- Email: [email protected]
Phonon emission from coupled quantum dots
Physica B 272 (1999) 42}44
Phonon emission from coupled quantum dots Tobias Brandes*, Bernhard Kramer University of Hamburg, 1. Inst. Theor. Physik, Jungiusstr. 9, D-20355 Hamburg, Germany
Abstract We present calculations for the non-linear current through double quantum dots in the presence of a coupling to phonons. The tunnel process between the dots is dominated by an orthogonality catastrophe of the phonon bath which strongly renormalizes the Lorentz-peak of the current. We show that absorption and emission current data can be scaled to the Bose distribution n and n#1 over a wide energy interval in agreement with a recent experiment. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Phonons in double dots; Spontaneous emission; Non-linear transport in quantum dots
1. Introduction
2. Results
The spontaneous emission of phonons from double quantum dots has been observed recently in mK non-linear transport experiments [1]. Even close to zero temperature, the tunnel current I(e) as a function of the di!erence e"e !e between the L R local chemical potentials of the left and the right dot showed a broad shoulder for e'0, whereas absorption of phonons (e(0) considerably changed the current only at higher temperatures. We performed calculations which show that the coupling to bulk piezoelectric acoustic phonons can lead to these e!ects. In addition, the interference between electron}phonon matrix elements in the two dots leads to oscillations of I(e) on a scale u " : c /d, where c is the speed of sound and d the d 4 4 distance between the centers of the two dots.
Our model starts from two coupled dots [2] &left' L and &right' R, where we de"ne operators n " L DLTSLD, n "DRTSRD, p"DLTSRD, s "D0TSLD, R L s "D0TSRD, and the total Hamiltonian system R H as the sum of the dot, the phonon, the reservoir and the electron}phonon interaction
* Corresponding author. E-mail address: [email protected] Brandes)
(T.
H
" H@ #H #H #H , 0 T V ab
H@ 0
" e n #e n #H #H , L L R R p 3%4
" ¹ (p#ps), H "+ u as a , # p Q Q Q (1) Q H " + (a n #b n )(a #as ), ab Q L Q R ~Q Q Q H " + (< cs s #= dss #c.c.), k k R V k k L k H " + eLcsc #+ eRdsd . k k k k k k 3%4 k k The tunneling between the left and right dots is described by a single tunnel matrix element ¹ . In # H T
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 4 8 - 3
T. Brandes, B. Kramer / Physica B 272 (1999) 42}44
the standard tunnel Hamiltonian H , < and V k = couple the dot to a continuum of channels k of k the left and right electron reservoir H . The spin of 3%4 the electron does not play any role here and is suppressed. The term H describes the lattice vibrap tions in harmonic approximation; the creation operator for a phonon of mode Q is as . The elecQ tron}phonon matrix elements are de"ned by a " : j SLDeiQrDLT and b " : j SRDeiQrDRT, where Q Q Q Q j is the matrix element for the interaction of Q 2DEG electrons and phonons. The phonons are assumed to be piezoelectric acoustical bulk (threedimensional) modes with an interaction matrix element Dj D2"j2/
G
C
A
C CH C B " : R (1#C C ) ~e # e 1# L e R e 2C 2C C L R R
(2)
P
= 0
dt e~(z~ie)tSX Xs T t 0 0
Fig. 2. Real part of C as a function of the energy di!erence e e between the left and right dot ground state energies. Energies +u "10 leV, +d"1 leV, cuto! +u "1 meV. $ #
and the correlation function SX Xs T "e~U(t~t{) t 0 0 with U(t) " : :=du o(u) 0
G
,
where C and C are the tunnel rates to the right R L and the left reservoir, C "C (dP0) with e e C (z) " : e
Fig. 1. Stationary tunnel current as a function of the chemical potential di!erence e between left and right dot. Dimensionless electron}phonon coupling parameter: g"0.05. Inset: e!ective density of states o(u) of phonon modes. C , C : electron tunnelR L ing rates to right/left leads, ¹ interdot tunneling parameter. #
] (1!cos ut)coth
BDH
(3)
43
H
+u #i sin ut , 2k ¹ B
(4)
Da !b D2 Q d(u!u ). o(u)" + Q Q u2 Q The real part RC , Fig. 2, is proportional to the e probability density for inelastic tunneling from the left to the right dot with energy transfer e [4]. In the limit u "0, one "nds RC "(2p/C(g))eg~1e~eh(e) d e
44
T. Brandes, B. Kramer / Physica B 272 (1999) 42}44
Fig. 3. Absorption (lower branch) and emission rates (upper branch), Eq. (5), from the data of Fig. 1.
at ¹"0, where only spontaneous phonon emission is possible. This shows that the e!ect is nonperturbative in the electron}phonon coupling g.
Here, ¹ is the reference temperature which we 0 chose as ¹ "10 mK because with our parameters 0 the currents practically do not change any longer for lower temperatures. Also note that we use the full current I(e, ¹ ) and not I (e) as the reference 0 %function on the absorption site e(0. Fig. 3 shows that the data can well be scaled to the Bose distribution function n(x)"1/(ex!1), i.e. N(e, ¹)" n(DeD/k ¹) for absorption e(0 and to N`(e, ¹)" B 1#n(e/k ¹) for emission e'0 over an energy B window 220 leV'DeD'20 leV. For very low temperatures and larger DeD, there are deviations on the absorption side which we will discuss elsewhere. Here, we point out that the analysis in terms of Einstein coe$cients works remarkably well, as was the case in the experiment [1].
Acknowledgements This work has been supported by the DFG project Kr 627/9-1, and Br 1528/4-1.
3. Scaling with temperature and energy Fujisawa et al. [1] used the Einstein relations between emission and absorption rates to demonstrate scaling of their data as a function of the ratio between temperature and energy k ¹/DeD. We B use our theoretical results to perform a scaling analysis by de"ning the spontaneous emission rate (we set the electron charge e"1) A(e'0)" I(e'0, ¹ )!I (e'0). Here, I (e) is the elastic 0 %%part of the current, i.e. the current for vanishing electron}phonon coupling g"0. One furthermore de"nes the relative emission N and absorption N`, N(e, ¹) " : [I(e, ¹)!I (e)]/A(e), e'0, %N`(e, ¹) " : [I(e, ¹)!I(e, ¹ )]/A(DeD), e(0. 0
(5)
References [1] [2] [3] [4]
T. Fujisawa et al., Science 282 (1998) 932. T.H. Stoof, Yu.V. Nazarov, Phys. Rev. B 53 (1996) 1050. S.M. Girvin et al., Phys. Rev. Lett. 64 (1990) 3183. G.-L. Ingold, Y.V. Nazarov, in: H. Grabert, M.H. Devoret (Eds.), Single Charge Tunneling, Vol. 294, NATO ASI Series B, Plenum Press, New York, 1991, p. 21. [5] R.H. Dicke, Phys. Rev. 93 (1954) 99. [6] R.G. DeVoe, R.G. Brewer, Phys. Rev. Lett. 76 (1996) 2049. [7] M. Benedict et al., Super-Radiance, Institute of Physics, Bristol, 1996.
Phonon emission from coupled quantum dots Tobias Brandes*, Bernhard Kramer University of Hamburg, 1. Inst. Theor. Physik, Jungiusstr. 9, D-20355 Hamburg, Germany
Abstract We present calculations for the non-linear current through double quantum dots in the presence of a coupling to phonons. The tunnel process between the dots is dominated by an orthogonality catastrophe of the phonon bath which strongly renormalizes the Lorentz-peak of the current. We show that absorption and emission current data can be scaled to the Bose distribution n and n#1 over a wide energy interval in agreement with a recent experiment. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Phonons in double dots; Spontaneous emission; Non-linear transport in quantum dots
1. Introduction
2. Results
The spontaneous emission of phonons from double quantum dots has been observed recently in mK non-linear transport experiments [1]. Even close to zero temperature, the tunnel current I(e) as a function of the di!erence e"e !e between the L R local chemical potentials of the left and the right dot showed a broad shoulder for e'0, whereas absorption of phonons (e(0) considerably changed the current only at higher temperatures. We performed calculations which show that the coupling to bulk piezoelectric acoustic phonons can lead to these e!ects. In addition, the interference between electron}phonon matrix elements in the two dots leads to oscillations of I(e) on a scale u " : c /d, where c is the speed of sound and d the d 4 4 distance between the centers of the two dots.
Our model starts from two coupled dots [2] &left' L and &right' R, where we de"ne operators n " L DLTSLD, n "DRTSRD, p"DLTSRD, s "D0TSLD, R L s "D0TSRD, and the total Hamiltonian system R H as the sum of the dot, the phonon, the reservoir and the electron}phonon interaction
* Corresponding author. E-mail address: [email protected] Brandes)
(T.
H
" H@ #H #H #H , 0 T V ab
H@ 0
" e n #e n #H #H , L L R R p 3%4
" ¹ (p#ps), H "+ u as a , # p Q Q Q (1) Q H " + (a n #b n )(a #as ), ab Q L Q R ~Q Q Q H " + (< cs s #= dss #c.c.), k k R V k k L k H " + eLcsc #+ eRdsd . k k k k k k 3%4 k k The tunneling between the left and right dots is described by a single tunnel matrix element ¹ . In # H T
0921-4526/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 2 4 8 - 3
T. Brandes, B. Kramer / Physica B 272 (1999) 42}44
the standard tunnel Hamiltonian H , < and V k = couple the dot to a continuum of channels k of k the left and right electron reservoir H . The spin of 3%4 the electron does not play any role here and is suppressed. The term H describes the lattice vibrap tions in harmonic approximation; the creation operator for a phonon of mode Q is as . The elecQ tron}phonon matrix elements are de"ned by a " : j SLDeiQrDLT and b " : j SRDeiQrDRT, where Q Q Q Q j is the matrix element for the interaction of Q 2DEG electrons and phonons. The phonons are assumed to be piezoelectric acoustical bulk (threedimensional) modes with an interaction matrix element Dj D2"j2/
G
C
A
C CH C B " : R (1#C C ) ~e # e 1# L e R e 2C 2C C L R R
(2)
P
= 0
dt e~(z~ie)tSX Xs T t 0 0
Fig. 2. Real part of C as a function of the energy di!erence e e between the left and right dot ground state energies. Energies +u "10 leV, +d"1 leV, cuto! +u "1 meV. $ #
and the correlation function SX Xs T "e~U(t~t{) t 0 0 with U(t) " : :=du o(u) 0
G
,
where C and C are the tunnel rates to the right R L and the left reservoir, C "C (dP0) with e e C (z) " : e
Fig. 1. Stationary tunnel current as a function of the chemical potential di!erence e between left and right dot. Dimensionless electron}phonon coupling parameter: g"0.05. Inset: e!ective density of states o(u) of phonon modes. C , C : electron tunnelR L ing rates to right/left leads, ¹ interdot tunneling parameter. #
] (1!cos ut)coth
BDH
(3)
43
H
+u #i sin ut , 2k ¹ B
(4)
Da !b D2 Q d(u!u ). o(u)" + Q Q u2 Q The real part RC , Fig. 2, is proportional to the e probability density for inelastic tunneling from the left to the right dot with energy transfer e [4]. In the limit u "0, one "nds RC "(2p/C(g))eg~1e~eh(e) d e
44
T. Brandes, B. Kramer / Physica B 272 (1999) 42}44
Fig. 3. Absorption (lower branch) and emission rates (upper branch), Eq. (5), from the data of Fig. 1.
at ¹"0, where only spontaneous phonon emission is possible. This shows that the e!ect is nonperturbative in the electron}phonon coupling g.
Here, ¹ is the reference temperature which we 0 chose as ¹ "10 mK because with our parameters 0 the currents practically do not change any longer for lower temperatures. Also note that we use the full current I(e, ¹ ) and not I (e) as the reference 0 %function on the absorption site e(0. Fig. 3 shows that the data can well be scaled to the Bose distribution function n(x)"1/(ex!1), i.e. N(e, ¹)" n(DeD/k ¹) for absorption e(0 and to N`(e, ¹)" B 1#n(e/k ¹) for emission e'0 over an energy B window 220 leV'DeD'20 leV. For very low temperatures and larger DeD, there are deviations on the absorption side which we will discuss elsewhere. Here, we point out that the analysis in terms of Einstein coe$cients works remarkably well, as was the case in the experiment [1].
Acknowledgements This work has been supported by the DFG project Kr 627/9-1, and Br 1528/4-1.
3. Scaling with temperature and energy Fujisawa et al. [1] used the Einstein relations between emission and absorption rates to demonstrate scaling of their data as a function of the ratio between temperature and energy k ¹/DeD. We B use our theoretical results to perform a scaling analysis by de"ning the spontaneous emission rate (we set the electron charge e"1) A(e'0)" I(e'0, ¹ )!I (e'0). Here, I (e) is the elastic 0 %%part of the current, i.e. the current for vanishing electron}phonon coupling g"0. One furthermore de"nes the relative emission N and absorption N`, N(e, ¹) " : [I(e, ¹)!I (e)]/A(e), e'0, %N`(e, ¹) " : [I(e, ¹)!I(e, ¹ )]/A(DeD), e(0. 0
(5)
References [1] [2] [3] [4]
T. Fujisawa et al., Science 282 (1998) 932. T.H. Stoof, Yu.V. Nazarov, Phys. Rev. B 53 (1996) 1050. S.M. Girvin et al., Phys. Rev. Lett. 64 (1990) 3183. G.-L. Ingold, Y.V. Nazarov, in: H. Grabert, M.H. Devoret (Eds.), Single Charge Tunneling, Vol. 294, NATO ASI Series B, Plenum Press, New York, 1991, p. 21. [5] R.H. Dicke, Phys. Rev. 93 (1954) 99. [6] R.G. DeVoe, R.G. Brewer, Phys. Rev. Lett. 76 (1996) 2049. [7] M. Benedict et al., Super-Radiance, Institute of Physics, Bristol, 1996.