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Electronic properties of aperiodic quantum dot chains
Physica E 44 (2012) 1580–1584
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Electronic properties of aperiodic quantum dot chains P.Yu. Korotaev a,n, Yu.Kh. Vekilov a, N.E. Kaputkina b a b
Department of Theoretical Physics and Quantum Technologies, National University of Science and Technology ‘‘MISIS’’, 119049 Moscow, Leninskiy pr. 4, Russia Department of Physical Chemistry, National University of Science and Technology ‘‘MISIS’’, 119049 Moscow, Leninskiy pr. 4, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2012 Received in revised form 27 March 2012 Accepted 28 March 2012 Available online 4 April 2012
The electronic spectral and transport properties of aperiodic quantum dot chains are investigated. The systems with singular continuous energy spectrum are considered: Thue–Morse chain, double-periodic chain, Rudin–Shapiro chain. The influence of electronic energy in quantum dot on the spectral properties, band structure, density of states and spectral resistivity, is discussed. Low resistivity regions correspond to delocalized states and these states could be current states. Also we discuss the magnetic field application as the way to tune electronic energy in quantum dot and to obtain metallic or insulating conducting states of the systems. & 2012 Elsevier B.V. All rights reserved.
1. Introduction The quasiperiodic systems as the systems which are not periodic nor disordered became the object of numerous investigations after the discovery of quasicrystals [1]. Among all quasiperiodic strictures we shall consider one-dimensional structures. Pioneering work of Anderson [2] and a set of following works (see for example Ref. [3]) gave a result that all states in one-dimensional system are exponentially localized for any value of disorder. This result can be described as interference of the waves reflected from the random potential barriers. From the other hand for one-dimensional periodic system any value of perturbation leads to system falls into insulating state due to Peierls transition. Therefore periodic and disordered one-dimensional systems are insulators. But this is not the case for quasiperiodic systems. Dunlap et al. proposed the dimer model where local correlations of potential lead to the absence of localization [4]. After that a numerous quantity of works about localization properties in one-dimensional aperiodic systems were done (see for example Refs. [5,6] and related papers). Now aperiodic structures are widely used in different areas of science (see Refs. [7–10] and related papers). In the present paper we consider aperiodic chain of quantum dots (QDs). QD is quasi-0 dementional quantum structure, the artificial atom, consisting of 103 105 atoms is widely used in optics, nanoelectronics, etc. (for the brief review see Ref. [11]). There are studies of aperiodic atomic chains [12,13]. The reason to choose QD instead atoms is in sensitivity of electronic states in QD to external fields, particularly magnetic. In present work Thue–Morse chain, double-periodic chain, Rudin–Shapiro chains
n
Corresponding author. Tel./fax: þ 7 495 638 45 06. E-mail address: [email protected] (P.Yu. Korotaev).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.03.031
of single electronic QDs were considered. As we show, the electronic energy in QD has an influence on spectral and transport properties of QDs aperiodic chains, and the magnetic field can be used to change electronic energy and obtain insulating or metallic conducting regime of the system. Till now such an investigation was absent (only electronic spectrum of Fibonacci QD chains was studied [14]). This paper is organized as follows. In Section 2 we introduce the model and the methods that we used to describe the system of QDs. In Section 3 we discuss the results of calculations of energy spectra, spectral resistivity and density of states (DOS) and Section 3.4 is related to magnetic field influence on spectral conductivity of QDs aperiodic chains.
2. The model and methods 2.1. The model One-dimensional chain can be described by tight-binding Hamiltonian: ^ ¼ H
N X n¼1
9nSen /n9 þ
N X
9nSt n,n þ 1 /n þ 19 þ 9nSt n,n1 /n19,
ð1Þ
n¼1
where n is the site number, en is electronic energy on site n, t n,n 7 1 is hopping matrix element between sites n and n 71 and N is the chain length. From Eq. (1) one can obtain one-dimensional tight¨ binding Schrodinger equation:
cn1 t n1 þ cn þ 1 tn þ 1 þ cn en ¼ Ecn ,
ð2Þ
here cn is an on-site wave function. The model under investigation is one-dimensional sequence of two types of single-electronic QDs with electronic energies eA and
P.Yu. Korotaev et al. / Physica E 44 (2012) 1580–1584
1581
Fig. 1. The example of aperiodic chain of QDs with electronic energies arranged in the Thew–Morse sequence, N ¼ 8.
eB and hopping energy t. The electronic energies are arranged
2.3. The trace map, calculation of the energy spectra and the DOS
according to aperiodic sequence code (see Section 3.1 for definition of the sequences and Fig. 1). We assume t n,n þ 1 ¼ t n,n1 ¼ t for any n. By shifting zero energy level by ðeA þ eB Þ=2 one can suppose eA ¼ eB ¼ e. Eq. (2) usually presented in the terms of transfer matrix: ! ! ! n Y cn þ 1 cn c1 Mi ¼ Mn ¼ , ð3Þ
For each aperiodic sequence there is unique recursive relation between trace of transfer matrix of the n-th generation and traces of previous generations. This relation is the so-called trace map. It can be very useful to obtain spectral properties of aperiodic sequences [16]. The bands of energy spectra were obtained via equation:
cn
cn1
i¼1
c0
here Mn is the so-called transfer matrix: 0 1 Een t n,n1 @ t n,n þ 1 A: M n ¼ t n,n þ 1 1 0
9xn ðEÞ9 r 1 xn ðEÞ ¼ 12trM n ðEÞ, ð4Þ
2.2. Calculation of the transmittance and the resistance To obtain the transmission coefficient of QD sequence we suppose the electron incident from left to right on the aperiodic chain. The particle energy in tight-binding model is E ¼ 2t cosðkaÞ. The n-th state was assumed as superposition of two plane waves:
cn,k ¼ an eink þbn eink ,
ð5Þ
here an and bn are amplitudes of the transmitted and reflected wave, respectively, and k is a wave vector. After a procedure which is similar to the one proposed by Lui and Chao [15] one can obtain the relation between amplitudes of waves at the site n and at the initial sites: ! ! b0 bn ¼ T n,k : ð6Þ a0 an The transmission matrix T n,k is defined as: ! n Y eikn 0 Q 1 Mi Q k , T n,k ¼ k ikn 0 e i¼1
Qk ¼
eik
eik
1
1
ð7Þ
,
1 @
DðEÞ ¼
p @E
r=t þ
ðr=t þ Þn
1=t nþ
! ,
ð8Þ
here r and t þ are relation between the amplitudes on site n and on the initial site of reflected and transmitted waves, respectively. Therefore the transmission coefficient is given by:
t ¼ 9aN þ 1 =a0 9 ¼ 9t þ 92 :
ðImðln t Nþ ÞÞ,
gðEÞ ¼ lim
1
N-1 N
lnð:M N ðEÞ:Þ,
3. Quantum dot aperiodic chains
The aperiodic chains investigated here are the substitutional sequences with two-letter alphabet A,B. It means that its generations can be obtained by the substitution of elements of previous generations according to defined rule. The Thue–Morse sequence is generated by substitution: B-BA:
A-AB,
B-AA:
To obtain the resistance of the chain we used Landauer formula:
BA-BBAB,
BB-BBBA:
e2 2p_
1t
t
:
ð10Þ
ð15Þ
and generations are: A,AB,ABAA,ABAAABAB, . . .. Finally, for Rudin– Shapiro sequence is generated as: AB-AABA,
r¼
ð14Þ
The generations of the Thue–Morse chains are A,AB,ABBA, ABBABAAB, . . . . The substitution rules for double-periodic sequence are:
AA-AAAB,
1
ð13Þ
here :MN ðEÞ: is the matrix norm.
ð9Þ
ð12Þ
here tNþ is the ratio between incident state and transmitted state t Nþ ¼ t þ eikN obtained from Eq. (8). The inverse localization length was calculated using expression:
A-AB,
and has the form: T n,k ¼
here tr is trace of matrix. The states cn that correspond to obtained energy are not grow exponentially, and the gap of spectrum corresponds to exponentially growth of states. The density of states (DOS) was obtained using Thouless result for Green function matrix elements [17] and relation of Kirkman and Pendry [18]:
3.1. Definition
!
1=t þ
ð11Þ
ð16Þ
with AA as the first element. We assume A-eA and B-eB for electronic energy in QDs (see Section 2.1 and Fig. 1). For calculations of spectral properties we used periodic boundary conditions, assuming A as the initial and the last element.
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Fig. 2. The energy spectrum of aperiodic chains of QDs for N ¼ 16 for different values of electronic energy in QD e1 ¼ 1 eV, e2 ¼ 0:5 eV, e3 ¼ 0:25 eV and t ¼ 1 eV. (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain.
3.2. The energy spectrum and the DOS The energy spectrum of one-dimensional aperiodic chains has a band structure, and reveals the general features of sequence definition. To illustrate the influence of electronic site energy on the energy spectrum we have used the value of hopping energy t¼1 eV (Fig. 2). One can see that decreasing in the value D ¼ 9eA eB 9 leads to increasing subband width and decreasing gap width. Total bandwidth goes to 4t as expected for periodic system. This is a common feature of investigated aperiodic chains. For the Thue– Morse chain the spectrum is symmetric under zero energy level and the number of bands is less than the number of elements due to degeneration. It can be clearly seen from trace map analysis [19]. The spectra of double-periodic and Rudin–Shapiro chains contain the number of bands equal to the number of elements. The spectra of investigated chains are singular continuous indicating an existence of critical states, i.e. the states that are neither extended nor localized. In a critical state the localization length is larger than interatomic distance that corresponds to a localized state. At Fig. 3 the inverse localization length along subband in energy spectrum of aperiodic chains of QDs is plotted. The distribution of the localization degree along subband is inhomogeneous. For the Thue–Morse chain mostly delocalized states are located at the center of the band, for other chains the mostly delocalized states occur as at the subband edges as inside the subband. Delocalization that corresponds to the absence of reflected wave is clearly seen from the resistance analysis (see Section 3.1). The DOS was obtained using Eq. (12), the results are plotted at Fig. 4. The DOS consists of a lot of gaps and pseudogaps corresponding to rich structure of energy spectrum. Note, that the band and location are determined by electronic energy in QD and by number of QDs in the chain. Therefore by changing these parameters one can change position of bands relatively to the Fermi energy and obtain metallic or insulating conducting properties. 3.3. The resistance At Fig. 5 we have plotted the spectral dependence of resistance of the QDs chain at the different values of electronic energy. There is a set of pikes of low resistivity corresponding to the existence of delocalized states in the energy bands. At the inset of the figure we have shown the resistance behavior near energy value, corresponding to delocalized state. Vanishing or low value of the resistance take place due to the absence of localization in aperiodic chains which is result of weakening of reflected wave. For the Thew–Morse chain the lowest resistivity occurs at the values of
Fig. 3. The distribution of inverse localization length along subband of energy spectrum of aperiodic chain of QDs. Arrows indicate subband edges. (a) Thue– Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain; t ¼ 1 eV, e ¼ 0:25 eV, N ¼ 16.
energy corresponding to so-called lattice-like functions. Some authors mentioned that these states are extended in the limit of large N, therefore the spectrum of Thew-Morse sequence consists of singular continuous and absolutely continuous parts [19]. One can see that the resistance is vanishing, and it is another cause to
P.Yu. Korotaev et al. / Physica E 44 (2012) 1580–1584
Fig. 4. The DOS of aperiodic sequence of QD for N ¼ 128. Electronic energy e ¼ 0:33 eV, t¼1 eV; (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin– Shapiro chain.
classify these states as extended. Another result is reducing mean resistance with decreasing D due to reducing confining potential.
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Fig. 5. Spectral dependence of resistivity for aperiodic chain of QDs for N ¼ 32, t¼ 1 eV. (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain. Two values of electronic energy e are considered: e ¼ 0:5 eV (black line) and e ¼ 0:25 eV (gray line). The inset shows the behavior of resistance near energy value corresponding to delocalized state.
3.4. The influence of magnetic field
electronic energy: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9m9 þ1 a oc _ En,m ¼ _ , þ þ o2c n þ 2 mn 2
We already mentioned that the electronic energy e has an influence on band location, its width and on the localization properties. As one of the possible ways to tune the electronic energy the applying of external magnetic was considered. The magnetic field required to change electronic energy in QD is much weaker than for an atom and there is a reason to consider chains of QDs instead of atomic chains. The influence of magnetic field was considered as follows. It was assumed that QD creates parabolic confining potential with steepness a, and Hamiltonian of electron in QD is:
here cyclotronic frequency is oc ¼ eB=mn c, B is magnetic field. Using Eq. (18) we have calculated effective electronic energy in QD e ¼ eðBÞ. At Fig. 6 spectral conductivity of aperiodic chain of QDs (only part of the spectrum is considered) is plotted . Applying magnetic field leads to shift of bands location and their width. Therefore by changing the value of magnetic field one can change the location of bands relatively to Fermi energy. It will change the localization degree of electrons at the Fermi energy and therefore to obtain metallic or insulating conducting properties of the system.
H¼
_ A 2 ire þ ar2 , 2mn c
ð17Þ
here r is a radius-vector of electron in QD, A is a vector potential of magnetic field, mn is effective electron mass, e is ¨ electron charge. The solution of Schrodinger equation leads to
ð18Þ
4. Conclusion The electronic spectral properties of several aperiodic chains of QDs: Thew–Morse chain, double-periodic chain and Rudin– Shapiro chain were investigated in single-electron tight-binding approximation using transfer matrix technique. The spectra have
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P.Yu. Korotaev et al. / Physica E 44 (2012) 1580–1584
states and the degree of localization is inhomogeneous along the subband, that is follows from calculations of inverse localization length and spectral resistivity. We have shown thereby that by tuning electronic energy in QD (or number of QDs in the chain) one can change the electronic localization degree at the Fermi energy and obtain metallic or insulating behavior of conductivity. The influence of magnetic field is considered as the method to tune electronic energy in QD, and the spectral conductivity calculations confirm that the relatively small magnetic field (about several Tl) change the conductive properties of the system.
Acknowledgments The research is carried out with financial support of the Programme of Creation and Development of the National University of Science and Technology ‘‘MISIS’’ and the Russian Foundation for Basic Researches projects (11-02-00604-a). We are also grateful to B. Kheyfets for helpful notes. References
Fig. 6. Spectral dependence of conductivity of aperiodic chain of QDs for N ¼ 64. (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain. t ¼15 meV, e ¼ 5 meV. The influence of magnetic field is shown by black line for B ¼ 0 and gray line for B ¼ 4 Tl.
band structure and the band location and width are affected by electronic energy value in QD and by the chain length. The correlations of electronic potentials lead to existing of delocalized
[1] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Physical Review Letters 53 (1984) 1951. [2] P.W. Anderson, Physical Review 109 (1958) 1492. [3] N. Mott, W. Twose, Advances in Physics 10 (1961) 107. [4] D.H. Dunlap, H. Wu, P.W. Phillips, Physical Review Letters 65 (1990) 88. [5] F.M. Izrailev, A.A. Krokhin, Physical Review Letters 82 (1999) 4062. [6] U. Kuhl, F.M. Izrailev, A.A. Krokhin, H.J. Stockmann, Applied Physics Letters 77 (2000) 633. [7] E. Macia, Reports on Progress in Physics 69 (2006) 397. [8] L. Moretti, I. Rea, L. De Stefano, I. Rendina, Applied Physics Letters 90 (2007) 191112. [9] J. Li, R.L. Zhang, R.W. Peng, X. Wu, D. Li, Q. Hu, Y. Qiu, M. Wang, Journal of Applied Physics 105 (2009). 07D537–07D537–3. [10] H. Aynaou, E.H. El Boudouti, Y. El Hassouani, A. Akjouj, B. Djafari-Rouhani, J. Vasseur, A. Benomar, V.R. Velasco, Physical Review E 72 (2005) 056601. [11] M. Henini, Materials Today 5 (2002) 48. [12] M. Kohmoto, B. Sutherland, C. Tang, Physical Review B 35 (1987) 1020. [13] E. Albuquerque, M. Cottam, Physics Reports 376 (2003) 225. [14] M. Bakhtiari, P. Vignolo, M. Tosi, Physica E: Low-dimensional Systems and Nanostructures 28 (2005) 385. [15] Y. Liu, K.A. Chao, Physical Review B 34 (1986) 5247. [16] J.M. Luck, Physical Review B 39 (1989) 5834. [17] D.J. Thouless, Journal of Physics C: Solid State Physics 5 (1972) 77. [18] P.D. Kirkman, J.B. Pendry, Journal of Physics C: Solid State Physics 17 (1984) 4327. [19] C.S. Ryu, G.Y. Oh, M.H. Lee, Physical Review B 46 (1992) 5162.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Electronic properties of aperiodic quantum dot chains P.Yu. Korotaev a,n, Yu.Kh. Vekilov a, N.E. Kaputkina b a b
Department of Theoretical Physics and Quantum Technologies, National University of Science and Technology ‘‘MISIS’’, 119049 Moscow, Leninskiy pr. 4, Russia Department of Physical Chemistry, National University of Science and Technology ‘‘MISIS’’, 119049 Moscow, Leninskiy pr. 4, Russia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 7 March 2012 Received in revised form 27 March 2012 Accepted 28 March 2012 Available online 4 April 2012
The electronic spectral and transport properties of aperiodic quantum dot chains are investigated. The systems with singular continuous energy spectrum are considered: Thue–Morse chain, double-periodic chain, Rudin–Shapiro chain. The influence of electronic energy in quantum dot on the spectral properties, band structure, density of states and spectral resistivity, is discussed. Low resistivity regions correspond to delocalized states and these states could be current states. Also we discuss the magnetic field application as the way to tune electronic energy in quantum dot and to obtain metallic or insulating conducting states of the systems. & 2012 Elsevier B.V. All rights reserved.
1. Introduction The quasiperiodic systems as the systems which are not periodic nor disordered became the object of numerous investigations after the discovery of quasicrystals [1]. Among all quasiperiodic strictures we shall consider one-dimensional structures. Pioneering work of Anderson [2] and a set of following works (see for example Ref. [3]) gave a result that all states in one-dimensional system are exponentially localized for any value of disorder. This result can be described as interference of the waves reflected from the random potential barriers. From the other hand for one-dimensional periodic system any value of perturbation leads to system falls into insulating state due to Peierls transition. Therefore periodic and disordered one-dimensional systems are insulators. But this is not the case for quasiperiodic systems. Dunlap et al. proposed the dimer model where local correlations of potential lead to the absence of localization [4]. After that a numerous quantity of works about localization properties in one-dimensional aperiodic systems were done (see for example Refs. [5,6] and related papers). Now aperiodic structures are widely used in different areas of science (see Refs. [7–10] and related papers). In the present paper we consider aperiodic chain of quantum dots (QDs). QD is quasi-0 dementional quantum structure, the artificial atom, consisting of 103 105 atoms is widely used in optics, nanoelectronics, etc. (for the brief review see Ref. [11]). There are studies of aperiodic atomic chains [12,13]. The reason to choose QD instead atoms is in sensitivity of electronic states in QD to external fields, particularly magnetic. In present work Thue–Morse chain, double-periodic chain, Rudin–Shapiro chains
n
Corresponding author. Tel./fax: þ 7 495 638 45 06. E-mail address: [email protected] (P.Yu. Korotaev).
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.03.031
of single electronic QDs were considered. As we show, the electronic energy in QD has an influence on spectral and transport properties of QDs aperiodic chains, and the magnetic field can be used to change electronic energy and obtain insulating or metallic conducting regime of the system. Till now such an investigation was absent (only electronic spectrum of Fibonacci QD chains was studied [14]). This paper is organized as follows. In Section 2 we introduce the model and the methods that we used to describe the system of QDs. In Section 3 we discuss the results of calculations of energy spectra, spectral resistivity and density of states (DOS) and Section 3.4 is related to magnetic field influence on spectral conductivity of QDs aperiodic chains.
2. The model and methods 2.1. The model One-dimensional chain can be described by tight-binding Hamiltonian: ^ ¼ H
N X n¼1
9nSen /n9 þ
N X
9nSt n,n þ 1 /n þ 19 þ 9nSt n,n1 /n19,
ð1Þ
n¼1
where n is the site number, en is electronic energy on site n, t n,n 7 1 is hopping matrix element between sites n and n 71 and N is the chain length. From Eq. (1) one can obtain one-dimensional tight¨ binding Schrodinger equation:
cn1 t n1 þ cn þ 1 tn þ 1 þ cn en ¼ Ecn ,
ð2Þ
here cn is an on-site wave function. The model under investigation is one-dimensional sequence of two types of single-electronic QDs with electronic energies eA and
P.Yu. Korotaev et al. / Physica E 44 (2012) 1580–1584
1581
Fig. 1. The example of aperiodic chain of QDs with electronic energies arranged in the Thew–Morse sequence, N ¼ 8.
eB and hopping energy t. The electronic energies are arranged
2.3. The trace map, calculation of the energy spectra and the DOS
according to aperiodic sequence code (see Section 3.1 for definition of the sequences and Fig. 1). We assume t n,n þ 1 ¼ t n,n1 ¼ t for any n. By shifting zero energy level by ðeA þ eB Þ=2 one can suppose eA ¼ eB ¼ e. Eq. (2) usually presented in the terms of transfer matrix: ! ! ! n Y cn þ 1 cn c1 Mi ¼ Mn ¼ , ð3Þ
For each aperiodic sequence there is unique recursive relation between trace of transfer matrix of the n-th generation and traces of previous generations. This relation is the so-called trace map. It can be very useful to obtain spectral properties of aperiodic sequences [16]. The bands of energy spectra were obtained via equation:
cn
cn1
i¼1
c0
here Mn is the so-called transfer matrix: 0 1 Een t n,n1 @ t n,n þ 1 A: M n ¼ t n,n þ 1 1 0
9xn ðEÞ9 r 1 xn ðEÞ ¼ 12trM n ðEÞ, ð4Þ
2.2. Calculation of the transmittance and the resistance To obtain the transmission coefficient of QD sequence we suppose the electron incident from left to right on the aperiodic chain. The particle energy in tight-binding model is E ¼ 2t cosðkaÞ. The n-th state was assumed as superposition of two plane waves:
cn,k ¼ an eink þbn eink ,
ð5Þ
here an and bn are amplitudes of the transmitted and reflected wave, respectively, and k is a wave vector. After a procedure which is similar to the one proposed by Lui and Chao [15] one can obtain the relation between amplitudes of waves at the site n and at the initial sites: ! ! b0 bn ¼ T n,k : ð6Þ a0 an The transmission matrix T n,k is defined as: ! n Y eikn 0 Q 1 Mi Q k , T n,k ¼ k ikn 0 e i¼1
Qk ¼
eik
eik
1
1
ð7Þ
,
1 @
DðEÞ ¼
p @E
r=t þ
ðr=t þ Þn
1=t nþ
! ,
ð8Þ
here r and t þ are relation between the amplitudes on site n and on the initial site of reflected and transmitted waves, respectively. Therefore the transmission coefficient is given by:
t ¼ 9aN þ 1 =a0 9 ¼ 9t þ 92 :
ðImðln t Nþ ÞÞ,
gðEÞ ¼ lim
1
N-1 N
lnð:M N ðEÞ:Þ,
3. Quantum dot aperiodic chains
The aperiodic chains investigated here are the substitutional sequences with two-letter alphabet A,B. It means that its generations can be obtained by the substitution of elements of previous generations according to defined rule. The Thue–Morse sequence is generated by substitution: B-BA:
A-AB,
B-AA:
To obtain the resistance of the chain we used Landauer formula:
BA-BBAB,
BB-BBBA:
e2 2p_
1t
t
:
ð10Þ
ð15Þ
and generations are: A,AB,ABAA,ABAAABAB, . . .. Finally, for Rudin– Shapiro sequence is generated as: AB-AABA,
r¼
ð14Þ
The generations of the Thue–Morse chains are A,AB,ABBA, ABBABAAB, . . . . The substitution rules for double-periodic sequence are:
AA-AAAB,
1
ð13Þ
here :MN ðEÞ: is the matrix norm.
ð9Þ
ð12Þ
here tNþ is the ratio between incident state and transmitted state t Nþ ¼ t þ eikN obtained from Eq. (8). The inverse localization length was calculated using expression:
A-AB,
and has the form: T n,k ¼
here tr is trace of matrix. The states cn that correspond to obtained energy are not grow exponentially, and the gap of spectrum corresponds to exponentially growth of states. The density of states (DOS) was obtained using Thouless result for Green function matrix elements [17] and relation of Kirkman and Pendry [18]:
3.1. Definition
!
1=t þ
ð11Þ
ð16Þ
with AA as the first element. We assume A-eA and B-eB for electronic energy in QDs (see Section 2.1 and Fig. 1). For calculations of spectral properties we used periodic boundary conditions, assuming A as the initial and the last element.
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P.Yu. Korotaev et al. / Physica E 44 (2012) 1580–1584
Fig. 2. The energy spectrum of aperiodic chains of QDs for N ¼ 16 for different values of electronic energy in QD e1 ¼ 1 eV, e2 ¼ 0:5 eV, e3 ¼ 0:25 eV and t ¼ 1 eV. (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain.
3.2. The energy spectrum and the DOS The energy spectrum of one-dimensional aperiodic chains has a band structure, and reveals the general features of sequence definition. To illustrate the influence of electronic site energy on the energy spectrum we have used the value of hopping energy t¼1 eV (Fig. 2). One can see that decreasing in the value D ¼ 9eA eB 9 leads to increasing subband width and decreasing gap width. Total bandwidth goes to 4t as expected for periodic system. This is a common feature of investigated aperiodic chains. For the Thue– Morse chain the spectrum is symmetric under zero energy level and the number of bands is less than the number of elements due to degeneration. It can be clearly seen from trace map analysis [19]. The spectra of double-periodic and Rudin–Shapiro chains contain the number of bands equal to the number of elements. The spectra of investigated chains are singular continuous indicating an existence of critical states, i.e. the states that are neither extended nor localized. In a critical state the localization length is larger than interatomic distance that corresponds to a localized state. At Fig. 3 the inverse localization length along subband in energy spectrum of aperiodic chains of QDs is plotted. The distribution of the localization degree along subband is inhomogeneous. For the Thue–Morse chain mostly delocalized states are located at the center of the band, for other chains the mostly delocalized states occur as at the subband edges as inside the subband. Delocalization that corresponds to the absence of reflected wave is clearly seen from the resistance analysis (see Section 3.1). The DOS was obtained using Eq. (12), the results are plotted at Fig. 4. The DOS consists of a lot of gaps and pseudogaps corresponding to rich structure of energy spectrum. Note, that the band and location are determined by electronic energy in QD and by number of QDs in the chain. Therefore by changing these parameters one can change position of bands relatively to the Fermi energy and obtain metallic or insulating conducting properties. 3.3. The resistance At Fig. 5 we have plotted the spectral dependence of resistance of the QDs chain at the different values of electronic energy. There is a set of pikes of low resistivity corresponding to the existence of delocalized states in the energy bands. At the inset of the figure we have shown the resistance behavior near energy value, corresponding to delocalized state. Vanishing or low value of the resistance take place due to the absence of localization in aperiodic chains which is result of weakening of reflected wave. For the Thew–Morse chain the lowest resistivity occurs at the values of
Fig. 3. The distribution of inverse localization length along subband of energy spectrum of aperiodic chain of QDs. Arrows indicate subband edges. (a) Thue– Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain; t ¼ 1 eV, e ¼ 0:25 eV, N ¼ 16.
energy corresponding to so-called lattice-like functions. Some authors mentioned that these states are extended in the limit of large N, therefore the spectrum of Thew-Morse sequence consists of singular continuous and absolutely continuous parts [19]. One can see that the resistance is vanishing, and it is another cause to
P.Yu. Korotaev et al. / Physica E 44 (2012) 1580–1584
Fig. 4. The DOS of aperiodic sequence of QD for N ¼ 128. Electronic energy e ¼ 0:33 eV, t¼1 eV; (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin– Shapiro chain.
classify these states as extended. Another result is reducing mean resistance with decreasing D due to reducing confining potential.
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Fig. 5. Spectral dependence of resistivity for aperiodic chain of QDs for N ¼ 32, t¼ 1 eV. (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain. Two values of electronic energy e are considered: e ¼ 0:5 eV (black line) and e ¼ 0:25 eV (gray line). The inset shows the behavior of resistance near energy value corresponding to delocalized state.
3.4. The influence of magnetic field
electronic energy: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 9m9 þ1 a oc _ En,m ¼ _ , þ þ o2c n þ 2 mn 2
We already mentioned that the electronic energy e has an influence on band location, its width and on the localization properties. As one of the possible ways to tune the electronic energy the applying of external magnetic was considered. The magnetic field required to change electronic energy in QD is much weaker than for an atom and there is a reason to consider chains of QDs instead of atomic chains. The influence of magnetic field was considered as follows. It was assumed that QD creates parabolic confining potential with steepness a, and Hamiltonian of electron in QD is:
here cyclotronic frequency is oc ¼ eB=mn c, B is magnetic field. Using Eq. (18) we have calculated effective electronic energy in QD e ¼ eðBÞ. At Fig. 6 spectral conductivity of aperiodic chain of QDs (only part of the spectrum is considered) is plotted . Applying magnetic field leads to shift of bands location and their width. Therefore by changing the value of magnetic field one can change the location of bands relatively to Fermi energy. It will change the localization degree of electrons at the Fermi energy and therefore to obtain metallic or insulating conducting properties of the system.
H¼
_ A 2 ire þ ar2 , 2mn c
ð17Þ
here r is a radius-vector of electron in QD, A is a vector potential of magnetic field, mn is effective electron mass, e is ¨ electron charge. The solution of Schrodinger equation leads to
ð18Þ
4. Conclusion The electronic spectral properties of several aperiodic chains of QDs: Thew–Morse chain, double-periodic chain and Rudin– Shapiro chain were investigated in single-electron tight-binding approximation using transfer matrix technique. The spectra have
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states and the degree of localization is inhomogeneous along the subband, that is follows from calculations of inverse localization length and spectral resistivity. We have shown thereby that by tuning electronic energy in QD (or number of QDs in the chain) one can change the electronic localization degree at the Fermi energy and obtain metallic or insulating behavior of conductivity. The influence of magnetic field is considered as the method to tune electronic energy in QD, and the spectral conductivity calculations confirm that the relatively small magnetic field (about several Tl) change the conductive properties of the system.
Acknowledgments The research is carried out with financial support of the Programme of Creation and Development of the National University of Science and Technology ‘‘MISIS’’ and the Russian Foundation for Basic Researches projects (11-02-00604-a). We are also grateful to B. Kheyfets for helpful notes. References
Fig. 6. Spectral dependence of conductivity of aperiodic chain of QDs for N ¼ 64. (a) Thue–Morse chain; (b) double-periodic chain; (c) Rudin–Shapiro chain. t ¼15 meV, e ¼ 5 meV. The influence of magnetic field is shown by black line for B ¼ 0 and gray line for B ¼ 4 Tl.
band structure and the band location and width are affected by electronic energy value in QD and by the chain length. The correlations of electronic potentials lead to existing of delocalized
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