A Dielectric function of disordered electron gas

Interaction of fast charged projectiles with two-dimensional electron gas: Interaction and disorder effects

Abstract

The results of a theoretical investigation on the stopping power of ions moving in a disordered two-dimensional degenerate electron gas are presented. The stopping power for an ion is calculated employing linear response theory using the dielectric function approach. The disorder, which leads to a damping of plasmons and quasiparticles in the electron gas, is taken into account through a relaxation time approximation in the linear response function. The stopping power for an ion is calculated in both the low- and high-velocity limits. In order to highlight the effects of damping we present a comparison of our analytical and numerical results, in the case of point-like ions, obtained for a non-zero damping with those for a vanishing damping. It is shown that the equipartition sum rule first formulated by Lindhard and Winther for three-dimensional degenerate electron gas does not necessarily hold in two-dimensions. We have generalized this rule introducing an effective dielectric function. In addition some new results for two-dimensional interacting electron gas have been obtained. In this case the exchange-correlation interactions of electrons are considered via local-field-corrected dielectric function.

pacs:
52.40.Mj, 52.25.Mq, 73.50.Mx, 52.27.Gr

I Introduction

There is an ongoing interest in the theory of interaction of swift charged projectiles with condensed matter. Although most theoretical works have reported on the energy loss of ions in a target medium which is modelled as a three-dimensional (3D) electron gas, the two-dimensional (2D) case has not yet received as much attention as the 3D case. A 2D electron system is now experimentally realizable in a laboratory. In the last three decades or so many interesting and intriguing properties of a 2D electron gas have been explored. For a recent update on some of these developments we refer to Refs. (1); (2). A widely used 2D electron system is realized at the interface between GaAs and GaAlAs, and in the interface metal-oxide-semiconductors (MOS). The interaction of charged particles with an electron gas is an important probe of many-body interactions in the target electron medium. It is known that many-body properties of an electron gas vary in notable aspects with spatial dimensions. It is therefore of interest to make a detailed study of interaction of charged particles with a 2D electron gas. This theoretical study is also of relevance to device applications e.g. in using ion implantation in devices which involve 2D electron systems.

In general, interaction of charged projectiles with condensed matter can be studied by means of the stopping power (SP) of the target medium. The SP accounts for the energy loss by an external charged projectile as it passes through and interacts with matter. And the SP of a medium can be used to construct diagnostic tools for studying this kind of physical systems. There have been several theoretical approaches to the energy loss and SP for 3D systems, and some of these approaches have been applied also to a 2D electron gas. Among previous theoretical works on a 2D electron gas some are based on the linear response dielectric function method (3); (4); (5); (6); (7); (8) and quantum scattering theory (9); (10); (12); (11). Further works have dealt with some nonlinear screening effects through a quadratic response approach within the random-phase approximation (13), the employment of density functional theory (14) and in a method based on frequency moments of the energy loss function (8).

In this paper we shall consider fast charged projectiles and hence a linear response theory to calculate energy loss is expected to be adequate. Previously, within this approach, Bret and Deutsch calculated the SP of an ion (3); (4) and a dicluster (5) in a 2D electron gas for any degeneracy. Their results show some interesting differences with the corresponding results for a 3D case. Of special interest is their finding that the leading term of the asymptotic expansion of the SP in a high-velocity limit decreases as , where is the projectile velocity, which differs from the well-known form predicted by the Bethe-Bloch formula (15); (16); (17) in the 3D case. The calculations in Refs. (3); (4); (5) are based on the random-phase approximation (RPA) which works well if electron-electron interaction can be neglected. Now, in 2D systems, electron density can be varied. For moderate values of electron density e.g. in semiconductors electron-electron interaction may not be negligible and going beyond RPA is desirable.

Our objective is to consider two physically motivated aspects of a 2D electron gas in the context of energy loss. For the first part of our study we consider a disordered electron gas which contains impurities. The effect of these impurities is included through a phenomenological relaxation time for electrons due to scattering by impurities. For this disordered system we use a linear response dielectric function in RPA and in a number-conserving relaxation time approximation (RTA), which was first considered by Mermin (18) and then by Das (19) for a 3D electron gas. This RTA formulation has not yet been extended beyond RPA. The effect of disorder which leads to a damping of excitations enters the RPA dielectric function, for a given electron-impurity collision frequency, through , where is used as a model parameter. For a degenerate electron gas (DEG) and for a given electron density, the damping parameter can be assumed to be a constant to a good approximation. The disorder-inclusive dielectric function, with the collision frequency as a free parameter, allows some physical insight and useful numerical estimates of the influence of disorder on energy loss in a DEG. In 3D the predicted effect is a shorter life time with a smaller propagation wavelength of plasmons resulting considerable modifications of the SP (see, e.g., Refs. (20); (21); (22); (23); (24) and references therein). For the stopping of a single ion, the broadening of the plasmon peak with increasing shifts the threshold for energy loss by plasmon excitation towards lower projectile velocities. It now becomes possible for low-velocity projectile ions to excite plasmons (in addition to single-particle excitations). This increases the SP of 3D electron gas at low projectile velocities, compared to the disorder-free RPA result (22); (23); (24). The situation with a 2D electron gas will be discussed in detail in the following sections.

The second objective of our study is to investigate the influence of exchange-correlation interaction (i.e. beyond RPA) in an electron gas on the SP. For a 3D electron system it has been shown (25); (26) that the SP in low and intermediate velocity regimes shows a definite increase due to this interaction. A similar result has been reported for a 2D system (6); (7). However let us note that if an asymptotic expansion of the SP in a high-velocity regime is considered then it has been shown previously that the first term in this expansion is unaffected by electron-electron interaction. In this paper we calculate the next non-vanishing term of this asymptotic expansion and show that it behaves as , where exchange-correlation interactions are involved in and . These functions depend on the target density through Wigner-Seitz density parameter , where and are electron gas density and Bohr radius, respectively. The details are presented in Sec. III.

The plan of the paper is as follows. In Sec. II.1 we derive analytical expressions for the disorder-inclusive dielectric function (DF) for a 2D degenerate electron gas (DEG). We would like to mention that an alternative but equivalent derivation is presented in Appendix A. The latter derivation contains certain attractive features. Through this alternative formulation we consider a small- approximation for the DF, and this approximate result is used in Sec. II.1. In a small- approximation the plasmon dispersion for a disordered two-dimensional DEG exhibits a constraint not present in 3D. This behavior has been previously discussed in the literature (27); (28); (29). We revisit this approximation through our formulation in Appendix A. The exact plasmon dispersion relations for an interacting DEG (including exchange-correlation effects) are derived in Sec. II.2 by employing local-field corrections to the RPA dielectric function. In Sec. III we briefly outline the general linear response function formalism of the 2D stopping power of a point-like ion. After dealing with the excitation equipartition in Sec. III.1 we develop, in Secs. III.2 and III.3, some analytical techniques to calculate the SP of an ion in low- and high-velocity regimes. The two particular cases studied in these sections are (i) low-velocity limit of the SP for an ion moving in a disordered DEG, and (ii) high-velocity limit for a strongly interacting DEG. Sec. IV contains systematic numerical calculations for the SP. The results are summarized in Sec. V which also includes discussion and outlook. Appendix A to which we draw the reader’s attention presents the above-mentioned alternative derivation of the DF for the disorder-inclusive case in RPA, which is also valid in the complex –plane. In Appendix B we provide some technical details for an evaluation of the asymptotic SP.

Ii Dielectric function and dispersion relations for 2D electron gas

In the linear response theory, the stopping power (SP) of an external projectile moving in a medium is related to the dielectric function of the medium. Both the single-particle and collective excitations (i.e. the plasmons) contribute to the SP and these contributions are contained in (see, e.g., Eq. (20) below). In our study the two-dimensional (2D) target medium is assumed to be disordered due to impurities etc. We shall incorporate effects of disorder in in a somewhat phenomenological manner. This is to include disorder through a relaxation time such that the particle number is conserved. For a three-dimensional (3D) medium this was done first by Mermin (18) and then by Das (19) in the random phase approximation (RPA) and in relaxation time approximation (RTA). We refer the reader to (18); (19) for details of this formalism. For , this linear response function reduces to the Lindhard dielectric function (16); (17). The dielectric function is understood to contain () as a damping parameter due to disorder. The form of is to be specified shortly for a 2D electron gas.

It is convenient to introduce the dimensionless Lindhard variables , , where  and  () are, respectively, the Fermi velocity and wave number of the target electrons. Also we introduce the density parameters . In our calculations and (or ) serve as a measure of electron density. (Note that the density parameter introduced above differs from usual definition by a factor see, e.g., Refs. (3); (4); (5); (6)).

ii.1 Disordered electron gas: RPA

Let us now specify the disorder-inclusive dielectric function for 2D zero-temperature (degenerate) electron gas (DEG). This has been done previously in Refs. (27); (28); (29) employing small- approximation. Here within RPA and RTA we derive the disorder-inclusive dielectric function (DF) without further approximations on the energy-momentum spectrum i.e. on and . As pointed out in Ref. (28) the physical arguments for deriving number-conserving DF by Mermin (18) and Das (19) in 3D are independent of dimensionality. Therefore with the notations introduced in the preceding paragraph, the DF for 2D DEG reads

(1)

where , being the Fermi energy with as the effective mass. The quantity (or ) is a measure of damping of excitations in the disordered electron gas. is the longitudinal dielectric function of DEG in the RPA derived in 2D by Stern (30). is the static dielectric function. We have analytically evaluated the disorder-inclusive for which the results, presented below, appear to be new and we have utilized them in our numerical investigation.

Let us recall the Lindhard (RPA) expression for the longitudinal dielectric function (16). In variables and and in 2D it reads as (30)

(2)

where we have split explicitly the DF into the real and imaginary parts and have introduced the real functions (for real and ) and as in the usual RPA expression of longitudinal dielectric function.

Performing the and integrations in Eq. (2) (31) we obtain, for a non-zero damping,

(3)
(4)

with ,

(5)

In the case of vanishing damping ( and ) the expressions (1)-(5) coincide with the Stern result (30) with

(6)
(7)
(8)

Here is the Heaviside unit-step function. The static DF involved in Eq. (1) can be found either from Eqs. (3) and (4) at the limits , or from Eqs. (6)-(8) at . The result reads

(9)

with

(10)

To demonstrate the effect of the damping in Fig. 1 we show the contour plots of the energy loss function without (left panel) and with (right panel) damping. The plasmon dispersion function  in the left panel is also shown as a dashed line (the explicit derivation of the plasmon dispersion curve without damping is given below in Sec. II.2, see Eqs. (16), (19) and Fig. 3). The single-particle excitations energies (or in dimensionless units) are demonstrated as thick solid lines. As expected the energy loss function in the case of vanishing damping (left panel) is localized in the domains with , as well as on the plasmon curve  where the function behaves as a Dirac -function and becomes infinity. In the case of non-zero damping (right panel) the energy loss function is broadened due to the damping and becomes non-zero also in the domains and .

Figure 1: (Color online) The contour plot of as a function of the variables and for and without (left panel) and with (right panel) damping ( eV). The dashed line in left panel shows the plasmon dispersion curve (see the text for details) with . The numbers indicate the values of .

Equations (1)-(4) constitute the number-conserving DF for a 2D disordered electron gas. Deriving these expressions we have explicitly split the DF into real and imaginary parts assuming real variables and . An alternative (but equivalent) expression for this DF is derived in Appendix A which is valid for any complex and . With this exact (within RPA and RTA) expression in Appendix A we then calculate the DF within small -approximation obtained previously in Refs. (27); (28); (29) and revisited in Appendix A. The basic feature of this approximation is the prediction of the threshold condition for plasmon propagation which is absent in 3D (see, e.g., Ref. (22)). Indeed the solution of the dispersion equation , where is given by Eq. (54), reads (27); (28); (29)

(11)

where is the plasma frequency for a 2D electron gas. The condition that has a real part (for plasmon propagation) leads to , where

(12)

with . Thus, within small -approximation, disorder in 2D electronic systems considerably softens plasmons; they cannot propagate for and their dispersion relation is strongly altered relative to the collisionless case. However, since these results were obtained in small –domain one can expect some modifications for large momentum transfers at . Figure 2 shows the real (left panel) and imaginary (right panel) parts of the solutions of the dispersion equations with approximate (Eq. (11)) and exact dielectric functions, Eqs. (49)-(52)). For simplicity we consider the case when the function in Eq. (51) vanishes. Note that the condition together with the inequality requires that with eV. It is seen that the slope of the imaginary part of  (right panel) is dramatically changed at some value of where the expression under square root in Eq. (11) changes the sign. For small -approximation this value of is given by Eq. (12). As pointed out in Appendix A the approximation (11) is valid when one neglects the single-particle energy with respect to . Therefore, in general, we expect good agreement between approximate and exact  for small momentum , as shown in Fig. 2. However, with increasing the approximate dispersion relation (11) fails to predict correctly. As shown in Fig. 2 (left panel, dotted curve with eV) at small () the energy of plasmons is not exactly zero as predicted by Eq. (11) although the probability of plasmon generation is strongly reduced due to the relation . Moreover, in contrast to the predictions of approximation (11) in this case with increasing momentum the real part of vanishes and plasmons cannot propagate any more.

Figure 2: (Color online) Real (, left panel) and imaginary (, right panel) parts (in a.u.) of exact (the lines without symbols) and approximate (the lines with symbols) solutions of dispersion equation vs (in a.u.) for , , eV (solid lines), eV (dashed lines), eV (dotted lines).

ii.2 Strongly coupled electron gas: Beyond RPA

In this section we consider exchange-correlation interaction effects via local-field corrected (LFC) DF but we neglect the disorder (i.e. ). To include disorder in a fully interacting electron gas at a microscopic level is rather involved, and no analytical calculations of without restrictions on and are still available. An attempt to involve strong correlations in RTA and within small -approximation (see Eq. (11)) has been done in Ref. (28). Instead we employ here the LFC dielectric function and demonstrate some useful results which have not been considered previously. Our discussion below is based on the LFC dielectric function of a fully DEG see, e.g., Ref. (6) (in dimensionless variables and )

(13)

where  is the polarizability of the free-electron gas obtained in RPA by Stern (30)

(14)

with , where , and are given by Eqs. (2), (6) and (7), respectively. Note that our definition of the functions and differs from the definition given in Refs. (6); (7) by a factor of . is the LFC function, which includes the effects of exchange-correlation interactions. Within a sum-rule version of the self-consistent approach, Gold and Calmels presented (32) a parameterized expression for the 2D electron gas,

(15)

The coefficients , and are determined by , , , with , , and the parameters , and ,  can be found in Ref. (32).

Now we consider the exact solution of the dispersion equation for an interacting electron gas when the DF is given by LFC expression (13). From Eqs. (6)-(8) and (13), (14) it is seen that the collective plasma modes (plasmons) can propagate with the frequency and momentum and (or and ) which lie in the domain where and . In this domain the dispersion equation has an exact analytical solution which, in Lindhard’s dimensionless variables, is given by

(16)

with and . It is straightforward to check that the solution (16) indeed satisfies the condition for arbitrary . However, an inspection of the dispersion equation shows that this solution exists only for the wave numbers from the domain (or ) where the critical wave number is obtained from an equation , i.e. in this point the plasmon curve touches to the boundary of the single-particle continuum . Explicitly, the critical wave numbers are determined from transcendental equation

(17)

Table 1 shows the quantity and the minimum of the dispersion function with  for some values of the density parameter . The critical wave numbers and the quantities (labeled as and , respectively) are also shown for non-interacting electron gas, i.e. with and . These quantities are important for evaluation of the SP in Sec. III.

0.10 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
0.126 0.288 0.390 0.457 0.507 0.546 0.579 0.606 0.629
0.135 0.345 0.510 0.638 0.748 0.845 0.932 1.013 1.089
1.116 1.264 1.358 1.421 1.469 1.507 1.538 1.565 1.588
1.122 1.304 1.440 1.543 1.629 1.704 1.770 1.831 1.886
Table 1: The critical wave numbers (dimensionless) and the minimum values of the dispersion function for some values of the density parameter . and represent the smae quantities but for non-interacting 2D electron gas.

We can present the dispersion expression (16) obtained above, in the usual form

(18)

which for vanishing exchange-correlation interactions (i.e. at ) reads

(19)

This exact (within the employed model) dispersion relation may be compared with an approximate result derived by Fetter within a hydrodynamical approach (33). Equation (19) agrees with the hydrodynamic result if the last term (the single-particle energy) in this expression is neglected and the coefficient at is replaced by a constant factor . It should be emphasized that in general and at long wavelengths from Eq. (18) for an interacting 2D electron system varies like independently of the LFC and in contrast to the 3D case. This latter behavior seems first to have been suggested by Ferrell (34) and later investigated in more detail by Stern (30) (see also the review paper (35)). It arises from the electromagnetic fields in the vacuum surrounding the plane, with an associated reduction in the screening. Since increases monotonically from zero, an external perturbation of arbitrarily low frequency can always excite collective modes. Hence, the characteristic 3D absorption edge at constant 3D is here entirely absent. Moreover, the group and phase velocities both diverge like as .

Figure 3: (Color online) The dispersion curve for free (lines with symbols) and interacting (lines without symbols) electron gas vs for (dashed lines) and (solid lines). The oblique solid line corresponds to (or ).

Figure 3 shows the plasmon dispersion curve for interacting (the lines without symbols) and non-interacting (the lines with symbols) electron gas, i.e. Eqs. (16) and (19), respectively. The points where the plasmon curves touch the single-particle excitation boundary are given by or , see Table 1. It is seen that the exchange-correlation interaction may strongly reduce the values of . It must be pointed out a technical but important detail which, to our knowledge, has not been yet discussed in the literature. From Fig. 3 it is seen that in plane the plasmon curve has a minimum which is absent in usual units where is a monotonic increasing function. By interchanging the and axes in Fig. 3 one obtains the plasmon dispersion curve which, however, in contrast to the 3D case has two different branches with increasing () and decreasing () dispersion functions (at the minimum of both and curves contact each other). The dispersion relations and can be provisionally treated as the ”single-particle” and ”plasmonic” relations, respectively. Therefore when one attempts to perform -integration in Eq. (20) before -integration, as was done in Ref. (6), the double integration in the SP is reduced to two line integrations along the contours and and both of them contribute to the SP. In other words in this case the energy loss function introduced above contains two Dirac -functions. In fact, we see from our numerical calculations that near the SP maximum the contribution of is not necessarily small compared to the contribution of the other one, . Although the total contributions of both in the SP are in general much smaller than the purely single-particle contributions. This is a violation of the Lindhard-Winther equipartition sum rule (17) which we further discuss in Sec. III.1. To avoid this technical problem in the numerical calculations it is easier to perform first in Eq. (20) the -integration and then using the dispersion function given by Eq. (16).

Iii Stopping power

With the theoretical formalism presented so far, we now take up the main topic of this paper. This is to study the stopping power (SP) of a point-like ion in a 2D degenerate electron gas as well as to show how collective and single-particle excitations in the target medium DEG contribute to the SP. And, as in the previous section, we shall present new theoretical results within the linear response approach. We consider two models for a DEG in 2D. (i) A disordered DEG for which we use a number-conserving DF given in Eqs. (1)-(5). For this case we present analytical calculations and new results for the SP in a low-velocity limit. (ii) A strongly coupled DEG with a DF which includes LFC, Eqs. (13) and (14). This case has been studied in Refs. (6); (7) where the leading term in a high-velocity limit of the SP is calculated using a plasmon-pole approximation. This calculation is supported by a more rigorous treatment, again for the leading term only, in Ref. (8) which is based on a method of moments and includes electron-electron interactions. Now the leading term happens not to depend on electron-electron interaction. It is then of interest to calculate analytically the next non-vanishing terms of the high-velocity SP. As shown below these terms are significantly modified by electron-electron interaction and thus are more involved than the leading term.

We consider an external point-like projectile of carge moving with velocity in a homogeneous and isotropic 2D electron medium characterized by the dielectric function or . Then in the linear response theory the SP which is the energy loss per unit length by this projectile is given by (3); (6)

(20)

Here , GeV/cm eV/Å. We have used the Lindhard variables and introduced in Sec. II. In our calculations we shall consider the range of for which the linear response theory is found to be adequate (36).

iii.1 Equipartition sum rule

With the theoretical formalism presented so far, we now take up one of the main topic of this paper. This is to study how collective and single-particle excitations in the 2D electron gas contribute to the SP. This problem was first addressed by Lindhard and Winther (17) (LW) for a 3D degenerate electron gas without damping (). They formulated an equipartition sum rule which states that an integral proportional to that in Eq. (20)

(21)

receives equal contributions from plasmon () (with ) and from single-particle excitations () (with ), respectively. The functions and may then be written as

(22)
(23)

Here  is the solution of the dispersion equation (the inverse of the dispersion function ). This equipartition rule is valid for sufficiently large , , where the threshold value in 3D case is obtained from the equation . In recent works (20); (23); (37) we have shown that the LW equipartition rule does not necessarily hold for an extended charged projectile e.g. a diproton cluster in a 3D degenerate electron gas without disorder () as well as for a point-like ion in a disordered DEG. We have established some generalized stopping power sum rules. In this section we briefly show that the LW equipartition rule is also violated for a 2D electron gas. In the present context it should be emphasized that the plasmon contribution given by Eq. (22) contains indeed two terms, with and , as discussed above. The existence of both branches requires the threshold condition , where  is the minimum value of the dispersion function shown, e.g. in Fig. 3. However, it is clear that the contribution of vanishes at , where  (the point where the plasmon curve touches to the single-particle excitations boundary). For simplicity we consider below only the domain where only contributes to the SP integral (22). As an example we employ the DF (13) together with Eqs. (6)-(8) and (14) for an interacting DEG. The simplest way to show the violation of the LW equipartition rule in 2D is to calculate the asymptotic values of the contributions  and at . The inverse dispersion function for 2D interacting DEG is evaluated in Appendix B, see Eqs. (59)-(60). Using these expressions it is straightforward to calculate the single-particle and collective contributions to the SP integral which at  become

(24)
(25)

Here and are defined in Appendix B. From the above expressions it is clear that the contribution of the collective excitations is much smaller than the contribution from single-particle excitations, , which indicates the violation of the LW equipartition rule. A similar result has been found numerically in Ref. (6) and is supported by our own numerical calculations. Of course, Eqs. (24) and (25) are not strong results. An exact treatment can be developed on the basis of the integration contour on the complex -plane suggested by LW (17) and investigated in details in Ref. (23). The technique developed in (23) is independent of the dimensionality of electron gas but requires a necessary analytic continuation of the DF in the complex -plane, that is , where the asterix indicates a complex conjugate quantity. It is easy to see that this condition is violated for a 2D electron gas. For simplicity let us consider non-interacting DEG with the DF given by Eq. (2) in the integral form and with . In this case one can easily check that (a similar equation can be obtained for an interacting electron gas). Therefore an arbitrary function of the form

(26)

with an arbitrary constant defines an effective DF of a 2D electron gas which satisfies the required condition, i.e. . Applying now the contour integration technique developed in Ref. (23) one can strongly prove that the single-particle and collective excitations contribute equally to the SP integral (21) where the DF  is replaced by the effective one, , given by Eq. (26). Thus the LW equipartition rule holds also in 2D treating the effective DF instead of . In this case it is straightforward to check that at  the leading order terms of the collective and single-particle excitations are given by . The physical origin of the modification of the equipartition rule in 2D is the change of the nature of the Coulomb potential (in Fourier space it behaves as in 2D) and as a consequence the long-wavelength dispersion relation: the plasma frequency behaves as in this limit. Technically this modification introduces an extra non-compensated variable as a prefactor in Eq. (2), first line, which changes the analytical properties of the DF. Introducing an effective DF (26) we formaly replace the 2D Coulomb potential by the 3D one without affecting the polarizability of the 2D system. This recovers formally the 3D-type dispersion relation with constant plasma frequency and hence the equpartition rule.

iii.2 Low-velocity limit

Let us consider SP for slow projectiles, with . A consequence of the 3D linear response theory, confirmed by experiments, is that for ion velocities low compared to the Fermi velocity , the stopping power is proportional to (see, e.g., the latest experiment (38)). The coefficient of proportionality may be called a friction coefficient. A similar linear behavior of the SP, , is expected in 2D case Refs. (3); (4); (5); (6); (7). Using analytical results obtained for the general expressions for SP follow from Eqs. (20)-(5):

(27)

where the dimensionless friction coefficient depends on the target properties and hence also on the dimensionless damping parameter . We have introduced the following functions

(28)
(29)
(30)

The static screening function  is determined from Eq. (10).

When the damping vanishes () Eq. (29) becomes

(31)

where is the Heaviside unit-step function. Therefore

(32)

and from Eq. (27) we find

(33)

with

(34)

The last expressions (33) and (34) are known results derived previously within RPA in Refs. (3); (6). Interestingly, in a low-velocity limit this SP completely agrees with the result obtained within a binary collision approach Ref. (9). In left panel of Fig. 4 we show the ratio of the disorder-inclusive friction coefficient and  vs damping parameter for two values of the density parameter and . To gain more insight in right panel of Fig. 4 we show the friction coefficient vs for some values of the damping parameter . As expected, the friction coefficient and hence the SP at low velocities increase with an increasing damping parameter ; this was previously reported for 3D in Refs. (22); (23); (24). The behavior of at fixed and at increasing density parameter is particularly noteworthy. At small damping the friction coefficient decays monotonically with while at large it may also increase for large . We will further discuss this behavior in Sec. IV.

The approximation (27) implies that the SP is proportional to velocity. The velocity region in which the linear proportionality between SP and the projectile velocity holds may be inferred from the numerical calculations (see Sec. IV). It is seen from those results that the approximation (27) remains quite accurate even when becomes as large as .

Figure 4: Left panel, the ratio of the friction coefficients with and without damping vs damping parameter (in eV) for various densities. (solid line), (dashed line). Right panel, the friction coefficient vs density parameter for (solid line), eV (dashed line), eV (dotted line) and eV (dash-dotted line).

iii.3 High-velocity limit

Consider next the limit of large projectile velocities in the case of strongly interacting DEG with the dielectric function Eqs. (13)-(15). In this limit the general expression (20) for point-like projectiles with charge moving in either interacting or free electron gas reduces to the simple formula (3)

(35)

which does not contain the gas electron mass anymore; and also the effects of electron-electron interactions appear only in the higher terms of the expansion. The other main discrepancy between the 2D and the 3D results is that the stopping power decreases as instead of behaving as in the 3D case. In the presence of interactions the next order terms are shown to be significantly modified. We derive below a generalized expression for SP, in a high-velocity limit, for point-like ions. In order to show how SP in a high-velocity limit is affected we consider expression (20) rewritten as follows:

(36)

where

(37)

and is the total contribution of the collective and single-particle excitations to the SP integral defined in Sec. III.1 (see Eqs. (21)-(23)). For further progress it is imperative to calculate the asymptotic behavior of the function at . For collective and single-particle excitations these asymptotic forms are given by Eqs. (24) and (25), respectively. Using these expressions we arrive at

(38)

for and with the expansion coefficients

(39)

Here the parameter  depends on the exchange-correlation interactions and is given explicitly in Appendix B.

Below we calculate the SP up to the order  thus neglecting the terms with . First the SP (36) can be represented in the equivalent form

(40)

which is convenient for further calculations. Here is a constant

(41)

and the other quantities are function of the ion velocity:

(42)
(43)

For the derivation of Eq. (40) we have used some elementary integrals (31). In Appendix B we prove that , see Eq. (62). This relation can be regarded as another SP sum rule for an interacting DEG.

For a calculation of the SP up to fourth order we need the asymptotic behavior of  up to the first order () which can be obtained from Eqs. (38) and (43)

(44)

and only the leading term of . We denote this leading term by and using Eq. (42) we obtain

(45)

The coefficient is explicitly evaluated in Appendix B and entirely depends on the density parameter , see Eq. (68). Thus substituting Eqs. (44) and (45) into (40) and setting we finally obtain

(46)

It is seen that in the correction term (the second term in Eq. (46)) the mass of electron enters through the Fermi velocity . A limit to the non-interacting DEG is performed by taking the limit , i.e. setting  (see Eq. (39)). In this limit the coefficient is given by Eq. (69). In the general case of non-vanishing exchange-correlation interactions it is too difficult to draw some conclusions from Eq. (46) about how these interactions affect the high-velocity SP. Numerical calculations of Refs. (6); (7) show that these interactions strongly increase the SP up to the intermediate velocity range with . We support this conclusion by our own calculations (not shown here) which also indicate that the asymptotic SP (46) remains quite accurate also in the intermediate velocity range.

Figure 5: Left panel, the SP (in a.u. and in logarithmical scale) of a proton vs moving in an interacting electron gas for (dotted line), (dashed line), (solid line). The lines with and without symbols correspond to the numerical evaluation of Eq. (20) with Eqs. (13)-(15) and asymptotic expression (46), respectively. Right panel, the dashed and solid lines (with symbols) from the left panel in linear scale.

We close this section with the following two remarks. First, the high-velocity SP Eq. (46) is also valid for a general LFC function . The derivations above and in Appendix B show that only the asymptotic values of at and contribute to Eq. (46). At short wavelengths is constant (see, e.g., Ref. (32)). At long wavelengths the LFC function behaves as , where the constant is related to the compressibility of a 2D electron gas through compressibility sum rule. The latter for a 3D electron gas is discussed in (39), and for a 2D electron gas in (40). Thus in the general case of arbitrary the quantities and in Eq. (46) are replaced by and