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Analytic expression for electronic density of states in random media with weak scattering potential
Solid State Communications 148 (2008) 42–45
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Analytic expression for electronic density of states in random media with weak scattering potential Udomsilp Pinsook ∗ , Virulh Sa-yakanit Center of Excellence in Forum for Theoretical Science, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
article
info
Article history: Received 25 February 2008 Received in revised form 6 June 2008 Accepted 13 July 2008 by F. Peeters Available online 18 July 2008 PACS: 71.22.+i 71.23.An 71.55.-i
a b s t r a c t We evaluate the electronic density of states (DOS) in random media by using the expression from the variational path integral theory. The scattering potential is modeled by a Gaussian function. By imposing the limit of weak scattering, the full spectrum DOS can be approximated by an analytical √ method. The solution has several features; in the extended states, it is essentially proportional to E. In the localized states, it resembles an exponential tail. However, this tail has less population than that of the compatible Kane DOS. The total energy of the system is lowered by Eα , depending linearly on the density of scatterers. Our results give good description to the photoluminescence spectra of Si:P and the tunneling measurement of GaAs. © 2008 Elsevier Ltd. All rights reserved.
Keywords: A. Semiconductors C. Impurities in semiconductors D. Electronic band structure
1. Introduction Random media are materials of which some elements are spatially disordered. The examples are heavily doped semiconductors, amorphous materials, nanoporous materials and large-molecule organic crystals. We are interested in the electronic properties of these materials as they exhibit some common experimentalobserved features [1–5], such as bandtails, an energy shift or band gap narrowing, and a sharp turning point at the intersection between the band edge and the tail. Furthermore, these features affect the physical properties, such as the specific heat and the optical absorption of the materials, and the diffusion of minority carriers in semiconductors [6]. On the theoretical side, the calculations of the electronic properties and the electronic density of states (DOS) in random media have a long history. The theory that gives the whole spectrum of the DOS was proposed by Kane [7]. However, the Kane DOS is known to overestimate the tail. Halperin and Lax (HL) proposed a wave-mechanical model which gives better description of the tail [8]. Nevertheless, the HL model gives poor description in the free electron region and the DOS is available in a tabular form only, not in an analytic form. Sa-yakanit and his collaborators [9– 11] have proposed a series of study using the variational path
∗
Corresponding author. Tel.: +66 2218 5104; fax: +66 2253 1150. E-mail address: [email protected] (U. Pinsook).
0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.07.017
integral method. The advantages are that this method provides some analytic solutions, the system parameters and their effects on final results can be examined, the analytic forms reduce to the Kane DOS on the same conditions, and it gives as good description of the tail region as that of the HL model. However, the whole spectrum of the DOS with correct description has never been achieved in analytic forms before. In this research, we propose a certain limit of the variational path integral theory which gives an analytic expression for the whole spectrum of the DOS. Moreover, the special features, such as a bandtail, an energy shift and a sharp turning point, come out immediately from the theory. 2. Electronic density of states from path integral method The variational path integral theory is a unique method for studying a system of independent electrons interacting with nonlocal effective potentials [12]. The effective interaction between an electron and random scatterers in random media is an example of a non-local effective potential. In this work, we use a Gaussian function for representing the one-scatterer potential. This is because the Gaussian interaction can be handled easily and requires only two physical parameters for specifying the interaction. We have also used the screened Coulomb interaction. Even though the final expressions are different but general conclusions in terms of the DOS’s features are the same. Thus we limit ourselves to the Gaussian interaction only.
U. Pinsook, V. Sa-yakanit / Solid State Communications 148 (2008) 42–45
In the limit of dense scatterers, the potential averaged over all possible scatterer configurations is in terms of a correlation function of the one-scatterer potential. Due to the Gaussian interaction, the correlation function can be written out explicitly as [9]
ρη2 (X(τ ) − X(σ ))2 exp − , π 3/2 L3 L2
W (X(τ ) − X(σ )) =
(1)
where X are the electron coordinates, ρ is the density of the scatterers, η is the strength of the interaction of one scatterer, and L is the scatterer–scatterer correlation length, τ and σ are timelike integral variables. The form of this equation is said to be nonlocal in time because the interaction at time τ still has some effects at another time σ . The validity of this equation also requires that α = ρη2 is finite. By using Eq. (1), the system’s action, S, can be explicitly written as T
Z
dτ
S= 0
1 2
T
Z
i
mX˙ 2 (τ ) +
2h¯
T
Z
0
dτ dσ W (X(τ ) − X(σ )),
(2)
0
where T is also a time-like integral variable and m is the electron mass. This action describes classical movement of an electron in vicinity of dense random scatterers. In order to obtain the quantum description, we must perform the integration of the phase factor i
In this expression, the spin degeneracy is included. The DOS is of interests because it leads to the calculations of other physical properties. Most of all, it can be probed directly by several experimental methods, such as photoluminescence [1], photocurrent measurements [2], thermoelectric force microscopy [3], or tunneling measurements [4]. It has been showed that in the semiclassical limit, the DOS derived from Eq. (6) resembles the Kane DOS [13,14]. Moreover, by taking appropriate limits, Sa-yakanit found that, at the band edge, several important tail regimes come out naturally from his theory [9]. The great advantage of the path integral theory is that it provides an analytic solution where the relation between the system parameters and the final solutions can be readily examined [9–11]. 3. Weak scattering limit In the present work, we propose a certain limit in which the theory can provide the full spectrum DOS. It was shown that the variational parameter ω has a very close relation with α , i.e. the units of (h¯ ω)2 is the same as Lα3 as they relate to the interaction strength [9,14]. The trend is that if α is small and L is large, then ω is small. Our significant step is to expand Eq. (6) in a series of ω, and we find that
S
e h¯ over all possible paths, i.e. G(X2 , X1 , T ) =
X2
Z
D[X(T )] exp
i h¯
X1
S
G(T ) ≈
.
(3)
The solution of this equation is in term of the one-electron propagator. However, this equation is very complicated and cannot be integrated out exactly. We resort to using a non-local harmonic trial action, i.e. T
Z
dτ
S0 = 0
1 2
ω2
mX˙ 2 (τ ) −
T
Z
2T
T
Z
0
dτ dσ (X(τ ) − X(σ ))2 ,
(4)
0
where ω is a variational parameter which can be adjusted to give the best description to Eq. (2). The solution of the harmonic action is exactly solvable, and it can be used further to approximate the solution of the present problem, i.e. i
G ≈ G0 e h¯
hS −S0 iS0
,
(5)
where G0 is the one-electron propagator with the non-local harmonic potential. By using Eq. (5), Samathiyakanit (Sa-yakanit) [9] already showed that the solution in terms of the one-electron propagator can be written as G(T ) =
3/2
m
1 2
2π ih¯ T
× exp
1 2
sin
3
1
2
2
αT − 16π 3/2 h¯ 2
ωT cot
ωT
T
Z
!3
ωT 1
2
−3/2
dτ A
−1
,
(6)
ω(T − τ ) .
(7)
0
where A=
L2 4
+
ih¯ sin
1 2
ωτ sin
mω
sin
1 2
1 2
ωT
The density of states (DOS) is simply the Fourier transform of G(T ), i.e. D(E ) =
1
π h¯
Z
∞
i
dT e h¯ −∞
ET
G(T ).
m
3/2
exp −
2π ih¯ T
mα T 2
π 3/2 h¯ 2 (2mL3 + ih¯ LT )
+ O[ω2 ]. (9)
This expression should give a reasonable description of a random system with dense, weak scatterers. Although, it seems like a simple expression, its Fourier transform is very complicated and contains some anomalies [14]. By inspection, we find that G(T ) contains some rapid phase changes, which should be one of the cause of the reported anomalies. These rapid phase changes have to be removed. In order to overcome this problem, we examine the argument of the exponential term. It can be decomposed into f re + if im . First, we examine f im . We find that limT →0 f im is proportional to T 3 . As T becomes larger, only the contribution from Eα T , where Eα = 3/m2α 2 , survives. In fact, the higher-order terms of T lead to π h¯ L the rapid phase changes, and have negligible contributions to the DOS in Eq. (8). We need to remove these terms by hand. Thus, we i
E T
simply replace eifim by e h¯ α . Next, we examine two important limits of f re and their physical meanings. In the limit of T → 0, f0re = − 3α/2T 2 3 . This limit is 2π h¯ L the semiclassical or the fast process limit. If we take only this limit into the account, we get a DOS which is compatible to the Kane DOS [7]. Another important regime is the limit where T → ∞, 2
Lα re we find that f∞ = − π2m 3/2 h4 , i.e. independent of T. This limit is the ¯ slow process limit. This limit leads to a free particle propagator, and hence it gives a DOS which is proportional to the free electron DOS. It is worth mentioning here that, unlike the Kane DOS, the free electron DOS is not differentiable at the band edge, i.e. the bottom of the band is well-defined. In some previous works [9–11], either the limit of T → 0 or T → ∞ was considered separately. The authors reported only the tail region of the DOS. The description in the free electron region is incorrect [6]. As the actual f re includes those two limits of T and a very smooth link in between, thus a calculation that accounts for both limits must significantly improve the result of the theory. However, the Fourier transform of Eq. (9) is not known. In order to allow analytic manipulation, one of us (U.P.) suggests an ansatz, 2
ωT
43
(8)
ef
re
re
re
re
≈ eC ·f0 + ef∞ − eC ·f0
re +f∞
+ d1 T 3/2 e−d2 |T | ,
(10)
44
U. Pinsook, V. Sa-yakanit / Solid State Communications 148 (2008) 42–45
re
Fig. 1. shows ef (thick line) and the ansatz (thin line) as functions of T . The difference between both functions is also shown (dashed line).
where C is a free parameter adjusted to give correct behavior at re the two limits of ef , d1 and d2 can be adjusted to give the best fit. The comparison between efre and the ansatz is shown in Fig. 1. We find that the last term of the ansatz is quite small and gives small contribution to the DOS. Furthermore, we have not found any analytic relation of this last term to the system’s parameters. Thus we omit this correction term from further discussion. i E T We replace the exponential term in Eq. (9) by e h¯ α and the first three terms of the ansatz. Then we perform the Fourier transform, and get the analytic DOS as D(E ) = D1(E ) + D2(E ) − D3(E ), where 8µν 1/4 π1 F1 (− 14 ; 21 ; −
D1(E ) =
(E +Eα )2 ) 4ν
Γ ( 14 )
+
2µ(E + Eα )π1 F1 ( 14 ; 32 ; −
−
2m2 Lα π 3/2 h¯ 4
−
2m2 Lα π 3/2 h¯ 4
D2(E ) = e D3(E ) = e
(E +Eα )2 ) 4ν
,
(11)
2m3/2 p E + Eα H (E + Eα ), π 2 h¯ 3
(12)
√
ν 1/4 Γ ( 43 )
D1(E ),
m3/2 , (2π h¯ )5/2
(13) Cα , 2π 3/2 h¯ 2 L3
H (y) is the heaviside step ν = function, and 1 F1 (a; b; z ) is the confluent hypergeometric function.
µ
=
Fig. 2. shows D(E ) (thick line) and its components D1(E ) (long dashed line), D2(E ) (thin line), and D3(E ) (dashed line). D1(E ) and D3(E ) resemble the Kane DOS. D2(E ) is the scaled non-interacting electron DOS. E = 0 is set as a reference point. The energy shift and the non-differentiable point are clearly observable.
A proper combination of the confluent hypergeometric function and the Gamma function is equivalent to the parabolic cylinder function [15]. The parabolic cylinder function has been used for describing similar bandtail structures in the literature [10,11,13]. 4. Results and discussion A typical DOS is shown in Fig. 2. D1(E ) and D3(E ) have similar 2
behavior to the Kane DOS, i.e. it has e−E tail at large negative E and proportional to the free electron DOS at some positive E. D2(E ) is the scaled non-interacting electron DOS. The effects of D3(E ) are very interesting. In the tail region, the contribution of D3(E ) is to reduce the population of the total DOS. Thus, the tail of the total DOS has less population than that of the compatible Kane DOS. The shape of the tail resembles the exponential tail. In the extended states, D3(E ) almost cancels with D2(E ). Thus, in this region, the contribution from D1(E ) is dominant. We find that the total DOS resembles the free electron DOS at some positive E. The total DOS also exhibit a non-differentiable point at the intersection between the tail states and the free electron states. This point manifests itself as a sharp turning point. It is the reflection of the behavior of D2(E ). We believe that this point could be used to locate the mobility edge. In fact, this sharp turning behavior at the band edge has been observed in wide varieties of materials [1–3].
Fig. 3. shows the conduction band of Si:P (ρ = 1.1 × 1019 cm−3 , open circles) extracted from high power photoluminescence [1], compared with our theory (thick line), where α = 0.28, L = 47, C = 1.44 in Rydberg units. The inset shows a log-scale plot. Note that the uncertainty in the experimental data is quite large in the region of low population.
Another important feature from this theory is that the total energy of the system is lowered by Eα , which is depending on ρ and η2 . The effects of the random scatterers to the energy shift were pointed out before by Hwang [16], and the energy shift can be observed experimentally by analyzing the photo absorption spectra [17]. In addition, it should be noted here that our theory suggests a shift toward the band gap, i.e. the contribution to the band gap shrinkage. At this stage, we apply our theory to explain some experimental data by adjusting α and L, and scaling the calculated DOS accordingly, in order to give the best fit. In Fig. 3, the conduction band of phosphorous-doped crystalline silicon (Si:P) extracted from high power photoluminescence [1] is compared with our theory, where α = 0.28, L = 47, C = 1.44 in Rydberg units. In this case, an extra energy shift of 9 meV is added. The fitting is good except in the region of very low population. However, the uncertainty in the experimental data is also quite large in that region (see the log-plot in the inset of Fig. 3). In addition, we reinterpret the valence band of GaAs, derived from tunneling experiments [4]. By using our theory with α = 0.8, L = 36, C = 1.08, we find that the tail states of the DOS can be decomposed into the exponential bandtail region and the socalled impurity band, as shown in Fig. 4. The impurity band can be described very well by a separate Gaussian fit, as suggested by a number of theoretical and experimental studies [3,18]. Another
U. Pinsook, V. Sa-yakanit / Solid State Communications 148 (2008) 42–45
45
of the tail region. The energy shift of Eα comes out immediately from the theory and contributes to the band gap shrinkage. The final expression gives good description to the photoluminescence spectra of Si:P and the tunneling measurement of GaAs. This theory can be easily modified to include screened Coulomb interaction [10,11] and to model lower dimensions [19]. We believe that other materials like nanoporous, organic crystals, and amorphous solids can be explained as random media, and can be modeled by our theory as well. Acknowledgement The authors would like to express their gratitude to Dr. P.P. Altermatt for kindly providing data used in figure 3. U.P. wish to thank the Thailand Research Fund (TRF) for funding. Fig. 4. shows the valence band of GaAs (ρ = 9.9 × 1018 cm−3 , filled diamonds), derived from tunneling experiments [4], compared with our theory (thick line), where α = 0.8, L = 36, C = 1.08. The difference between the experimental data and the theoretical fitting is also shown (open triangles). It can be described very well by a separate Gaussian fit (thin line).
explicit example of the band decomposition and the Gaussian impurity bands was reported by Altermatt et al. [1]. The next stage is to determine whether the system parameters, α and L, give a sensible representation of the experimental data. In order to verify this, we compare the results with the work of Sritrakool, et al. [11], which also studied the effects of impurities to the valence band structure. They used the screened Coulomb interaction to represent the electron-impurity interaction. Despite different models, we found that the coupling energy and the length scale from our work and the work of Sritrakool, et al. [11] are quite similar. For example, for ρ = 2.4 × 1019 cm−1 , Sritrakool, et al. [11] reported the screening length Q −1 = 13.4 Å, and the coupling energy ξQ = 49.3 meV, whereas for ρ = 9.9 × 1018 cm−1 , we report the correlation length L = 19.04 Å, and the coupling energy p α/L3 = 56.32 meV. These results between the two works are indeed consistent. In conclusion, we use the expression from the variational path integral theory to evaluate the whole spectrum of the DOS in random media. The expression of the DOS is in the analytic form. It describes several important features correctly, such as the contribution of the free electron region and the description
References [1] P.P. Altermatt, A. Schenk, G. Heiser, J. Appl. Phys. 100 (2006) 113714; B. Bergersen, J.A. Rostworowski, M. Eswaran, R.R. Parsons, P. Jena, Phys. Rev. B 14 (1976) 1633; M. Eswaran, B. Bergersen, J.A. Rostworowski, R.R. Parsons, Solid State Commun. 20 (1976) 811. [2] D.V. Lang, X. Chi, T. Siegrist, A.M. Sergent, A.P. Ramirez, Phys. Rev. Lett. 93 (2004) 086802. [3] O. Tal, Y. Rosenwaks, Y. Preezant, N. Tessler, C.K. Chan, A. Kahn, Phys. Rev. Lett. 95 (2005) 256405. [4] G.D. Mahan, J.W. Conley, Appl. Phys. Lett. 11 (1967) 29. [5] P.N. Rao, E.A. Schiff, L. Tsybeskov, P. Fauchet, Chem. Phys. 284 (2002) 129. [6] E.O. Kane, Solid State Electron. 28 (1985) 3. [7] E.O. Kane, Phys. Rev. 131 (1963) 79. [8] B.I. Halperin, M. Lax, Phys. Rev. 148 (1966) 722. [9] V. Samathiyakanit (or V. Sa-yakanit), J. Phys. C: Solid State Phys. 7 (1974) 2849. [10] V. Sa-yakanit, W. Sritrakool, H.R. Glyde, Phys. Rev. B 25 (1982) 2776. [11] W. Sritrakool, V. Sa-yakanit, H.R. Glyde, Phys. Rev. B 32 (1985) 1090. [12] R.P. Feynman, A.R. Hibbs, Quantum Mechanics and Path Integral, McGraw-Hill, New York, 1965. [13] P.V. Mieghem, Rev. Mod. Phys. 64 (1992) 755. [14] D.C. Khandekar, S.V. Lawande, Phys. Rep. 137 (1986) 115. [15] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972. [16] C.J. Hwang, J. Appl. Phys. 41 (1970) 2668. [17] A. Iribarren, R. Castro-Rodríguez, V. Sosa, J.L. Peña, Phys. Rev. B 58 (1998) 1907; M.D. Sturge, Phys. Rev. 127 (1962) 768; H.C. Casey, D.D. Sell, W. Wecht, J. Appl. Phys. 46 (1975) 250. [18] K. Koughia, Z. Shakoor, S.O. Kasap, J.M. Marshall, J. Appl. Phys. 97 (2005) 033706; W. Fuhs, L. Korte, M. Schmidt, J. Optoelectron. Adv. Mater. 8 (2006) 1989. [19] A. Thongnum, U. Pinsook, S. Khan-ngern, V. Sa-yakanit, Solid State Commun. 145 (2008) 207.
Contents lists available at ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Analytic expression for electronic density of states in random media with weak scattering potential Udomsilp Pinsook ∗ , Virulh Sa-yakanit Center of Excellence in Forum for Theoretical Science, Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
article
info
Article history: Received 25 February 2008 Received in revised form 6 June 2008 Accepted 13 July 2008 by F. Peeters Available online 18 July 2008 PACS: 71.22.+i 71.23.An 71.55.-i
a b s t r a c t We evaluate the electronic density of states (DOS) in random media by using the expression from the variational path integral theory. The scattering potential is modeled by a Gaussian function. By imposing the limit of weak scattering, the full spectrum DOS can be approximated by an analytical √ method. The solution has several features; in the extended states, it is essentially proportional to E. In the localized states, it resembles an exponential tail. However, this tail has less population than that of the compatible Kane DOS. The total energy of the system is lowered by Eα , depending linearly on the density of scatterers. Our results give good description to the photoluminescence spectra of Si:P and the tunneling measurement of GaAs. © 2008 Elsevier Ltd. All rights reserved.
Keywords: A. Semiconductors C. Impurities in semiconductors D. Electronic band structure
1. Introduction Random media are materials of which some elements are spatially disordered. The examples are heavily doped semiconductors, amorphous materials, nanoporous materials and large-molecule organic crystals. We are interested in the electronic properties of these materials as they exhibit some common experimentalobserved features [1–5], such as bandtails, an energy shift or band gap narrowing, and a sharp turning point at the intersection between the band edge and the tail. Furthermore, these features affect the physical properties, such as the specific heat and the optical absorption of the materials, and the diffusion of minority carriers in semiconductors [6]. On the theoretical side, the calculations of the electronic properties and the electronic density of states (DOS) in random media have a long history. The theory that gives the whole spectrum of the DOS was proposed by Kane [7]. However, the Kane DOS is known to overestimate the tail. Halperin and Lax (HL) proposed a wave-mechanical model which gives better description of the tail [8]. Nevertheless, the HL model gives poor description in the free electron region and the DOS is available in a tabular form only, not in an analytic form. Sa-yakanit and his collaborators [9– 11] have proposed a series of study using the variational path
∗
Corresponding author. Tel.: +66 2218 5104; fax: +66 2253 1150. E-mail address: [email protected] (U. Pinsook).
0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.07.017
integral method. The advantages are that this method provides some analytic solutions, the system parameters and their effects on final results can be examined, the analytic forms reduce to the Kane DOS on the same conditions, and it gives as good description of the tail region as that of the HL model. However, the whole spectrum of the DOS with correct description has never been achieved in analytic forms before. In this research, we propose a certain limit of the variational path integral theory which gives an analytic expression for the whole spectrum of the DOS. Moreover, the special features, such as a bandtail, an energy shift and a sharp turning point, come out immediately from the theory. 2. Electronic density of states from path integral method The variational path integral theory is a unique method for studying a system of independent electrons interacting with nonlocal effective potentials [12]. The effective interaction between an electron and random scatterers in random media is an example of a non-local effective potential. In this work, we use a Gaussian function for representing the one-scatterer potential. This is because the Gaussian interaction can be handled easily and requires only two physical parameters for specifying the interaction. We have also used the screened Coulomb interaction. Even though the final expressions are different but general conclusions in terms of the DOS’s features are the same. Thus we limit ourselves to the Gaussian interaction only.
U. Pinsook, V. Sa-yakanit / Solid State Communications 148 (2008) 42–45
In the limit of dense scatterers, the potential averaged over all possible scatterer configurations is in terms of a correlation function of the one-scatterer potential. Due to the Gaussian interaction, the correlation function can be written out explicitly as [9]
ρη2 (X(τ ) − X(σ ))2 exp − , π 3/2 L3 L2
W (X(τ ) − X(σ )) =
(1)
where X are the electron coordinates, ρ is the density of the scatterers, η is the strength of the interaction of one scatterer, and L is the scatterer–scatterer correlation length, τ and σ are timelike integral variables. The form of this equation is said to be nonlocal in time because the interaction at time τ still has some effects at another time σ . The validity of this equation also requires that α = ρη2 is finite. By using Eq. (1), the system’s action, S, can be explicitly written as T
Z
dτ
S= 0
1 2
T
Z
i
mX˙ 2 (τ ) +
2h¯
T
Z
0
dτ dσ W (X(τ ) − X(σ )),
(2)
0
where T is also a time-like integral variable and m is the electron mass. This action describes classical movement of an electron in vicinity of dense random scatterers. In order to obtain the quantum description, we must perform the integration of the phase factor i
In this expression, the spin degeneracy is included. The DOS is of interests because it leads to the calculations of other physical properties. Most of all, it can be probed directly by several experimental methods, such as photoluminescence [1], photocurrent measurements [2], thermoelectric force microscopy [3], or tunneling measurements [4]. It has been showed that in the semiclassical limit, the DOS derived from Eq. (6) resembles the Kane DOS [13,14]. Moreover, by taking appropriate limits, Sa-yakanit found that, at the band edge, several important tail regimes come out naturally from his theory [9]. The great advantage of the path integral theory is that it provides an analytic solution where the relation between the system parameters and the final solutions can be readily examined [9–11]. 3. Weak scattering limit In the present work, we propose a certain limit in which the theory can provide the full spectrum DOS. It was shown that the variational parameter ω has a very close relation with α , i.e. the units of (h¯ ω)2 is the same as Lα3 as they relate to the interaction strength [9,14]. The trend is that if α is small and L is large, then ω is small. Our significant step is to expand Eq. (6) in a series of ω, and we find that
S
e h¯ over all possible paths, i.e. G(X2 , X1 , T ) =
X2
Z
D[X(T )] exp
i h¯
X1
S
G(T ) ≈
.
(3)
The solution of this equation is in term of the one-electron propagator. However, this equation is very complicated and cannot be integrated out exactly. We resort to using a non-local harmonic trial action, i.e. T
Z
dτ
S0 = 0
1 2
ω2
mX˙ 2 (τ ) −
T
Z
2T
T
Z
0
dτ dσ (X(τ ) − X(σ ))2 ,
(4)
0
where ω is a variational parameter which can be adjusted to give the best description to Eq. (2). The solution of the harmonic action is exactly solvable, and it can be used further to approximate the solution of the present problem, i.e. i
G ≈ G0 e h¯
hS −S0 iS0
,
(5)
where G0 is the one-electron propagator with the non-local harmonic potential. By using Eq. (5), Samathiyakanit (Sa-yakanit) [9] already showed that the solution in terms of the one-electron propagator can be written as G(T ) =
3/2
m
1 2
2π ih¯ T
× exp
1 2
sin
3
1
2
2
αT − 16π 3/2 h¯ 2
ωT cot
ωT
T
Z
!3
ωT 1
2
−3/2
dτ A
−1
,
(6)
ω(T − τ ) .
(7)
0
where A=
L2 4
+
ih¯ sin
1 2
ωτ sin
mω
sin
1 2
1 2
ωT
The density of states (DOS) is simply the Fourier transform of G(T ), i.e. D(E ) =
1
π h¯
Z
∞
i
dT e h¯ −∞
ET
G(T ).
m
3/2
exp −
2π ih¯ T
mα T 2
π 3/2 h¯ 2 (2mL3 + ih¯ LT )
+ O[ω2 ]. (9)
This expression should give a reasonable description of a random system with dense, weak scatterers. Although, it seems like a simple expression, its Fourier transform is very complicated and contains some anomalies [14]. By inspection, we find that G(T ) contains some rapid phase changes, which should be one of the cause of the reported anomalies. These rapid phase changes have to be removed. In order to overcome this problem, we examine the argument of the exponential term. It can be decomposed into f re + if im . First, we examine f im . We find that limT →0 f im is proportional to T 3 . As T becomes larger, only the contribution from Eα T , where Eα = 3/m2α 2 , survives. In fact, the higher-order terms of T lead to π h¯ L the rapid phase changes, and have negligible contributions to the DOS in Eq. (8). We need to remove these terms by hand. Thus, we i
E T
simply replace eifim by e h¯ α . Next, we examine two important limits of f re and their physical meanings. In the limit of T → 0, f0re = − 3α/2T 2 3 . This limit is 2π h¯ L the semiclassical or the fast process limit. If we take only this limit into the account, we get a DOS which is compatible to the Kane DOS [7]. Another important regime is the limit where T → ∞, 2
Lα re we find that f∞ = − π2m 3/2 h4 , i.e. independent of T. This limit is the ¯ slow process limit. This limit leads to a free particle propagator, and hence it gives a DOS which is proportional to the free electron DOS. It is worth mentioning here that, unlike the Kane DOS, the free electron DOS is not differentiable at the band edge, i.e. the bottom of the band is well-defined. In some previous works [9–11], either the limit of T → 0 or T → ∞ was considered separately. The authors reported only the tail region of the DOS. The description in the free electron region is incorrect [6]. As the actual f re includes those two limits of T and a very smooth link in between, thus a calculation that accounts for both limits must significantly improve the result of the theory. However, the Fourier transform of Eq. (9) is not known. In order to allow analytic manipulation, one of us (U.P.) suggests an ansatz, 2
ωT
43
(8)
ef
re
re
re
re
≈ eC ·f0 + ef∞ − eC ·f0
re +f∞
+ d1 T 3/2 e−d2 |T | ,
(10)
44
U. Pinsook, V. Sa-yakanit / Solid State Communications 148 (2008) 42–45
re
Fig. 1. shows ef (thick line) and the ansatz (thin line) as functions of T . The difference between both functions is also shown (dashed line).
where C is a free parameter adjusted to give correct behavior at re the two limits of ef , d1 and d2 can be adjusted to give the best fit. The comparison between efre and the ansatz is shown in Fig. 1. We find that the last term of the ansatz is quite small and gives small contribution to the DOS. Furthermore, we have not found any analytic relation of this last term to the system’s parameters. Thus we omit this correction term from further discussion. i E T We replace the exponential term in Eq. (9) by e h¯ α and the first three terms of the ansatz. Then we perform the Fourier transform, and get the analytic DOS as D(E ) = D1(E ) + D2(E ) − D3(E ), where 8µν 1/4 π1 F1 (− 14 ; 21 ; −
D1(E ) =
(E +Eα )2 ) 4ν
Γ ( 14 )
+
2µ(E + Eα )π1 F1 ( 14 ; 32 ; −
−
2m2 Lα π 3/2 h¯ 4
−
2m2 Lα π 3/2 h¯ 4
D2(E ) = e D3(E ) = e
(E +Eα )2 ) 4ν
,
(11)
2m3/2 p E + Eα H (E + Eα ), π 2 h¯ 3
(12)
√
ν 1/4 Γ ( 43 )
D1(E ),
m3/2 , (2π h¯ )5/2
(13) Cα , 2π 3/2 h¯ 2 L3
H (y) is the heaviside step ν = function, and 1 F1 (a; b; z ) is the confluent hypergeometric function.
µ
=
Fig. 2. shows D(E ) (thick line) and its components D1(E ) (long dashed line), D2(E ) (thin line), and D3(E ) (dashed line). D1(E ) and D3(E ) resemble the Kane DOS. D2(E ) is the scaled non-interacting electron DOS. E = 0 is set as a reference point. The energy shift and the non-differentiable point are clearly observable.
A proper combination of the confluent hypergeometric function and the Gamma function is equivalent to the parabolic cylinder function [15]. The parabolic cylinder function has been used for describing similar bandtail structures in the literature [10,11,13]. 4. Results and discussion A typical DOS is shown in Fig. 2. D1(E ) and D3(E ) have similar 2
behavior to the Kane DOS, i.e. it has e−E tail at large negative E and proportional to the free electron DOS at some positive E. D2(E ) is the scaled non-interacting electron DOS. The effects of D3(E ) are very interesting. In the tail region, the contribution of D3(E ) is to reduce the population of the total DOS. Thus, the tail of the total DOS has less population than that of the compatible Kane DOS. The shape of the tail resembles the exponential tail. In the extended states, D3(E ) almost cancels with D2(E ). Thus, in this region, the contribution from D1(E ) is dominant. We find that the total DOS resembles the free electron DOS at some positive E. The total DOS also exhibit a non-differentiable point at the intersection between the tail states and the free electron states. This point manifests itself as a sharp turning point. It is the reflection of the behavior of D2(E ). We believe that this point could be used to locate the mobility edge. In fact, this sharp turning behavior at the band edge has been observed in wide varieties of materials [1–3].
Fig. 3. shows the conduction band of Si:P (ρ = 1.1 × 1019 cm−3 , open circles) extracted from high power photoluminescence [1], compared with our theory (thick line), where α = 0.28, L = 47, C = 1.44 in Rydberg units. The inset shows a log-scale plot. Note that the uncertainty in the experimental data is quite large in the region of low population.
Another important feature from this theory is that the total energy of the system is lowered by Eα , which is depending on ρ and η2 . The effects of the random scatterers to the energy shift were pointed out before by Hwang [16], and the energy shift can be observed experimentally by analyzing the photo absorption spectra [17]. In addition, it should be noted here that our theory suggests a shift toward the band gap, i.e. the contribution to the band gap shrinkage. At this stage, we apply our theory to explain some experimental data by adjusting α and L, and scaling the calculated DOS accordingly, in order to give the best fit. In Fig. 3, the conduction band of phosphorous-doped crystalline silicon (Si:P) extracted from high power photoluminescence [1] is compared with our theory, where α = 0.28, L = 47, C = 1.44 in Rydberg units. In this case, an extra energy shift of 9 meV is added. The fitting is good except in the region of very low population. However, the uncertainty in the experimental data is also quite large in that region (see the log-plot in the inset of Fig. 3). In addition, we reinterpret the valence band of GaAs, derived from tunneling experiments [4]. By using our theory with α = 0.8, L = 36, C = 1.08, we find that the tail states of the DOS can be decomposed into the exponential bandtail region and the socalled impurity band, as shown in Fig. 4. The impurity band can be described very well by a separate Gaussian fit, as suggested by a number of theoretical and experimental studies [3,18]. Another
U. Pinsook, V. Sa-yakanit / Solid State Communications 148 (2008) 42–45
45
of the tail region. The energy shift of Eα comes out immediately from the theory and contributes to the band gap shrinkage. The final expression gives good description to the photoluminescence spectra of Si:P and the tunneling measurement of GaAs. This theory can be easily modified to include screened Coulomb interaction [10,11] and to model lower dimensions [19]. We believe that other materials like nanoporous, organic crystals, and amorphous solids can be explained as random media, and can be modeled by our theory as well. Acknowledgement The authors would like to express their gratitude to Dr. P.P. Altermatt for kindly providing data used in figure 3. U.P. wish to thank the Thailand Research Fund (TRF) for funding. Fig. 4. shows the valence band of GaAs (ρ = 9.9 × 1018 cm−3 , filled diamonds), derived from tunneling experiments [4], compared with our theory (thick line), where α = 0.8, L = 36, C = 1.08. The difference between the experimental data and the theoretical fitting is also shown (open triangles). It can be described very well by a separate Gaussian fit (thin line).
explicit example of the band decomposition and the Gaussian impurity bands was reported by Altermatt et al. [1]. The next stage is to determine whether the system parameters, α and L, give a sensible representation of the experimental data. In order to verify this, we compare the results with the work of Sritrakool, et al. [11], which also studied the effects of impurities to the valence band structure. They used the screened Coulomb interaction to represent the electron-impurity interaction. Despite different models, we found that the coupling energy and the length scale from our work and the work of Sritrakool, et al. [11] are quite similar. For example, for ρ = 2.4 × 1019 cm−1 , Sritrakool, et al. [11] reported the screening length Q −1 = 13.4 Å, and the coupling energy ξQ = 49.3 meV, whereas for ρ = 9.9 × 1018 cm−1 , we report the correlation length L = 19.04 Å, and the coupling energy p α/L3 = 56.32 meV. These results between the two works are indeed consistent. In conclusion, we use the expression from the variational path integral theory to evaluate the whole spectrum of the DOS in random media. The expression of the DOS is in the analytic form. It describes several important features correctly, such as the contribution of the free electron region and the description
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