy of carriers in plane semiconductors for the various detunings of a separate impurity depth.
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
5.2. The electron scattering with m a0 in two- and l a 0 in threedimensional cases Consider broadening of the absorption line under scattering of slow electrons with nonzero projection of angular momentum and nonzero angular momentum on the impurity center in twoand three-dimensional semiconductors, correspondingly. Solving the Schrodinger equation for the electron with arbitrary m a0ðl a0Þ scattering on the two-(three)-dimensional rectangular potential well, we can find partial scattering amplitudes. Calculating partial scattering cross-sections and substituting them in Eq. (4), we can obtain relaxation time of electrons with m a0 ðl a 0Þ and get half-widths of the absorption lines in twoand three-dimensional semiconductors. At scattering of electrons with 9m9 ¼ 1 in two-dimensional and l ¼1 in bulk semiconductors on neutral impurities, the absorption line half-widths become:
n 4p2 n2d J2 ðKrc Þ 2 2 9m9 ¼ 1 i m do2d ¼ e , ð35Þ J0 ðKrc Þ ‘3
dol3d¼ 1 ¼
pffiffiffi
6 n 3=2 J5=2 ðKrc Þ 2 5=2 2 2pn3d i rc ðm Þ e : J1=2 ðKrc Þ ‘4
ð36Þ
In the three-dimensional case, the partial contribution of electrons with nonzero angular momentum in the total cross section is small compared with the contribution of electrons with l ¼0. The resonant scattering cross-section with lþ1 gives the same contribution of the order of magnitude as the non-resonant scattering cross-section with l [11] therefore on the background of dol3d¼ 0 (Eq. (26)) we are interested in additional broadening of the absorption line only due to the resonant scattering of electrons with l ¼1. Resonant electron scattering cross-section with l¼1 is calculated by analogy with resonant electron scattering cross-section with l ¼0. The levels with l a 0 escapes the three-dimensional rectangular potential well at pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Jl1=2 ð Ures =U0 Þ ¼ 0, ð37Þ so this equality defines the resonant depth of the impurity potential Ures for arbitrary l. Then we obtain half-width of the absorption curve in the case of resonant scattering of electrons with l ¼1 on neutral impurities in bulk semiconductors: pffiffiffi pffiffiffi 2 2p2 rc2 n3d e pffiffiffiffiffiffiffi i : ð38Þ dol3d¼ 1 ¼ mn ð1þ DU=eÞ2 þ5=4
4287
(Eq. (33)) at scattering of electrons with m¼ 0 on the real level. Broadening of the classical CR line due to scattering of electrons with 9m9 ¼ 1 on the real level (at DU 40) is less than to scattering of electrons on the virtual level (while DU o 0). Partial amplitudes of the scattering of electron with 9m9 Z 2 ðl Z2Þ are small compared to the amplitude of s-scattering, even if the weakly bound level with 9m9 Z 2 ðl Z2Þ exists in the potential well. We do not provide here the formulas for halfwidth of the absorption line at scattering of electrons with arbitrary 9m9 and l. We only give (see below) the main dependences of half-widths on energy and detuning of the potential at small DU for two-dimensional and bulk semiconductors: 9m9 Z 2
do2d
dol3dZ 2
e29m92 ðe=U0 Þ
29m92
þð1 þ DU=eÞ2
e29m93=2 ðe=U0 Þ
29m93=2
þ ð1 þ DU=eÞ2
,
ð41Þ
:
ð42Þ
6. Comparison with experiment and calculations by modified Erginsoy formula The experiment described in Ref. [1] was carried out at temperatures ð1:625:5Þ K in n-Ge samples doped with Sb at concentrations ð1:523:4Þ 1014 cm3 . Fig. 3 shows a comparison between inverse relaxation time obtained in the present work and one from Ref. [1]. The experimental data are given for a sample with concentration of neutral donors 2:6 1014 cm3 , effective ˚ and effective mass of carriers Bohr radius of impurities 44:6 A, mn ¼ 0:22me , where me is the mass of free electron. Temperature dependence of inverse relaxation time was obtained by averaging Eq. (34) with the use of Boltzmann distribution function. The temperature dependence of inverse relaxation time, obtained in the present work, with a good accuracy matches the experimental data [1]. We choose the value of detuning of the impurity potential DU equal to 6.6 meV. This allowed us to make a conclusion that scattering of electrons by neutral impurities, which takes place in the experimental conditions [1], is not strictly resonant. In the meantime, if the resonance character of scattering is not taken into account, it seems to be impossible to describe the experimental values of inverse relaxation time with good accuracy. According to our calculations, the resonant
In a bulk semiconductor under resonant scattering of electrons with l ¼1 the absorption line half-width is a monotonically rising pffiffiffi function of carrier energy dol3d¼ 1 e (this is true for all l a0), in contrast to half-width of the classical CR line dol3d¼ 0 at resonant electron scattering with l¼0, which has a sharp maximum. Consider scattering of electrons on a two-dimensional potential well with weakly bounded state with 9m9 ¼ 1 inside. For m a0, resonant depth Ures is defined by the condition: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð39Þ J9m91 ð Ures =U0 Þ ¼ 0: Using Eq. (39), we obtain the absorption line half-width at resonant scattering of electrons with 9m9 ¼ 1 on neutral impurities in two-dimensional semiconductors: 9m9 ¼ 1
do2d ¼ 2d 4‘ ni
e=U0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi : mn e=U0 þ ½DU=ðp Ures U0 Þð1 þ e=DUÞ þ e=U0 lnð4U0 =g2 eÞ2 ð40Þ
At resonant scattering of electrons with 9m9 ¼ 1 on the virtual or real level, half-width of the absorption line increases monotoni2 ¼0 cally with increasing energy 1=ln feg as it does dom ðeÞ 2d
Fig. 3. Dependence of inverse relaxation time on temperature at electrons scattering by neutral donors in bulk semiconductors. The experimental data of Ref. [1] are shown by crosses; solid curve is built using Eq. (34), averaged over energy; dashed curve represents the theoretical results [1].
4288
S.P. Andreev et al. / Physica B 406 (2011) 4283–4288
scattering length is 117 A˚ (see Eq. (34)), while the energy value of the bounded level in the impurity potential is 1.3 meV. Thus, if one can find experimentally the temperature dependence of relaxation time by neutral impurities, a value of the potential detuning and energy of weakly bounded state in the impurity potential can be obtained with good accuracy by Eq. (34).
[3] [4] [5] [6] [7]
References
[9]
[1] L.E. Blagosklonskaya, E.M. Gershenzon, Yu.P. Ladizhinskiy, A.P. Popova, Fiz. Tverd. Tela 11 (1969) 2967. [2] L.E. Blagosklonskaya, E.M. Gershenzon, Yu.A. Gurvich, N.G. Ptitzina, N.A. Serebryakova, Fiz. Tverd. Tela 8 (1966) 332; E.M. Gershenzon, Yu.A. Gurvich, S. L Orlova, N.G. Ptitzina, Zh. Eksp. Theor. Fiz. 67 (1974) 627.
[8]
[10] [11] [12]
E.J. Otsuka, J. Appl. Phys. Jpn. 25 (1986) 303. H. Kobori, T. Ohyama, E. Otsuka, J. Phys. Soc. Jpn. 59 (1990) 2141. S.P. Andreev, Usp. Fiz. Nauk. 143 (1984) 213. S.P. Andreev, T.V. Pavlova, Physica E 40 (2008) 1377. A.S. Andreev, S.P. Andreev, Yu.A. Gurvich, I.F. Podlivaev, Laser Phys. 6 (1996) 1214; A.S. Andreev, S.P. Andreev, Yu.A. Gurvich, Laser Phys. 7 (1997) 488. S.P. Andreev, T.V. Pavlova, Laser Phys. 12 (2002) 1381; S.P. Andreev, T.V. Pavlova, Laser Phys. 13 (2003) 897. K.M. Itoh, W. Walukiewicz, H.D. Fuchs, J.W. Beeman, E.E. Haller, J.W. Farmer, V.I. Ozhogin, Phys. Rev. B 50 (1994) 16995. P.S. Zyrianov, M.I. Klinger, Quantum Theory of Electron Kinetic Phenomena in Crystallite Semiconductors, Nauka, Moscow, 1976. L.D. Landau, E.M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, Pergamon, Oxford, 1977. B.J. Verhaar, L.P.H. de Goey, J.P.H.W. van den Eijnde, E.J.D. Vredenbregt, Phys. Rev. A 32 (1985) 1424.