Flexible quantum dots interacting with phonons: A quantum capacitive approach

Physica E 50 (2013) 6–10

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Flexible quantum dots interacting with phonons: A quantum capacitive approach J.C. Flores a,n, M. Calcina N. b a b

´, Casilla 7-D, Arica, Chile ´n IAI, Universidad de Tarapaca Instituto de Alta Investigacio ´, Casilla 7-D, Arica, Chile Departamento de Fı´sica, FACI, Universidad de Tarapaca

H I G H L I G H T S c c c c

Flexible quantum dots. Vibration degrees disturb the usual properties. Vibration enhances the charging energy. Vibration changes Coulomb blockade.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 October 2012 Received in revised form 3 February 2013 Accepted 12 February 2013 Available online 18 February 2013

Through a capacitive model, the properties of flexible quantum dots interacting with phonons are studied. Usual quantum dots properties like charging energy and bias window become deeply modified. In the phononic emission–absorption process, we show the existence of forbidden regions where neither absorption nor emission of phonons can happen. The forbidden regions are related to the regime where the phononic-band-size is smaller than the charging energy. Finally, we derive the conditions for electrical current existence through flexible quantum dots. & 2013 Elsevier B.V. All rights reserved.

1. Introduction: phonon–capacitance interaction The physics of quantum dots (qudots) [1] plays an important role, not only from a technological perspective, but also from a fundamental point of view [1–4]. Such devices, also called artificial atoms [5,6], can be thought as boxes filled with electrons occupying discrete quantum energy levels. Similarly, singlemolecule junctions reveal rich features related to quantum transport [1,7–9] being significant in quantum computing areas [10]. In this work we shall consider the Hamiltonian of a set of phonons interacting with quantized flexible LC-circuits with charge discreteness [11–15]. The LC-circuits describe quantum dots through a capacitive dynamics allowing us to study a flexible geometry. Similar considerations were carried out for experimental realizations in a single-C60 transistor showing evidence of coupling between mechanical degrees and single-electron hopping. As a consequence, single-electron charging can be used to excite and probe the mechanical motion of molecules [16–22]. Moreover, from a basic point of view, the study of flexible

n

Corresponding author. Tel.: þ56 58 7656452. E-mail address: cfl[email protected] (J.C. Flores).

1386-9477/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2013.02.011

electronic systems becomes interesting since the practical need to fold and unfold electronic devices in a near future. There is exhaustive research related to memory circuit elements [23] (mencapacitive, meninductive, etc.) where quantization is envisaged [24]. Memory circuit elements take into account internal degrees of freedom, which are coupled to electrical states. In fact, flexible circuits are closely related to these memory elements becoming a direct example of them. Consider a vibrational lattice having a LC-circuit per cell at position l (integer). Assume that mechanical vibrations distort the corresponding capacitance in each cell. For instance, for a parallelplate capacitor (area A) one has C ¼ eo A=ðx þ DxÞ with Dx being the operative size change. In this way, the electrical component becomes relocated due to the coupling with vibrational degrees. We shall consider the coupled vibration–capacitance Hamiltonian given by ! q  X q^ 2 2_2 l ^ ¼ ^ H þ 2 sin2 e f^ l V ðlÞ g ql 2C l Lqe 2_ l ! 2 X p^ 1 2 l þ Kðx^ l x^ l1 Þ , þ ð1Þ 2M 2 l where qe is the discrete charge parameter and the capacitance at the l-site becomes

J.C. Flores, M. Calcina N. / Physica E 50 (2013) 6–10

  x^ x^ 1 1 1 þ l l1 : ¼ Cl C d

ð2Þ

In the above expression, the distance parameter d describes 1 the distortion (i.e. when ðd Þ-0, the capacitances are rigid). Note that it generates the interaction term Dxq2 which differs from others ðDxqÞ reported in the literature [18]. Moreover, operators satisfy the usual canonical commutation rules: for charge and magnetic flux ½q^ l , f^s  ¼ i_dl,s and for position and momentum ½x^ l , p^s  ¼ i_dl,s (zero otherwise). The quantum mechanical coherent [25] gate voltage V ðlÞ g corresponds to the external voltage of a third conductor inserted in the electrical cell at l. Gate voltage is important in this work since we shall study its role face to flexible components. The mechanic and electromagnetic part of Eq. (1) become formally uncoupled by working as follows [26]. Consider the usual Fourier transform for the mechanical variables, 1 X ^ ikla x^ l ¼ pffiffiffiffi X ke , N k

1 X ^ ikla P ke p^ l ¼ pffiffiffiffi , N k

ð3Þ

where N b 1 is the total number of masses in the lattice (including the masses of plates) and a is the lattice elementary size. With this transformation, the Hamiltonian (1) becomes ! q  X q^ 2 2_2 X P^ k P^ k l ^ ¼ ^ þ 2 sin2 e f^ l V ðlÞ q H g l þ 2C 2_ 2M Lqe l k ! X q^ 2 X^ k M o2k ^ ^ l pffiffiffiffi eikla ð1eika Þ , þ ð4Þ X k X k þ 2 l,k 2Cd N pffiffiffiffiffiffiffiffiffiffiffi where ok ¼ 2 K=M 9sinðka=2Þ9 is the usual one-band phononic frequency spectrum [27,28]. Now, using the coordinate transformation, X X^ k ¼ X^ 0k  l

2 q^l

pffiffiffiffi eikla ð1eika Þ, 2CdM o2k N

ð5Þ

7

9fnl g; fsk gS, of nl charges at the l-cell and sk phonons with wavenumber k, has the energy !  ! X ðlÞ X 1 , ð9Þ Eo ðfnl g; fsk gÞ ¼ Enl þ _ok sk þ 2 fn g fs g l

k

where for simplicity, we have defined the auxiliary variable (electrical energy per site) ! q2e n2l q2e n2l ðlÞ 1 Enl ¼ ð10Þ V ðlÞ g qe nl : 2 2C 4CKd Without the inductive term in Eq. (8), the capacitance energy does not decay since the charged particles could not travel from one plate to another and annihilate each other. In this sense, it is necessary for the inductance to provide annihilation. The firstorder correction DEð1Þ of the spectrum due to magnetic induction is given formally by

DEð1Þ ¼ / . . . n0 n1 . . . sk 9H^ L 9 . . . n0 n1 . . . sk S ¼

X _2 l

Lq2e

,

ð11Þ

and proportional to the size N of the network. Finally, to end this section, remark that in Eq. (10) the physical condition to avoid plates superposition requires 14

q2e n2l 2

4CKd

,

ð12Þ

in the corresponding cell. From Eq. (10), the spectrum is bounded above which gives rise to a maximum number of allowed electrons in the flexible capacitance until it collapses due to excessive charge.

2. Phonons emission–absorption: general 1

the electrical and mechanical parts of the Hamiltonian (4) become formally uncoupled. It appears a nonlinear term q4 related to the interaction between the vibration and charge. The transformed Hamiltonians are ^ ¼H ^ o þH ^ L, H

ð6Þ

where ^o¼ H

X q^ 2 l 2C l

1

!

2 q^ l 2

4CKd

!

^ V ðlÞ g ql þ

! X P^ k P^ k M o2 k ^0 ^0 þ X k X k , 2M 2 k ð7Þ

and the inductive part remains as 2X q  ^ L ¼ 2_ sin2 e f^ l , H 2 2_ Lqe l

ð8Þ

which will be considered at first-order. Formally in the limit d-1, the capacitance-phonon interaction term (  q4 ) disappears as expected. In this work, we shall consider preferentially a capacitive ^ L is regime where formally L-1. The flux inductive part H considered as perturbation being necessary for electrical current b b existence. The gate voltage V ðlÞ g breaks the symmetry q l 2q l and then the configuration with zero charge does not necessarily corresponds to the background state. We must operate carefully since the new operators related to the transformation Eq. (5) are not independent (i.e. in the new representation the electrical and mechanical operators do not, in general, commute, where 0 ^  a 0Þ. Nevertheless, in the capacitive regime the inductive ½X^ , f term is negligible and then, the spectrum Eo of the purely capacitive Hamiltonian Eq. (7) is easily obtained. The eigenstate

When the gate voltage is zero and d ¼ 0 (rigid plates) the minimal energy in the electric component of Eq. (10) occurs when nl ¼ 0 in each cell. As said before, this energy state changes when V g a 0 because the minimal electrical energy becomes re-adapted. Consider each capacitance having originally a definite charge nl qe , we want to know the number of emitted (or absorbed) phonons when charge decays (or electrical-pairs creation) forming new charge states. To visualize this process, electrical currents are necessary and then the term (11) must be considered. This process is related to energy conservation. The interaction term Eq. (2) produces phonons when the original charge is annihilated (or the inverse). When charges re-organize we have the energy conservation equality Efnl a 0;sk ¼ 0g ¼ Efml a 0;sk a 0g :

ð13Þ

The zero-point energy and the perturbation term Eq. (11) disappear from both the sides of the Eq. (13) and we have X ðlÞ X ðlÞ X Enl ¼ Eml 7 _ok sk , ðemissionabsorptionÞ: ð14Þ fnl g

fml g

fsk g

The above energy-balance equation is the basic tool in this work. In Eq. (14) the frequency characterizing the phonon spectrum ok is generic.

3. One charged capacitance: distortion and gate voltage Consider one charged capacitor where nl ¼ ndl,0 and V gðlÞ ¼ V g dl,0 . In the charging or discharging process from n2ðn1Þ there is mechanical distortion. Assume two phonons (s¼2) of opposite wavenumber ko related to this process. From Eq. (14), one has the

8

J.C. Flores, M. Calcina N. / Physica E 50 (2013) 6–10

algebraic relation   q2e n2 q2 n2 q2 ðn1Þ2 1 e 2  e 2C 2C 4CKd

1

q2e ðn1Þ2 4CKd

!

Ko

V g qe ¼ 72_oko ,

2

4

ð15Þ 2

which defines the wavenumber ko and the gate voltage Vg. The signs 7 on the right side stand for charge or discharge. There are two cases corresponding to different behaviors which must be recognized separately. The first case corresponds to narrow-band and the second to wide-band. The parameter defining these behaviors becomes related to the rate between the size of the phononic-band-size the elementary charge energy pffiffiffiffiffiffiffiffiffiffiand ffi of the capacitance ðs  2_ K=MÞ= ðq2e =2CÞ. For the transition process (15), Fig. 1(a) shows the dimensionless phonon wavenumber ko as function of the dimensionless gate 0 2 voltagep Vffiffiffiffiffiffiffiffiffiffi g ¼ ðV ffi g qe Þ=ðqe =2CÞ. In this particular band regime we set ð2  2_ K=M Þ ¼ ðq2e =2CÞ and a negligible distortion parameter 1 d ¼ 0. We show only four channels which correspond to the values V 0g ¼ 3,1,1,3, when ko ¼ 0. Fig. 1(b) corresponds to the 1 same set of parameters than Fig. 1(a) but with distortion d 4 0. In this case, capacitance distortion produces gaps around voltages V 0g ¼ 2, 2 where neither absorption nor emission can happen. Thus, qudot distortion drastically changes the emission–absorption phonons diagram. Fig. 2(a) and (b), like Fig. 1(a) and (b), shows the same emission–absorption pffiffiffiffiffiffiffiffiffiffiffidiagram but in the narrow phononic-band regime ð2  2_ K=M Þ o ðq2e =2CÞ. Interestingly in this case, for 1 distortion parameter d Z0, there are always gaps around different gate voltages. Fig. 2(a), with a distortion parameter

-5

-4

-3

-2

-1

1

2

3

4

Vg

5

-2

-4

Ko

4

2

-5

-4

-3

-2

-1

1

2

3

4

Vg

5

-2

-4 pffiffiffiffiffiffiffiffiffiffiffi Fig. 2. (a) Narrow-band case ð2  2_ K=M o ðq2e =2CÞÞ with zero distortion para1 meter ðd ¼ 0Þ. There are gaps structures. (b) Narrow-band case ð2  pffiffiffiffiffiffiffiffiffiffi ffi 1 2_ K=M o ðq2e =2CÞÞ with distortion parameter d 4 0. Note the shift of the gaps.

Ko 4

1

2

-5

-4

-3

-2

-1

1

2

3

4

5

Vg -2

d ¼ 0, shows four channels which corresponds to the values V 0g ¼ 3,1,1,3 when k0 ¼ 0. Fig. 2(b) shows four channels with 1 distortion d 4 0. Note that the effect of the distortion is a shift of the channels but there are always gaps.pffiffiffiffiffiffiffiffiffiffiffi Finally, in the wide-band case ð2  2_ K=M Þ 4 ðq2e =2CÞ gaps do not exist and channels become superposed either with or without distortion.

-4

4. Falling to the background by phonons emission: N capacitances

Ko

For the case when V ðlÞ g ¼ 0, consider the transition from an excited channel nl ¼ n to nl ¼ 0 (background). For N charged capacitances (n charges each one) and 2s emitted phonons of wavenumber ko, the conservation energy Eq. (14) becomes   q2 n2 q2 n2 1 e 2 0 ¼ 2s_oko : ð16Þ N e 2C 4CKd

4 2

-5

-4

-3

-2

-1

1

2

3

4

5

Vg -2 -4 Fig. 1. (a) Emission–absorption process for small distortion parameter (1=d  0). It shows the dimensionless wavenumber ko as function of the dimensionless pffiffiffiffiffiffiffiffiffiffi ffi gate voltage V 0g ¼ ðV g qe Þðq2e =2CÞ. In this case we set 2  2_ K=M ¼ ðq2e =2CÞ. (b) Wavenumber ko as function of the gate voltage Vg, similar to Fig. 1(a), but with distortion parameter 1=d 40. Remark the presence of gaps where neither absorption nor emission can happen.

Fig. 3 showspthis ffiffiffiffiffiffiffiffiffiffiffitransition (with N ¼s) for different band sizes (i.e. ð2  2_ K=MÞ compared to ðq2e =2CÞ) as function of the 1 distortion parameter d . (a) The dashed-curve corresponds to the narrow-band case, (b) the solid-curve to the case when phononic-band and elementary charging energy have the same sizes, and (c) the dot-curve to a wide-band. The narrow-band case presents gaps as expected from the previous section. For small 1 parameter ðd  0Þ there is no emission and decaying state. Eq. (16) determines the wavenumber ko of 2s emitted phonons. In the limit N-1 one must take s-1. Moreover, similar to a QED-box, the wavenumber discretization can make Eq. (16) unviable and freeze the initial charge state.

J.C. Flores, M. Calcina N. / Physica E 50 (2013) 6–10

Ko

position l) when   qe q3 _ok V gðlÞ   2e 2 þ s: 2C 8C Kd qe

1.0

05

0.0 0.2

0.4

0.6

0.8

10

1.2

1/d 05

1.0 Fig. 3. The emitted wavenumber ko as a function of the dimensionless distortion 0 2 parameter 1=d ¼ q2e =4CKd . The narrow-band case (dashed-line) presents a gap for 0 1=d o 0:5. The solid-curve (where both the phononic-band and charge energy have the same values) does not present gaps. The wide-band case (pointed-curve) has a bounded wavenumber (order of ko o 0:3 in the figure).

5. Charging energy and distortion: bias window approach Incoherent quantum [25] transport occurs through a qudot when the differences between Fermi levels, around the dot, fall in the tunnel-resonant bias window [2,3]. Assume that the electrochemical potential background is m at the right of the dot. As usual, an electrical current could be eventually established by putting an electrochemical potential m þ ¼ m þ DV at the left of the dot. The condition to have one-charge electrical current transport through the flexible capacitance, at position l, becomes ! q2 ðn þ1Þ2 q2 ðn þ 1Þ2 1 e 0r e 2 2C 4CKd   q2 n2 q2 n2 1 e 2 V gðlÞ qe r DV l ,  e ð17Þ 2C 4CKd

9

ð21Þ

The above expression is quite significant and an important result in this work. The charging energy (qe V gðlÞ ) becomes deeply modified due to vibrations and plates distortion. The first term, in round parenthesis, corresponds to the modification of the charging energy by plates relocation and it reduces the charging potential. The second term tells us that vibrations enlarge the charging energy value. A similar conjecture was realized with respect to the enlarged bias voltage [18] for a specific microscopic model including vibrations with interaction like Dxq.

6. Simplicity being operative (n b 1) Consider a transition related to charges where the number of channels is large (n b 1). In this case we can replace a difference like En þ 1 En by the formal derivative @E=@n. In this way, the main formulas considered in this work become simplified. For the sake of simplicity we shall consider only one charged capacitor. Eq. (15), related to an emission–absorption of two phonons of wavenumber ko and written for 2s phonons, becomes   q2e q2e 3 n n ð22Þ V g qe ¼ 72s_oko , 2 C 2CKd and Eq. (18) for transmission through the qudot with a bias voltage DV is simplified as   q2 q2e 3 n ð23Þ V g qe r DV, 0 r e n 2 C 2CKd being Eqs. (22) and (23) easier to handle than the original ones Eqs. (15) and (18).

after simplifications ððn þ 1Þ4 n4 ¼ 4n3 þ6n2 þ 4n þ 1Þ, 0r

q2e q4 ð2n þ 1Þ 2 e 2 ð4n3 þ 6n2 þ 4n þ 1ÞV ðlÞ g qe r DV l : 2C 8C Kd

ð18Þ

Incoherent transmission through the chain requires the validity of Eq. (18) in every site. Note that the vibrations could be easily added to Eq. (18) by putting the corresponding positive term s_ok =qe together with V g qe term. As a conclusion, vibrations enlarge the bias voltage responsible for electrical currents as it occurs for some particular models in a single-molecule junction [18]. As expected, for large bias voltage DV l , more channels contribute to inequality (18). For small bias window (DV  0 þ ) and 1 rigid components (d ¼ 0), the gate voltage to produce electrical current through the qudot becomes V ðlÞ g  ðqe =CÞðn þ 1=2Þ (one 1 channel [1–3]). Nevertheless, when d 4 0, it changes to   qe 1 q3  2 e 2 ð4n3 þ6n2 þ 4n þ 1Þ, n þ V ðlÞ  ð19Þ g 2 C 8C Kd which must be complemented with the condition of no plates transposition (Eq. (12)). It could be easily understood since charge distorts the plates changing the capacitance value (recall Eq. (2)). For the ‘‘uncharged’’ system (n¼0), Eq. (19) becomes V ðlÞ g 

qe q3e  , 2C 8C 2 Kd2

ð20Þ

showing that distortion changes the usual qe =2C charging voltage [3,8,9]. When vibrational degrees are considered (s a0), for instance this one of the center of mass of a generic [1,16] qudot like C60, expression (20) must be still corrected to incorporate the new degrees of freedom. Assuming an excited vibrational mode of frequency ok , there is transmission through the qudot (n ¼0, at

7. Conclusions and open questions We have considered the Hamiltonian (1) composed of a line of LC quantum circuits modeling a set of qudots interacting with phonons. Geometrical distortion changes the usual properties of a qudot related to vibrations (Sections 3 and 4) and electrical current through the dot (Section 6). Particularly, Coulomb charging energy, usually being qe V g  q2e =2C, becomes deeply modified by relocation and vibrations (see Eq. (21)). The vibrational term is positive definite showing that the charging energy enlarges with it. A similar result operates for the bias voltage (Section 6) in accord with other models [18] with linear interaction. To end, through a quantum electrical capacitive approach, we have studied two separate kind of physical processes associated to qudots dynamics: (a) phonons dynamics and (b) electrical current through a qudot. Both the processes (Eqs. (22) and (23)) involve the same quantities which are related to energy. It suggests a deep correlation between the vibrational degrees and electrical current through the qudot which agrees with the work by Koch and von Oppen [18].

Acknowledgments MCN acknowledges financial support from Convenio de Desem˜o UTA-MINEDUC. pen References [1] M. Di Ventra, et al., (Eds.), Introduction to Nanoscale Sciences and Technology, Springer, 2004.

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