Local Field Eects

5.4 Local Field Eects 89

90 5 DIRTY QUANTUM WIRE where the current amplitudes ~Jn(~r) depend on position, on the driving electric eld, on the driving frequency, on the interaction strength, on temperature and so on, but not on time.

Following eqs. (83) and (84) in section 3.3, the local eld is related to the longi-tudinal and transverse parts of the current via

~EL(~q;!) = ~EL;ext(~q;!) + 1i0!~JL(~q;!); (208)

~ET(~q;!) = ~ET;ext(~q;!) + i0! g0(~q;!)~JT(~q;!): (209) Fourier transforming the current in eq. (207) to ~q-space, the transverse and longi-tudinal parts of the current can be obtained using eqs. (77) and (78). Performing also the Fourier transform to frequency one obtains

~EL(~q;!) = 2 ~EL;ext(~q)(!^;!ext) + 1i0! ^{2 X1}

n=^;1 ~JL;n(~q)(!^;n!ext);xxx(210)

~ET(~q;!) = 2 ~ET;ext(~q)(!^;!ext) (211) + i0! g0(~q;!)2 ^X1

n=^;1 ~JT;n(~q)(!^;n!ext):

Hence, a nite local eld is obtained for any ! =n!ext, n integer, if ~JL=T;n is nite for that value of n. The local longitudinal and transverse elds are not monochro-matic but contain all harmonics. Further, due the non-linearity of the system, a local transverse eld at frequency n!ext induces currents at all possible harmonics, m(n!ext),m integer. These currents again induce elds at frequencies ofm(n!ext).

As a consequence, a local eld at frequency k!ext is inuenced by any local eld with frequency n!ext ifk=nis an integer number. Finally, due to mixing, local elds with dierent frequencies n!ext and k!ext combine to drive currents at frequencies (nk)!ext and the corresponding higher harmonics. Thus, if e.g. n^;k= 1, they also inuence the fundamental mode.

Based on these considerations, it is obvious that the equations for the local elds in eqs. (210) and (211) do not decouple, to yield one separate equation for any frequencyn!ext, because the local elds at dierent frequencies inuence each other.

This cross-inuence leads to a set of innitely many coupled equations for the local elds which is in general impossible to solve.

One approach that simplies the task is the so-called parametric approximation.

This approximation neglects the coupling that allows a local eld at frequencyn!ext

to inuence another eld at a smaller frequencym!ext,m < n, i.e. it neglects mixing [2]. The parametric approach is justied when mixing is an inecient process such that the cross-inuence between the elds is small. But still the local eld equations are coupled as via harmonic generation a eld at frequency k!ext is inuenced by any eld at n!ext with k=n integer. This coupling might be neglected when also harmonic generation is inecient and the current amplitudes decrease rapidly with increasing harmonic number. Then, the impact of the fundamental mode, which contains also the external eld, on a eld at frequencyn!ext is much larger than the impact of a local eld at k!ext, n=k integer, on a eld at frequency n!ext.

5.4 Local Field Eects 91

5.4.1 Interaction vs. Local Longitudinal Field

The following paragraphs are dedicated to a comparison between the microscopic consideration of electron-electron interactions and the mean-eld like approach dis-cussed in sections 2.3.1, 3.3, and 4.4.1.

Choose a longitudinal external driving eld parallel to the wire and constant as a function of ~R, and neglect the induced transverse eld. For an innitely high tunneling barrier, ² = 0, the ac current in the wire is purely linear, see eq. (186),

J^x(x; ~R;!) =Jclean^x (x; ~R;!)^;r(x;!)Jclean^x (x= 0; ~R;!): (212) Here, J_xclean is the linear current in an innitely thin wire without a barrier, see eq. (132). The function r(x;!) is given in eq. (185). The induced longitudinal eld for an external longitudinal eld parallel to the wire and in the absence of a transverse eld is given in eq. (154) in section 4.4.1,

E_L;ind^x (~q;!) = q_2xVee^C(~q)

i!e² J^x(~q;!): (213) Inserting the expression for the current given in eq. (212) into eq. (213), multiplying both sides by B_x00(^;~Q)=(2)², integrating with respect to ~Q and dening the pro-jected eld as in eq. (140), one obtains for the propro-jected induced longitudinal eld in terms of the projected external eld [12]

E_L;ind;1b^x (qx;!) =

"

_1b;d=0^xx (qx;!)

_1b;non^{xx ;}_int;d=0(qx;!)^;1

#

E_L;ext;1b^x (qx;!) (214)

;

R dqx_1b;d=0^xx (qx;!)E_xL;ext;1b(qx;!) 2 _1b;d=0^xx (x= 0;!)

;

where _1b;d=0^xx is the conductivity of the clean single-channel quantum wire of zero diameter, see eq. (145), and ^xx_1b;non^;_int;d=0 is the conductivity of the non-interacting system which is obtained from eq. (145) by replacing !2LL(qx) with v2Fq2x. The rst term in the curly brackets in eq. (214) is equivalent to the one derived for the clean quantum wire in section 4.4.1, the second term represents the inuence of the barrier.

The same result for the projected induced longitudinal eld is obtained when driving the current with the projected local eld using the conductivity of the non-interacting system [12]. Hence, the two approaches, considering either electron-electron interactions or the induced longitudinal eld, are here equivalent. This result is in complete analogy to the one found in section 4.4.1 and might have been expected. For a barrier of nite height, however, the situation changes completely.

For a barrier of nite height, ^{2 6}= 0, the current in the wire is non-linear if and only if the current is evaluated based on a microscopic model that considers inter-acting electrons. As a consequence, one observes higher harmonic generation in ac transport as discussed in detail in section 5.3.2 and in particular one obtains induced elds at all harmonic frequencies. The mean-eld like approach that neglects inter-actions and considers self-consistently the induced longitudinal eld yields a current

92 5 DIRTY QUANTUM WIRE at the fundamental frequency provided by the external eld as for non-interacting electrons, the dirty quantum wire exhibits a linear-current voltage characteristic and hence does not show harmonic generation. As Maxwell equations are linear, the induced eld also contains only the fundamental mode.

Hence, the two approaches, considering either the interaction or the longitudi-nal induced eld lead in the case of a dirty quantum wire with a barrier of nite height to qualitatively completely dierent results. Entering into the details of the evaluation of the non-linear current, see appendix B.1, it turns out that it is crucial for the appearance of a non-linear current-voltage characteristic that the interaction is represented by a term in the Hamiltonian that contains the product of two density operators, see eq. (14). From the discussion of the mean-eld like approach in section 2.3.1, it is known that the Hamiltonian taking into account the self-consistent induced eld, see eq. (4), contains the product of one density oper-ator and an expectation value of another density operoper-ator. This seemingly slight dierence leads here to two qualitatively completely dierent results: a linear vs. a non-linear current-voltage characteristic. Choosing a mean-eld like approximation, one therefore misses important features of the system.

5.4.2 Induced Transverse Field

In the following, the magnitude of the local transverse eld in the dirty single-channel quantum wire is estimated. Consider a current amplitude at frequency n!ext that is driven by the external eld alone. For n > 1, this current is a pure tunneling current proportional to the tunneling probability ². When the driving electric eld is not localized at the barrier, a displacement current that is independent of ² is also present, see the discussion in section 5.1, but it contributes only to the fundamental mode, n = 1. The induced eld at n!ext is thus also proportional to ² for n > 1. The correction to the tunneling current at another frequency k!ext

due to this induced eld at frequency n!ext is of correspondingly higher order in ². As the total current was evaluated in section 5.1 only in order ² it would be inconsistent to take into account corrections of higher order. Hence, harmonics induced by local elds of frequency n!ext,n >1, and mixing have to neglected. The induced transverse eld at n!ext, however, also drives the displacement current at the same frequency and this current is linear in the driving eld. The corresponding correction to the total current is exactly of order ²and has to be taken into account.

In appendix B.4 it is shown how to evaluate this induced transverse eld in lowest order in ². An innitely thin wire is assumed. The induced eldE_xT;ind(~r;!) consists of two terms, one diverges at^j~R^j= 0 while the other remains nite \within"

the wire. Now, as for an innitely thin wire the non-linear ac current J^x(~r;!) is proportional to (~R), one indeed expects the induced eld to reect this delta-function. As we are interested in a correction to the current caused by the induced transverse eld, the term that does not diverge at ^j~R^j= 0 is neglected with respect to the other term. Then, the current J^x(~r;!) and the induced eldE_xT;ind(~r;!) are averaged with respect to ~R. The resulting expression for the induced eld is for

5.5 Emitted Electromagnetic Fields 93