J Dubois

T Jullien

C Grenier

P Degiovanni

P Roulleau

D C Glattli

Integer and fractional charge Lorentzian voltage pulses analyzed in the framework of Photon-assisted Shot Noise

Keywords: numbers: 73.23.-b, 73.50.Td, 42.50.-p, 42.50.Ar

The periodic injection n of electrons in a quantum conductor using periodic voltage pulses applied on a contact is studied in the energy and time-domain using shot noise computation in order to make comparison with experiments. We particularly consider the case of periodic Lorentzian voltage pulses. When carrying integer charge, they are known to provide electronic states with a minimal number of excitations, while other type of pulses are all accompanied by an extra neutral cloud of electron and hole excitations. This paper focuses on the low frequency shot noise which arises when the pulse excitations are partitioned by a single scatterer in the framework of the Photo Assisted Shot Noise (PASN) theory. As a unique tool to count the number of excitations carried per pulse, shot noise reveals that pulses of arbitrary shape and arbitrary charge show a marked minimum when the charge is integer. Shot noise spectroscopy is also considered to perform energy-domain characterization of the charge pulses. In particular it reveals the striking asymmetrical spectrum of Lorentzian pulses. Finally, time-domain information is obtained from Hong Ou Mandel like noise correlations when two trains of pulses generated on opposite contacts collide on the scatterer. As a function of the time delay between pulse trains, the noise is shown to measure the electron wavepacket autocorrelation function for integer Lorentzian thanks to electron antibunching. In order to make contact with recent experiments all the calculations are made at zero and finite temperature.

This paper addresses the noiseless injection of a small finite number of electrons in a quantum conductor. Indeed, quantum effects become more and more accessible when only few degrees of freedom are controlled. During the last thirty years, research in this direction has lead to the possibility to manipulate quantum states with several degrees of freedom and to entangle particles making possible simple quantum information processing. Up to now most advances have been obtained in quantum optics with the manipulation of single photons emitted by atoms or semiconductor quantum dots, and in atomic physics with optical arrays of trapped cold atoms or ions More recently the manipulation of quantum states has become available in condensed matter systems using superconducting circuits and semiconductor quantum dots.

A recent approach is the manipulation of single charges injected in a quantum ballistic conductors. Realizations of time controlled single charge sources have been reported in [1][2][3][4][5] and considered theoretically in [6][7][8][9][10][11][12][13] with a particular focus on the energy resolved single electron source based on a quantum dot [1]. Injecting more than one electron requires a different practical approach which is the subject of this paper. We will consider the more general case of coherent trains of few undistinguishable electrons [14,15] which opens the way to entangle several quasi-particles but also to probe the full counting statistics [16] with a finite number of electrons [17]. The injection of single or of few electrons injected in a ballistic one dimensional conductor lets envisage to parallel the flying qubit approach developed for photons [18][19][20]. As opposed to Bose statistics, the Fermi statistics makes entanglement of several electrons injected in a quantum conductor more favorable and the Coulomb interaction, when not screened, the possibility to make non-linear interaction more easier than the Kerr effect used for photons. The ballistic conductors can be realized using high mobility 2D electrons confined in high quality III-V semiconductor heterojunctions. In addition, applying a perpendicular magnetic field gives rise to one dimensional chiral propagation along the edge of the samples in the Quantum Hall regime. This, combined with nano-lithographied split gates, allows to implement quantum gates similar to that used in quantum optics: electronic beam splitters, electronic Fabry-Perot and Mach-Zehnder interferometers [21,22].

But how to inject single to few electrons in a quantum conductor? Single electron pumps based on the Coulomb blockade of tunneling have been first realized using ordinary metallic systems [23,24]. They have potential application in quantum metrology but are not suitable for our purpose. First the conductor is usually not ballistic. Second the electrons are sequentially injected. The lack of quantum coherence between electron tunneling events makes the injection of trains of few undistinguishable electrons impossible. More recently itinerant quantum dots obtained using surface acoustic waves [25] have been shown able to transport single to two electrons along a depleted electron channel [4,5]. These systems may also have application in metrology. Because of the long coherence time of the spin, they could be used for spin based qubit operation and to carry quantum information between distant dots [4,5]. Finally, the on-demand injection of a single electron emitted from a single energy level suddenly risen at a definite energy above the Fermi sea of the leads has been realized using a quantum mesoscopic capacitor [26] in a non-linear regime [1]. The system can be viewed as the electronic analog of a frequency resolved single photon source and opens the way for new quantum experiments where single electrons are injected at tunable energy well above the Fermi energy. However this kind of on-demand emission is not generalizable to a coherent train of several undistinguishable electrons.

We consider here a powerful and simpler way to inject a coherent train of n electrons in a single quantum channel of a ballistic conductor. Here spin is disregarded for simplicity. The principle, proposed by Levitov et al. in [14,15] uses voltage pulses V (t) applied to one contact. For a quantized action e ∞ -∞ V (t)dt = nh, n integer and h the Planck constant, exactly n charges are injected. As the current I(t) at the injecting contact is at all time equal to I(t) = e 2 h V (t) this is equivalent to write ∞ -∞ I(t)dt = ne. This is the simplest injection of charge that we can imagine. On the experimental level, it does not require the implementation of a quantum dot nor the combination of tunnel barriers or of several Quantum Point Contacts. For quantum bit applications this will improve the reliability as the delicate circuitry simplifies and reduces only to that necessary for quantum logic gate implementation. On the conceptual level, this apparently naive approach involves a non trivial Physics related to the deep properties of the Fermi sea. Indeed, injecting n charges in general does not mean injecting n electrons alone but also involves a collective excitation of the whole Fermi sea: a neutral cloud of electrons and holes. However the beautiful theoretical observation of ref. [14] is that a voltage pulse with Lorentzian shape carrying one electron is a minimal excitation state free of these neutral electron-hole excitations. More generally a combination of n Lorentzian having arbitrary shape and position in time but all carrying a unit charge of same sign is a minimal excitation state. The remarkable result of ref. [14,15] has triggered several relevant theoretical contributions [9,17,[27][28][29][30] in which the property and potential use of the Lorentzian voltage pulses are discussed. An experimental implementation has been recently done [31].

In the following, we explore some properties of the coherent injection of several indistinguishable electrons by periodic voltage pulses applied on an Ohmic contact and we consider their characterization in energy and time domain using shot noise. We compare different pulse shapes and also the departure from integer charge values and we include finite temperature effects with the aim to provide direct comparison to experiments [31]. The goal being to produce injection with minimal excitation, we will only consider the injection of charges with same sign as alternate sign charge injection was found in general not suitable (see ref. [14]). Considering periodic pulses allows to use the powerful Floquet scattering theory approach of Ref. [32]. While Lorentzian shape pulses with n charges per period correspond to exactly n excitations [14,15], pulses of arbitrary shape contain more excitations [28] whose number is calculated for sine, square and rectangular wave shape. We also address the case of non-integer charge pulses which was shown in ref. [14,33] to be the dynamical analog of the Anderson orthogonality catastrophe problem [34]. Here the periodicity introduces of cut-off in the long time log-divergent increase of electron and hole excitations at long time. The Anderson physics reflects in a marked minimum of neutral excitations for integer charge per period as theoretically observed in Ref. [28].

In order to quantify the creation of extra electron and hole excitations, we need a physical quantity counting them. The right quantity is the low frequency shot noise of the electrical current already considered for dc transport [35][36][37][38][39][40] where electrons injected by a dc voltage biased contact are partitioned by a scatterer playing the role of an electronic beam splitter. In a quantum optics language, this can be viewed as a Hanbury-Brown Twiss experiment (HBT) with electrons. As remarked in the pioneering works or refs. [14,15,33], shot noise is also expected when the electron pulse trains pass through a beam splitter. Indeed all excitations are partitioned irrespective to their charge, and the shot noise counts the sum of electron and hole excitations while the current counts their difference. The current noise is then proportional to the number n of injected electrons, plus the number N e and N h of extra electron and hole excitations created per period, with N e = N h by definition. Using the separation of the voltage pulse into its mean value, or d.c. part < V (t) >= V dc and its ac component V ac (t) = V (t) -V dc as in ref. [28], the physics of voltage pulses can be conveniently discussed in the framework of Photo-Assisted Shot Noise (PASN) where both theoretical [41][42][43] and experimental results [45,48] are already available. At zero temperature, the excess noise ∆S I characterizing the extra excitations is then the difference between the PASN noise S P ASN I due to V (t) = V ac + V dc and the (transport) shot noise (TSN) S T SN I that we would have with only the dc voltage V (t) = V dc [35-40, 49, 50]. As ∆S I ∝ (N e + N h ) and the current is I = eν(N e -N h +n) = n, ν being the frequency, minimizing the noise at constant current implies N h = N e = 0. In terms of photo-assisted effect we will see below that this implies only absorption or only emission of photon, depending on the sign of the n charges injected.

The paper is organized as follows. In section I we introduce the basic Physics of the photo-assisted effects in a quantum conductor with a contact driven by a periodic voltage source in the Floquet scattering approach. We then consider the PASN and the competition between PASN and TSN when both a dc bias and an ac voltage are applied. We recall the remarkable result of a singularity appearing in the derivative of the noise when eV dc = nhν, i.e. for exactly n electrons in average injected per period T = 1/ν making direct connection with the problem of periodic voltage pulses carrying integer charges. We also give the expression of the photo-assisted current.

In section II we address the comparison between several types of integer charge pulses : the square, the sine, the rectangular and the Lorentzian. Using the PASN results of section I, we calculate the number of electronhole pairs excitations via the excess noise. For comparison with experiments the computation is done for both zero and finite temperature. The hierarchy of the charge pulses regarding noise production is compared with the hierarchy of the pulses based on photo-current production.

Section III addresses the case of non-integer charge pulses. We show that for all type of pulses, the number of electron-hole pairs rises for non-integer charges but is always minimal for integer charges (and even zero for integer Lorentzian pulses). For large non integer charge number, the number of neutral excitations quickly decreases for Lorentzian while it remains finite or increases for other type of pulses.

Section IV provides an energy domain characterization of the charge pulses using shot noise spectroscopy. We compute the PASN as a function of an arbitrary dc voltage bias which provides a direct measure of the photoabsorption probabilities [28,41,45,48,54]. In particular the Lorentzian pulses show an asymmetric PASN versus dc bias characteristic of the absence of hole type excitations while sine and square waves lead to symmetric PASN.

Finally, section V gives a time domain characterization of the pulses by looking at the shot noise generated by trains of electrons colliding on the scatterer. In analogy with Hong Ou Mandel (HOM) experiments [60], a time delay between the pulse trains emitted by opposite contacts controls the overlap of electronic wavefunctions which reflects in noise suppression. Here, the gauge invariance maps the HOM experimental scheme discussed here to a simpler Hanbury Brown Twiss (HBT) problem with only one driven contact. For colliding single

x 0 ) ( ˆε L a b QUANTUM CONDUCTOR energy ε ) (t V F E ) ( / ) ( t i t i x k i e e e φ ε ε - - h ε ν ε h + ν ε h - ν ε h 2 + ν ε h 2 - ( LEFT CONTACT ) ) (t V ) ( ˆ0 ε L a
FIG. 1: Under the effect of an ac voltage, electrons emitted far inside the contact acquire a time dependent phase on their way to the scattering region (the quantum conductor). For periodic voltage, frequency ν, the incoming electrons can be described by a quantum superposition of states at different energies ε + lhν.

charge Lorentzian pulses the noise is directly linked to the wavepacket overlap of the injected charges while for other pulse shapes the cloud of neutral excitations also contributes to HOM noise.

I Floquet scattering description of periodic voltage pulses applied on a contact

We consider a quantum conductor with one contact, say the left (L), periodically driven by a voltage V ac (t) of frequency ν = 1/T , see Fig. 1 . Here, without loss of generality we choose the approach of refs. [42,51] where the periodic potential is assumed to be screened in all other regions of the quantum conductor. The voltage drop is assumed sharp on the electron wavepacket length but smooth on the Fermi wavelength. The quantum conductor has a small width compared to that of the leads and a small length L compared with the electron phase coherence length l ϕ such that electrons can propagate coherently over a length l ϕ ≫ L on the leads. The region where electron loose coherence is called the reservoir or contact, following the standard description of the scattering theory of quantum transport. We assume that the electron coherence time l ϕ /v F is large compared with the period T . We consider an electron emitted by the left reservoir at energy ε in a state ∼ exp(ik(ε)x) exp(-iεt/ ) with occupation probability

f L (ε) = 1/(exp(ε/k B T e )+1)
where T e is the electronic temperature and the Fermi energy is the zero energy reference. From the left reservoir to the left entrance of the conductor, the electron experiences the potential V ac (t) and acquires an extra term exp(-iφ(t)) in its amplitude probability with time dependent phase:

φ(t) = 2π e h t -∞ V ac (t ′ )dt ′ (1)
The Fourier transform of:

exp(-iφ(t)) = +∞ l=-∞ p l exp(-i2πlνt) (2)
gives the probability amplitude p l for an electron to absorb (l > 0) or emit (l < 0) l photons. Eq. ( 2) expresses that an electron emitted at energy ε enters the conductor in a superposition of quantum states at different energies ε + lhν. The knowledge of the p l completely defines the states of the incoming electrons. The magnitude of the p l depends on the reduced quantity α = eV ac /hν where V ac is the characteristic amplitude of the ac voltage. Combined with the scattering properties of the conductor, all photo-assisted effects resulting from the absorption or emission of energy quanta hν such has the ac current, the photo-current and the photo-assisted shot noise, etc, can be calculated. The properties of the p l are best expressed in the frame of the Floquet scattering theory [32] in which the continuous energy variable describing states in the reservoir is sliced into energy windows of width hν, i.e. ε → ε + l ′ hν with ε now restricted to [-hν, 0]. We can view the photoassisted processes as the coherent scattering of electrons between the different energy windows. The scattering matrix S(ε) = {S ll ′ } relates the set of annihilation operators a 0 L (ε) = { a 0 L (ε + l ′ hν)} operating on the states of the left contact to the set of annihilation operators a L (ε) = { a L (ε + lhν)} of electrons incoming on the conductor :

a L (ε) = S(ε) × a 0 L (ε) (3)
with {S l ′ l } = p l-l ′ . The unitarity relations S † S = SS † = I gives useful relations for the amplitude probabilities :

+∞ l=-∞ p * l p l+k = δ k,0 (4)
In particular, the sum of the probabilities P l = |p l | 2 to absorb or emit photons or to do nothing is equal to unity. As shown below and used in Section IV, the probabilities P l can be inferred from Shot Noise spectroscopy, when in addition to the ac voltage a tunable dc voltage is applied between the contacts of the conductor. Note that the set of P l does not contain all the information on the system. Indeed, the products p * l p l+k , k = 0, i.e. the non-diagonal part of the matrix density, enters in the calculation of the coherence [54] as discussed in [53].

Photo-assisted shot noise :

In the following we calculate the photo-assisted shot noise which occurs when the conductor elastically scatters the electrons. For simplicity we will consider a single mode (or one-dimensional) quantum conductor with transmission probability D as shown in Fig. 2.

energy ε ) (t I ∆ F E ε ν ε h + ν ε h - ν ε h 2 + ν ε h 2 - ) ( t V ac F E ν h ν h ) ( ~ε f ) (D FIG. 2:
Electrons emitted by the left reservoir and pumped by the ac voltage in a superposition of states of different energy are scattered. The random partitioning between reflected and transmitted scattering states leads to current noise. The noise measures the sum of the number of holes and electrons which are photo-created. On the right is represented the energy distribution function f (ε). It should not be confused with an incoherent distribution function as the non-diagonal terms of the density matrix are non-zero.

Noise occurs only when an electron incoming from the left finds no incoming electron from the right or if a hole incoming from the left finds an incoming electron from the right. Indeed, because of the Pauli principle, or Fermionic anti-bunching, the case where two electrons or two holes are simultaneously incoming gives no noise. In the former case, the incoming charges are randomly partitioned by the conductor with binomial statistics. At zero temperature and for zero ac voltage the noise is zero. In presence of an ac voltage electron and hole excitations are created in the left lead. The photon-created electrons span energies above the right lead Fermi sea and the photon-created holes below the right lead Fermi sea. All the excitations contribute to partition noise, which is called the Photon-Assisted Shot Noise (PASN). According to ref. [41,42] the low frequency current noise spectral density due to photo-assisted process S P ASN I is, including the Fermi distributions f L,R (ε) of the left and right reservoirs :

S P ASN I = S 0 I dε hν +∞ l=-∞ P l {f L (ε -lhν)(1 -f R (ε)) +(1 -f L (ε -lhν))f R (ε)} (5)
where S 0 I = 2 e 2 h D(1 -D)hν is the typical scale of the PASN. The complete noise expression is obtained by adding the thermal noise of the reservoirs: 4k B T e D 2 e 2 h . Its origin is the thermal fluctuation of the population in the reservoir and is not related to partitioning nor to photo-assisted processes. In absence of ac voltage, P l = δ l,0 , the full noise reduces to thermal noise 4k B T e D e 2 h and vanishes at zero temperature, as discussed above.

In order to best extract the Physics, we first consider the zero temperature limit :

S P ASN I = S 0 I +∞ l=-∞ |l|P l (6)
The sum in the right hand side is directly proportional to the number of electrons and holes created respectively above and below the Fermi energy (chosen as the zero of energy). To understand this, let us consider for simplicity that only P 0 and P ±1 are important and first concentrate on the absorption process. Electrons below the Fermi surface at energy ε -khν, k positive integer, can be promoted to energy ε -(k -1)hν with probability P 1 . This global upward shift of the Fermi sea leaves the electron population unchanged below E F but fills the empty sates of energies ε ∈ [0, hν] with occupation probability P 1 , above E F . Similarly, in the emission process, electrons are displaced to energies ε -(k + 1)hν with probability P -1 . The downward shift of the Fermi sea gives no net change of the population of states with energy < -hν, while for the energy range [-hν, 0] the population is now 1 -P -1 . More generally the l-photon processes give electron excitations above the Fermi sea with population P l in the energy range [0, lhν], l > 0 and holes excitations below the Fermi sea in the energy range [-lhν, 0], see Fig. 2. As the current of electrons emitted by the left contact and able to create a l-photon electron excitations is (e/h)lhν = leν [48], the number of corresponding electrons created per period is lP l . The total number of electron excitations generated per period is thus:

N e = +∞ l=1 lP l (7)
and similarly the number of holes :

N h = -1 l=-∞ (-l)P l (8)
and, from Eq.( 6), the PASN is :

S P ASN I = S 0 I (N e + N h ) (9)
To conclude this part we must emphasize that the energy distribution function f L (ε) = +∞ l=-∞ P l f L (ε -lhν) depicted in Fig. 2 should not be confused with an incoherent non-equilibrium population. If this were the case, even for a perfect lead (D = 1) one would expect a current noise associated with the population fluctuation ∝ dε f L (ε)(1 -f L (ε)) as for thermal noise. In the present case the terms p * l p l+k contributing to the nondiagonal part of the density matrix should not be forgotten. For example they are important for the multiparticle correlations considered by Moskalets et al. [52] or for the concept of electron coherence defined in analogy with quantum optics by Grenier et al. [53,54]. These interference terms contribute to make the low frequency Photo Assisted Shot Noise strikingly vanishing at unit transmission. This was experimentally shown by Reydellet et al. [48]. In this work, the theory of quantum partition noise of photon-created electron-hole pairs, equations (6)(7)(8)(9), was experimentally checked from weak to large ac excitation and by varying the transmission. Motivated by this experiment, Rychkov et al [43] have theoretically shown that the electron and hole excitations contributing to noise in Eq.( 9) are not statistically independent. Eq.( 9) was derived by Lee et al. [44], Levitov et al. [14] and later by Keeling et al. [15]. In the different context of periodic injection of energy resolved single electron and single hole from a quantum dot, a similar equation has been derived and experimentally tested by Bocquillon et al. [3] for the partitioning of single charges. Finally, as discussed below it is important to note that Eq.( 9) measures the number of electron and hole excitations only accurately at zero temperature.

Photo-assisted and Transport Shot noise :

We consider now a dc voltage bias V dc > 0 added to the periodic ac voltage V ac (t) on the left contact while the right lead Fermi energy is kept to zero. Let q = eV dc /hν be the number of electrons emitted per period due to the dc bias. The total number of left electrons participating to noise is q + +∞ l=1 lP l = q + N e . The number of holes generating partition noise is however reduced by the positive shift of the left Fermi energy. For (n -1)hν < q < nhν the number of holes participating to noise is reduced to -n l=-∞ (-l -q)P l < N h . The shot noise expression ( 9) is then changed by replacing the |lhν| terms by |lhν + eV dc | as originally derived by Lesovik et al. and later by Pedersen et al. [41,42]. In absence of ac voltage one recovers the transport shot noise

S T SN I = 2 e 2 h D(1 -D)|eV dc |.
The mixed situation with both V ac and V dc leads to interesting effects due to the competition between PASN and TSN . They are best displayed using the excess noise where the pure TSN is subtracted from the total noise:

∆S I = 2 e 2 h D(1 -D)[ +∞ l=-∞ P l |lhν + eV dc | -|eV dc |] (10)
This would correspond to an experimental situation where the noise measured with V ac on is subtracted from the dc transport shot noise measured with V ac off while keeping the dc voltage on [28,31]. Finally, we can write reduced units for the excess shot noise ∆S I = S 0 I ∆N eh where :

∆N eh = +∞ l=-∞ |l + q|P l -|q| (11)
represents at zero temperature the total number ∆N eh of photon created electrons and holes not contributing to the TSN. The derivative of the excess shot noise ∆S I (or of ∆N eh ) with respect to q shows remarkable singularities each time q = eV dc /hν is an integer n. At the singularity, the change of slope of the variation of ∆S I with positive (negative) V dc is proportional to 2P -n ( 2P n ), for n = 0 and (2P 0 -1) for n = 0. Varying V dc thus provides direct information on the P l . Their direct measure is provided by the second derivative of the noise [45][46][47] :

∂ 2 ∆N e-h /∂q 2 = (2P 0 -1)δ q,0 + l =0 2P l δ q,-l (12)
Applying a monochromatic sine wave to a contact, the noise singularities at dc voltage multiple of the frequency has been observed in a diffusive metallic wire by Schoelkopf et al. [45] via the second derivative of the noise, giving the P l spectroscopy. Later the controlled suppression by a dc voltage bias of hole or electron excitation contribution to PASN has been discussed and observed in a Quantum Point Contact by Reydellet et al. [48]. The extracted P l quantitatively agrees with the expected Bessel function. Inferring the energy distribution of photoexcited electrons from shot noise spectrocopy has been discussed in [46] and was compared to tunnel spectroscopy by Shytov [47]. In the different context of the energy resolved single electron and hole source using a mesoscopic capacitor, a similar shot noise spectroscopy was proposed by Moskalets and Büttiker [9]. In this same context, the concept of shot noise spectroscopy and equation (12) has been extended to propose a full tomography of the electron and hole quantum states by Grenier et al. [54]. The Shot Noise spectroscopy is used in Section IV to analyze the excitation content of periodic charge voltage pulses. Finally, directly relevant to the topics of charge injection, the singularity at eV dc = nhν (q = n) corresponds exactly to the condition required for injecting n electrons per period. This is why the presentation of periodic charge injection using voltage pulses is particularly relevant and enlightening in the frame of PASN. This singularity is also a useful tool to characterize the carrier charge in interacting systems. The superconductingnormal junction where conjugated electron-hole Andre'ev pairs carry twice the electron charge have been studied by Torres et al [56]. The finite frequency PASN noise of charge e and charge e/3 partitioned by a QPC was studied in respectively the integer and the fractional quantum Hall regime at 1/3 Landau level filling factor by Crépieux et al [57] and later by Chevallier et al. [58].

For comparison with realistic experimental situation, we provide the excess noise formula at finite electron temperature T e . The excess noise is ∆S I (V ac , V dc , T e) = S 0 I ∆N eh (α, q, θ e ) where : FIG. 3: Excess electron and hole particle for sine and square wave pulses carrying q = n integer charges per period at zero temperature. The asymptotic log divergence of the square for large number, as defined in the main text, is shown as a dashed line with thermal excitations. This reduces the shot noise which therefore misses these excitations [55].

∆N eh (α, q, θ e ) = +∞ l=-∞ (l+q) coth( l + q 2θ e )P l (α)-q coth( q 2θ e ) (13
Photo-assisted Current : Finally, we consider an other photon-assisted effect, the photo-current, which also depends on the probabilities P l . We consider a weakly energy dependent transmission probability D(ε) ≃ D + ε∂D/∂ε and neglect the energy dependence of the transmission amplitude phase for simplicity. This situation is known to give a rectification effect characterized by a quadratic term in the low frequency I-V characteristic of the conductor. In terms of photo-assisted effect this leads to a dc photo-current whose expression is :

I ph = e 2 h (hν) 2 ∂D/∂ε +∞ l=-∞ l 2 P l (14)
One can see that I ph gives information on the P l and can discriminate between different types of ac signal. However as shown in the next sections it is not as useful as shot noise as it can not distinguish between electron and hole excitations and in particular can not identifies the minimal excitation Lorentzian pulses. Indeed, 2 and I ph gives the same information than a classical time averaging of V (t) 2 . This quantity is also proportional to the production rate of heat in the contacts

+∞ l=-∞ l 2 P l = 1 T T 0 |dϕ(t)/dt| 2 = (e/ ) 2 V ac (t)
I(t)V (t) ∝ V (t) 2 .

II Integer periodic charge injection

In this part, we give the expression of the probabilities P l associated with various type of pulses carrying q = n charges per period: the sine, the square and the Lorentzian. From the shot noise we establish the hierarchy of the pulses in terms of number of e-h excitations ∆N eh . We also calculate the photo-current and conclude that shot noise is the right quantity able to character-photocurrent (a.u.) ize the cleanliness of the charge pulses. We consider a periodic excitation carrying a charge q = ne per period

T = 1/ν. This occurs if 1/T t+T t V (t ′ )dt ′ = nhν.
It is convenient to decompose the voltage into its mean value V dc = nhν and its ac part V ac [28,31].

For the sine wave V (t) = V ac (1 -cos(2πνt)), we have V dc = V ac = nhν/e. The emission and absorption probabilities are given by integer Bessel functions with P l = J l (n) 2 where P l is calculated putting in the time dependent phase only the V ac term and not the dc voltage part. Then, we can directly use the PASN shot noise formula with dc bias (11) to calculate the number of excitations (n > 0). for the square wave V (t) = 2V dc = 2hν/e for t ∈ [0, T /2] and V (t) = 0 for t ∈ [T /2, T ] (mod T ). The probabilities are given by

P l = 4 π 2 n 2
(l 2 -n 2 ) 2 for odd l -n, P l = 0 for even l -n, and P n (n)=1/4. Both the sine and the square wave have symmetric variation around zero voltage which implies symmetric electron and hole excitation creation. With P l = P -l , equation (11) now writes :

∆N eh = 2 +∞ l=n+1 (l -n)P l (15)
The above expression is useful to provide the asymptotic expression of ∆N eh at large n for square wave pulses:

∆N eh (n) sq ≃ 1 π 2 (ln(n) + γ + 2 ln(2) -1)
where γ is the Euler constant. For the sine wave case, no log divergence occurs as P l ≃ 1 2πl (en/2l) l for l ≫ n. Figure 3 gives ∆N eh versus the number n of injected electrons for the square and the sine wave. Clearly the square wave pulse creates more excitations than the sine wave.

We now consider the Lorentzian pulse whose behaviour is quite different. Strikingly we will see below that the P l associated to the ac part of the voltage pulse are zero for l < -n. The expression for the periodic sum of Lorentzian pulses where each pulse carry n electrons and the Full Width at Half Maximum (FWHM) is 2W writes:

V (t) = V ac π +∞ k=-∞ 1 1 + (t -kT ) 2 /W 2 (16)
where eV ac = nhν. The total phase Φ(t), including the dc voltage part and using reduced time units u = t/T and η = W/T , gives:

exp(-iΦ(t)) = sin(π(u + iη)) sin(π(u -iη)) n (17)
Remarkably, the above expression has only simple poles in the upper complex plane. This implies that its Fourier transform contains only positive frequencies. In physical terms, the total voltage only allows electrons to absorb photons, i.e. it induces a global upward shift of the Fermi sea with no hole creation. Going back to the decomposition into dc and ac part of the voltage:

eV (t) = n hν 2 sinh(2πη) sinh(πη) 2 + sin(πu) 2 (18)
with eV dc = nhν, the amplitude probabilities associated with the only ac part are the Fourier transform of exp(-i(Φ(t) -nνt)) is:

p l = 1 0 du sin(π(u + iη)) sin(π(u -iη)) n exp(i(l + n)u) (19)
and the probabilities P l = |p l | 2 . From the above consideration on the analyticity of Φ we see that P (l<-n) = 0. Note that this is consistent with vanishing negative frequency Fourier component for the total phase Φ as the p l differs from them by of shift of l into l -n.

For n = 1 we find P l = 0 for l < -1, P -1 = exp(-4πη), and P l = exp(-l4πη)(1-exp(-4πη)) 2 . The expression of the P l for integer charge number > 1 involves the same exponential factor exp(-l4πη) and a Laguerre polynomial. The complete expression is given in the next part as a special case of fractional charges q when q = n. Using ∞ l=-n lP l = 0 and equation (11) gives:

∆N eh = 0 ( 20
)
The excess shot noise vanishes and only the transport shot noise of n electrons remains. The Lorentzian pulses are therefore minimal excitation states as no excess electron and hole excitations are created. The n electronic excitations are however not concentrated on the energy window [E F , E F + eV dc ] but they occupy states above the Fermi energy with a weight exponentially decaying on the scale ∼ /2W . We also compare the photo-current for different types of pulses, see Fig. 4. For the square and the sine wave with n electrons the sum +∞ l=-∞ l 2 P l is 1 and 1/2 respectively. probability P 0.99 FIG. 5: absorption l > 0 and emission l < 0 probabilities corresponding respectively to electron and hole particle creation for periodic Lorentzian pulses of width W/T = 0.1 and various charge q per pulse. The asymmetric spectrum for q = 0.99 reflecting the lack of hole creation quickly leads to a symmetric spectrum for q ≪ 1. Lines connecting the discrete P l values are guide for the eyes. This is consistent with the hierarchy found using shot noise which showed that the square contains more energetic excitations. For the Lorentzian, the sum is however non zero and given by coth(2πη) -1. It strongly increases with the sharpness of the Lorentzian shape. We conclude that the photocurrent can not characterize the neutral excitation content of charge pulses.

III Periodic injection of arbitrary charges

In this part, we calculate ∆N eh for arbitrary charge q carried per period. As in the previous part, we consider the square, sine and Lorentzian pulse shapes and also include rectangular pulses. We show that ∆N eh oscillates with q and is locally minimal for integer q = n. For the sine wave P l = J l (q) 2 while for the Lorentzian case the calculation of the P l for non integer charges is less trivial. Physically one may expect that carrying non integer charges will involve more complex excitations. Mathematically, we immediately see that the term in the right hand side of equation ( 17) is no longer analytic in the lower plane when q, replacing n, is not integer. We thus expect a proliferation of hole excitations contrasting with the integer case. The calculation is done in the appendix.

Figure 5 shows the P l for periodic Lorentzian pulses of width W/T = 0.1. The absence of components for l ≤ 1 when q = 0.99 ≃ 1 strikingly contrasts with the case of q < 1. For small q the P l spectrum is almost symmetrical with l signaling nearly equal electron and hole pair excitation creation. Figure 6 shows the evolution of the excess particle number (or excess noise) versus q from 0 to 3 for square and sine wave pulses and for Lorentzian pulses of width η = W/T = 0.1. We observe that, even if the square and the sine do not provide minimal excitations states, they do provide a minimum of excitations for integer charges. This seems to be a remarkable prop- erty of the Fermi sea. A similar figure can be found in Ref. [28]. Figure 7, left, shows how the excess particle number evolves for Lorentzian pulses of different width. We see that for W/T ≥ 0.1 the particle number becomes exponentially weak. Indeed the potential becomes close to a constant voltage V (t) ≃ V dc . For small width and half integer charge the electron hole excitation number is large but quickly decreases with q which contrast with the almost constant value found for the sine wave and the increasing value for the square wave. For comparison, the right graph of figure 7 shows the excess particle content of discrete Dirac pulses (or rectangular pulses) of similar width W (the voltage pulses have amplitude h/eW for the time duration W and zero otherwise). Here again the excitation content is much larger. It increases with q which contrasts with the Lorentzian pulse behavior which definitely shows the lowest noise.

It is interesting to see how the temperature affects the excess noise. As mentioned previously ∆N eh no longer measures the number of excess electron and hole quasiparticles with good fidelity. For simplicity we will keep the same notation but call it now the effective excess particle number. For sine and square waves there is only one energy scale to compare with the temperature, i.e. hν. For the Lorentzian case there are two energy scales hν and /2W . Figure 8 shows the sine wave case. We observe that the minima occur to higher q values. The effect is even more pronounced for the case of Lorentzian voltage pulses shown in figure 9 and is probably related to the stronger asymmetry of ∆N eh with q around q = n. for η = W/T = 0.1. The oscillations of ∆N eh are quickly damped by the temperature and when k B T e > 0.2hν are almost unobservable.

To end this section, it is worth to note that the non integer charge considered here are injected in a noninteracting Fermionic system (more precisely a good Fermi liquid where interaction give rise to Landau quasiparticles). Extension to fractional charges in Luttinger liquids has been considered by [15]. The e/3 fractional charge of fractional quantum Hall edge states has been considered by Jonckheere et al. [59].

IV Energy domain: spectroscopy of the e-h pairs excitations

In this part we use the Shot Noise spectroscopy tool discussed in I to analyze the periodic charged states in the energy domain. To do this, we compute ∆N e-h as a function of V dc . Here the dc bias q = eV dc /hν is no longer linked to the ac amplitude parameter α = eV ac /hν. As explained in I, this allows to make a spectroscopy of the electron and hole excitations and infer the P l from the excess noise variation with V dc or from the first or second noise derivative with respect to V dc . The calculations and graphs are done and displayed at zero and finite temperature.

Figures 10 and11 show the variations of ∆N e-h versus q at zero temperature for respectively a sine wave and a Lorentzian wave. For each case we show the curves for two different amplitudes V ac corresponding to α = 1 and 2. These amplitudes would correspond respectively to single and double charge voltage pulses when q = α = 1 or 2 respectively. For sine and square waves ∆N eh (α, q) is symmetric with q ( or V dc ) as P l = P -l . However it is asymmetric for the case of Lorentzian pulses. Such asymmetry is expected for pulses whose ac part of the voltage is not symmetrical with respect to zero voltage. But more relevant and striking, for the Lorentzian case the excess noise is zero for q > α (or eV dc > nhν), a direct consequence of zero P l for l < -n.

The effect of finite temperature is shown on Figure 12 for a sine wave of amplitude α = 1. Finite temperature calculations are also shown in figure 13 and figure 14 for Lorentzian pulses of width W/T = 0.1 with respectively α = 1 and 2. The relevant temperature needed to reveal the asymmetry of ∆N eh with dc bias for a Lorentzian is given by the width, the smaller the width, the higher the energy /W at which we find the contributions of the positive P l responsible for the long tail at negative voltages. The other temperature scale is given by the ratio k B T /hν which controls the smoothing of the singularities at integer q. The time periodicity imposes that only information on a discrete energy spectrum is available and full characterization must be completed with a dual time domain information within a period. Here again, shot noise is the useful tool to provide this information. Inspired by the optical Hong Ou Mandel (HOM) correlation experiment, an electronic HOM analog can be built as a useful tool to infer the time shape of wave-packets where electrons emitted from two contacts with a relative time delay collide on the scatterer. A clear relation between shot noise and wavepacket overlap can be made for a single charge Lorentzian pulse. For other pulse shapes, it is expected that the contribution of the neutral excitation cloud gives an extra contribution to the shot noise.

V Time domain: shot noise characterization using collision of periodic charge pulses

In a optical experiment, single photons are emitted from two distinct sources in each of the two input channels of a semi-transparent beam-splitter. Photon detectors are placed on the two respective outputs and a time-delay between them, sizable with the photon wavepacket, is introduced. For zero delay, Bose statistics implies a constructive two particle interference where the two photons bunch and exit, at random, in one of the two output channels. The coincidence events are zero and the particle fluctuations (the noise) is doubled with respect to a Hanbury Brown-Twiss (HBT) experiment where only one photon at a time arrives on the beamsplitter. The latter situation is recovered when the delay τ is much longer than the size of the wave-packets. For intermediate time delays the noise variation is directly related to the overlap of photon wavepackets [60,61]. A similar experiment could be done with electrons with an artificial scatterer in the form of a controllable beamsplitter, i.e. a Quantum Point Contact. For zero time delay, Fermi statistics leads to a destructive interference for the probability of finding two electrons in the same output channel. In terms of charges counted by the detector (here the contacts of a conductor) there is always a charge arriving in each contact and consequently no cur-rent fluctuations. The noise increases from zero at τ = 0 to the single particle noise value (similar to the photon HBT case) for time delay larger than the electron wavepacket. The difference between Fermion and Boson for HOM correlations has been discussed by Loudon [62]. An electronic HOM experiment has been proposed by Giovannetti et al. [63]. HOM shot noise correlation of electron-hole pairs using two phase shifted ac voltages of weak amplitude (eV ac < hν) has been theoretically considered by Rychkov et al. [43]. Up to now, the theoretical works have only addressed the case not discussed here of two single charges emitted from two ac driven quantum dot capacitors. In this case, the HOM noise has been calculated by Ol'khovskaya et al [7]. In the adiabatic regime the wavepackets mimic those emitted by a contact driven by Lorentzian voltage pulses carrying a single electron followed by a single hole (note that, except for well separated electron and holes, the alternate charge prevents to have clean excitations [14]). More recently, considering the same system Jonckheere et al. [13] have looked to similar HOM correlations and including the effect of electron and hole collisions. Their analysis used the quantum electron optics frame developed by Grenier et al. [53,54]. Other recent theoretical works discussed various related situations [8][9][10]64]. In particular ref. [9] considered a mixed situation where electrons emitted form a quantum dot capacitor collide with electrons from a voltage pulse source. Here we consider only voltage pulse electron sources. This regime was also addressed by in C. Grenier's thesis [54].

Practically, two generic situations, different in their geometry provide similar information: the zero magnetic field case for which the reflected path physically coincides with the input path (this corresponds in optics to the mirror facing the two sources and detectors); the high magnetic field case, using the edge states of the Quantum Hall regime where chiral propagation allows to geometrically separate input and output channels. The first case, provides simpler interpretation as the interaction are well screened (for example the electrons reservoirs formed by a 2DEG is a good Fermi liquid). The second case better mimics the optical geometry but the Coulomb interaction between co-propagating quantum Hall edge channels leads to fractionalization of the injected pulses, see [53].

In the following, the HOM shot noise of colliding trains of electron pulses is used as a tool to investigate the time shape of electron wavepackets. For simplicity we will consider, as above, a two terminal geometry in zero magnetic field. The result can be directly applied to the edge state geometry. Indeed, the noise of a voltage pulse electron source vanishes for unit transmission whatever the pulse shape is. The autocorrelation and cross-correlation are thus pure partition noise and have same amplitude. We consider the periodic injection of charges from both contacts with voltage V (t) and V (t + τ ) applied respectively on the left and right contacts. The scatterer has transmission D. In order to compute the Shot Noise, we use the above photon-assisted Floquet scattering description, with the left and right input operators defined immediately at the left and right entrance of the electronic beam-splitter (or scatterer) :

a L(R) (ε) = S(ε) L(R) × a 0 L(R) (ε) (21)
with

S L(R) l ′ l = 1 T T 0 dt exp(iφ L(R) (t)) exp(i2π(l -l ′ )νt) (22) and φ L (t) = 2π e h t -∞ V (t ′ )dt ′ and φ R (t) = 2π e h t+τ -∞ V (t ′ )dt ′ .
Then the shot noise can be calculated as in section II now using both the left and right amplitude probabilities.

Interestingly, in the present case of energy independent scattering, the gauge invariance property valid at all time, makes the electronic HOM problem formally equivalent to the calculation of the shot noise with a single contact, say the left, biased with the voltage difference V (t)-V (t+ τ ) while keeping the right contact at zero. It is then obvious to have a zero HOM shot noise for τ = 0, the case of maximal wavepacket overlap or maximal antibunching. However, the fact that the gauge invariance allows to map the problem to a simpler problem does not prevent the HOM Physics to be the underlying process. Indeed, we will see below that the HOM noise is directly related to the overlap of the electronic wavefunctions for periodic trains of wavepackets carrying a single electron. A HOM realization not reducible to voltage drop differences via a gauge transformation would for example require the use of two single electron sources based on quantum dots [1].

Figure 15 gives the calculated shot noise in reduced unit S 0 I versus the time delay τ /T for the collision of single charge pulses per period emitted by the left and right reservoirs. The left figure corresponds to Lorentzian pulses of different widths and the right figure to discrete Dirac pulses of similar corresponding widths (V (t) = h/eW if 0 ≤ t ≤ W , V (t) = 0 for W < t < T ). For both cases the noise starts from zero at τ = 0 (perfect antibunching) and is maximum at τ = T /2 where the wavepacket overlap is minimal. For the Lorentzian case and W/T ≪ 1, the wavepackets at τ = T /2 are well separated and the noise is doubled as two electrons (one from the left, one from the right) contribute independently to single particle noise at each period. We note however that the noise for rectangular pulses exceed the value 2. Indeed, rectangular pulses (which identify to square waves for W/T = 0.5) involve a large amount of neutral excitations which are measured in the HOM noise.

How far can the HOM noise correlation be used to infer the single charge wavepacket shape in the time domain? Ideally, one expects the noise to be given by 2(1 -C(τ )) where :

C(τ ) = | Ψ(t -τ -x/v F )|Ψ(t -x/v F ) | 2 (23)
and Ψ denotes the wavefunction of the excess electron injected right above the Fermi sea by the pulse. For Lorentzian pulses, which are clean excitations, a clear positive answer can be given. The shot noise of single electron colliding Lorentzian pulse is:

S Coll I (τ )/S 0 I = 8β 2 sin(πντ ) 2 1 -2β 2 cos(2πντ ) + β 4 ( 24
)
The noise is zero for τ = 0 and for the case of infinite width W (β = 0). The maximum noise is S 0

I 2 cosh(2πW/T ) 2 for τ = T /2.
For W/T → 0 (or β → 0), it goes to twice S 0 I as expected for the partition noise of well separate single charge pulses alternatively emitted by the left and right contacts. In this limit, we can use the known expression for the wavefunction generated by a single Lorentzian voltage pulse : Ψ(t) ∝ 1/(t + iW ) [15]. This yields for the square of the wavefunction overlap C(τ ) = 1 1+(τ /2W ) 2 and this agrees with the limiting expression of Eq.( 24):

S Coll I (τ )/S 0 I ≃ 2(1 - 1 1 + (τ /2W ) 2 ) (25)
In the appendix we show that Eq.( 24) is also proportional to 1 -C(τ ) even for the case of arbitrary overlap between pulses carrying single electrons. In the small W/T limit, close expressions have been obtained by Ol'khovskaya et al. [7] and by Jonckheere et al. [13] (see also C. Grenier Thesis [54]). In these works, single charges whose charge sign periodically alternates are emitted from an ac driven quantum dot capacitor and the trains of charges collide in a QPC. In the adiabatic ac drive limit, the wavepackets mimic those emitted by Lorentzian voltage pulses on a contact. Another striking property of the shot noise of colliding single charge Lorentzian pulses is the trivial effect of the temperature: The time delay variation and the temperature variation decouple: thermal fluctuations do not prevent to get accurate information on the time shape of the wavepacket but they only reduce the noise signal. As shown in the appendix this is not true for sine wave pulses and in general for other pulse shapes.

S Coll I (τ, T e ) = S Coll I (τ, 0)F (T e ) F (T e ) = 1 -β 2 β 2 l=∞ l=1 lβ 2l coth( lhν 2k B T e ) (26)
All theses remarkable properties demonstrate that, for integer charge Lorentzian, the HOM Shot Noise does give information on the time domain shape of the charge wavepacket. This property is no longer verified for the case of non Lorentzian pulses as the neutral excitations contributes. The left graph of Figure 16 shows the value of the HOM noise of Dirac pulses time shifted by T /2 normalized to the partition noise of single pulses, denoted S HBT I . We see that when the overlap between pulses becomes negligible, the ratio tends to twice the single pulse partition noise. This doubling is expected as now two separate pulses are partitioned per period. For large overlap, the ratio found in Fig. (16) is well smaller than two because of antibunching. An inherent large overlap is also found for a sinewave, right graph of Fig. (16).

Finally we have calculated the HOM noise of Lorentzian pulses for larger charge, including fractional. For non integer charge pulses we do not expect to get simple information on charge pulses in the time-domain because of the large content of neutral excitations expected even for the Lorentzian case. The left and right graphs of Figure 17 show the shot noise for colliding sine wave and Lorentzian pulses for q = 0.5, 1, 1.5 and 2. For both pulse shapes the variation with τ for q = 2 suggests a double structure related to the interference of several electrons. A detailed study of colliding pulses for large q will be given elsewhere [65].

To conclude this part, we have shown that the HOM electronic shot noise correlation provides meaningful time-domain information on the wavepacket. This is true however only for single charge Lorentzian which is a clean excitation. Other kind of electron generation is accompanied by a cloud of neutral excitations which contribute to the HOM shot noise and prevent to get time domain information on the charge part of the excitations.

Conclusion

We have studied in detail the physics of periodic voltage pulses carrying few electrons with the aim to give a useful basis for comparison with experiments [31]. Shot noise, as emphasized in the pioneering papers on Lorentzian voltage pulses, is a tool of choice to characterize the charge pulses. We have exploited the periodicity to show the intimate connection with the well established physics of Photo Assisted Shot Noise using the powerful Floquet scattering approach. We have used the known properties of PASN as a direct measure of the electron and hole excitation number and, when combined with transport shot noise, as a tool for spectroscopy of the excitations. In the later case this provides an energy domain characterization of the pulses accessible to experiments. This information has been supplemented by a time domain study of the charge wavepackets using HOM-like pulse collisions. The gauge invariance was used to map the charge pulse collision problem to the partitioning of neutral pulses. The comparison of different pulse shapes emphasizes the peculiar nature of Lorentzian pulses with integer charge as ideal electron source. We have also considered pulses with arbitrary charge and show that they always contain a large number of neutral excitations, even for Lorentzian, a manifestation of dynamical orthogonality catastrophe for Fermions. For large arbitrary q, the low noise property of Lorentzian remains remarkable as the neutral excitation content rapidly decreases with q while it shows a logarithm increases for rectangular shape pulses of same width, or remains almost constant for sine wave pulses.

Integer charge Lorentzian pulses deliver a noiseless source of few electrons. One expects many applications in quantum physics. For example by repeating the injection of clean coherent packets with same electron number and measuring the charge arrived in contacts, the full counting statistics of few electrons partitioned in quantum conductors become accessible. One may also envisage to explore the Fermionic statistics of multiple electrons colliding of a beam splitter or interfering in Mach-Zehnder interferometers. Integer charge Lorentzian pulses are also the appropriate source for realizing electronic flying qubits and to entangle few electrons in ballistic conductors. Finally the powerful approach of Lorentzian pulses could find similar applications in fermionic cold atom analogs.

Support from the ERC Advanced Grant 228273 MeQuaNo is acknowledged by D.C.G., J.D., T.J. and P.R.. The support from ANR-2010-BLANC-0412 is acknowledged by C.G. and P.D..

Appendix

Lorentzian voltage pulses carrying arbitrary charge:

In this part we calculate the amplitude probabilities p l for Lorentzian pulses with arbitrary charges. We start with equation (19) with n replaced by q :

p l = 1 0 du sin(π(u + iη)) sin(π(u -iη)) q exp(i2π(l -q)u) (27)
Here, the physical decomposition of the voltage into dc and ac part is essential to avoid ill mathematical behavior. While the Fourier transform of exp(-iΦ(t)) leads to divergent logarithmic terms as Φ(T -) -Φ(0 + ) = 2πq is not a multiple of 2π, that of the phase associated to the ac part is well defined as φ(T -) -φ(0 + ) = 0 . Defining z = exp(i2πu) and β = exp(-2πη)

p l = exp(i2πq) i2π dz z z l 1 -βz 1 -βz q (28)
As β < 1 and the integration contour over z is on the unit circle, both the numerator and the denominator are convergent series of positive powers of βz and βz respectively. The integral reduces the double series expansion to a single power series of β by imposing that the power of z under the integrand is zero. After some algebra we get :

p l = qe iqπ β l ∞ k=0 (-1) k (q + l + k -1)! (k)!(q -k)!(l + k)! β 2k (29)
for l > 0, and

p l = (-1) l qe iqπ β |l| ∞ p=0 (-1) p (q + p -1)! (p)!(q -|l| -p)!(|l| + p)! β 2p (30
) for l < 0. Interestingly, while for non integer q the sum are infinite (but quickly convergent), for q=n the sums reduce to the expected polynomial expressions of order n -1 and the terms for l < -n vanish. Indeed when the terms (q -k)! and (q -|l| -p)! in the denominator of respectively the first and second above expressions are negative integer, they take infinite values which therefore truncate the infinite series.

HOM Shot Noise calculation:

Calculating the HOM noise of colliding electron pulses generated at opposite contacts when applying voltage V L = V (t -τ /2) and V R = V (t + τ /2) on left and right contact respectively, is equivalent to calculate the noise when applying V (t-τ /2)-V (t+ τ /2) on the left contact only. The resulting zero temperature shot noise is then:

S I /S 0 I = +∞ k=-∞ |k||Π k | 2 (31)
where

Π k = 1 T T 0
dte -iϕ(t-τ /2) e iϕ(t+τ /2) e i2πkνt

Π k = e -i2πνkτ /2 ( l p l p * l-k e i2πνlτ ) (33) where p l takes the same values as in the main text and is associated to the unshifted potential V (t). Because the effective voltage V (t -τ /2) -V (t + τ /2) has symmetric weight for positive and negative values and is symmetric in τ we have Π k = Π * -k . The shot noise computation then reduces to calculate:

S I /S 0 I = 2 ∞ k=1 |k||Π k | 2 (34)

HOM Noise of periodic single charge Lorentzian pulses:

To calculate the Π k we use the exact values : p (l<-1) = 0, p -1 = -β and p (l≥0) = (1 -β 2 )β l obtained for single charge q = 1 Lorentzian voltage pulses . Injecting these values in Eq.( 33) we find:

Π 0 = 1 -(2 -e i2πτ )β 2 1 -β 2 e i2πτ (35)
Π l≥1 = β l (1 -β 2 )e -i2πlτ /2 e i2πτ -1 1 -

β 2 e i2πτ (36)
This allows to calculate the HOM shot noise for single charge colliding periodic Lorentzian pulses:

S Coll I /S 0 I = (2 ∞ l=1 lβ 2l ) e i2πτ -1 1 -β 2 e i2πτ 2 (1 -β 2 ) 2 (37)
which gives Eq.( 24) of the main text.

To calculate, the temperature dependence one have to replace in Eq. (36) |l| by l coth(lhν/2k B T e ) -2k B T e /hν. Interestingly, as Π l≥1 = Π 1 β 2l , we see that the τ dependence is only included in Π 1 . The temperature dependence of the HOM shot noise decouples from the τ dependence. The temperature only reduces the noise but does not affect the shape.

HOM Noise of periodic sine wave pulses:

Rather than calculating the Π l using Eq. (33), it is better to remark that the difference of two sine wave voltages, time shifted by ±τ /2, is also a sine wave. We get Π l = J l (2q sin(πντ )) for arbitrary charge q per period. This gives

S
The temperature dependence of the HOM signal can be calculated by replacing |l| by l coth(lhν/2k B T e ) -2k B T e /hν. We see however that the series of Bessel functions in the above expressions does not allow a decoupling of the τ and T e dependence and the HOM shape is expected to change with temperature. Indeed, the change affects electron and holes pairs with lhν k B T e which interact with thermal excitations.

Wavepacket interpretation of the Floquet scattering approach:

Here we show that the Floquet approach finds a useful wavepacket interpretation which allows to calculate the single electron wavefunction in the time domain for single charge pulses. The wavefunction overlap can then be computed and be compared with the HOM shot noise. This approach is restricted to zero temperature.

We start with the Martin Landauer wavepackets [39] defined in the energy bandwidth hν. Without loss of generality the wavefunctions describing the states of the, say left, reservoir ∼ e -iε(t-x/vF )/ and labeled by the continuous energy variable ǫ can be transform into a set of orthogonal wavepackets ϕ n,l (t -x/v F ) defined by integration over the energy window ε ∈ [lhν, (l + 1)hν]. Defining u = (t -x/v F )/T :

ϕ n,l (t-x/v F ) = 1 √ 2π V F sin(π(u -n))
π(u -n) e -i2πν(l+1/2)(u-n) (40) Using the annihilation operators a 0 n,l acting on the Fock states of the unperturbed reservoir, the Fermion operator before the action of the ac potential is:

Ψ 0 (t -x/v F ) = l n ϕ n,l (t -x/v F ) a 0 n,l (41)
with a 0 † n ′ ,l ′ a 0 n,l = δ n,n ′ δ l,l ′ f l and f l = 1 for l < 0 and f l = 0 for l ≥ 0. After the electrons have experienced the potential V (t), the electron Fermion operator becomes:

Ψ(t -x/v F ) = k l n
p k ϕ n,l (t -x/v F ) a 0 n,l-k (42) We see that the ac potential does not mix wavepackets of different n but only wavepackets of different energies and same n (we can also see that l p l ϕ n,l (t -x/v F ) forms a new orthogonal basis of wavefunctions).

Hong Ou Mandel correlation for single charge Lorentzian periodic pulses : We remind that the HOM noise associated with independent Lorentzian voltage pulses carrying one electron is directly related to the overlap of the electron wavefunction in the beam splitter. Here we would like to see if a generalization is possible in the case of a periodic train of overlapping Lorentzian pulses still carrying only one electron. To do that, instead of the wave function overlap we consider the time average hermitian product of the electron Fermion operators with a relative time-shift τ . Using the reduced variable θ = τ /T it is given , before time averaging, by: (45) For the presently considered case of periodic Lorentzian voltage pulses carrying unit charge and using the probabilities defined with respect to the total voltage V (t) = V dc + V ac (t), i.e P 0 -1 = β 2 -1 and P l = (1 -β 2 ) 2 β 2(l-1) for l ≥ 1, we get:

C(τ ) = (1 -β 2 ) 2 1 -2β 2 cos(2πτ /T ) + β 4 (46)
It is then straightforward to show that 1 -C(τ ) yields the HOM shot noise given by Eq.( 24), i.e. .

S HOM I = 2S 0 I (1 -C(τ )) (47)
Thus, even for overlapping pulses (but carrying a single electron), the HOM noise can still be interpreted as a measure of the overlap of the electronic wavefunction.

  ) and θ e = k B T e /hν is the temperature in frequency units. At finite temperature ∆N eh (α, 0, θ e ) no longer represents a direct measure of N e + N h as the excitations created in the energy range k B T e around the Fermi energy interfere square wave (exact) square wave (asymptotic)

FIG. 4 :

 4 FIG.4: Photocurrent in arbitrary unit versus charge per period. We see that this photo-assisted effect can not probe the hierarchy of excitation content of different type of pulses. The dashed and doted curves are guide for the eyes.

1 FIG. 6 : 5 FIG. 7 :

 1657 FIG.6: Excess electron and hole particle for square, sine and Lorentzian pulses carrying q charges per period. For the Lorentzian the ratio W/T = 0.1

FIG. 8 :FIG. 9 :

 89 FIG.8: Effective excess electron and hole particle for sine pulses carrying q charges per period and for different values of the electron temperature Te .

2 FIG. 10 :

 210 FIG.10: Zero temperature excess partition noise versus dc voltage in reduced units for sine amplitudes α = 1 and α = 2 corresponding to single and double charge pulses when respectively q = 1, 2.

FIG. 11 : 30 FIG. 12 :FIG. 13 : 5 FIG. 14 :

 11301213514 FIG.11: Zero temperature excess partition noise versus dc voltage in reduced units for Lorentzian amplitudes α = 1 and α = 2 corresponding to single and double charge pulses when respectively q = 1, 2.

5 FIG. 15 :

 515 FIG. 15: Electron shot noise versus the time delay τ for Hong Ou Mandel like correlations for colliding Lorentzian (left) and rectangular (right) pulses The noise is normalized to S 0 I .

FIG. 16 : 5 FIG. 17 :

 16517 FIG. 16: left figure: HOM noise at τ = T /2 for a rectangular pulse normalized to twice the partition noise of a single pulse, versus pulse width; right figure: HOM noise of sine pulses versus τ /T normalized to twice the partition noise of single charge (dashed line, right axis) and to twice the single pulse partition noise, solid line left axis

∞

  l=1 lJ l (2q sin(πντ ))2 2 ∞ l=1 lJ l (q) 2


SHAHZAD Muhammad Kashif
email: muhammadkashif.shahzad@st.com

Michel Tollenaere
email: michel.tollenaere@grenoble-inp.fr

Stéphane Hubac
email: stephane.hubac@st.com

Ali Siadat
email: ali.siadat@ensam.fr

CIGI 2011 Extension des méthodes DFM pour l'industrialisation de produits microélectroniques

L'industrie des semi-conducteurs (SMI) est caractérisée par de rapides évolutions sur une courte période de temps. Pour maîtriser les temps de développement (Time to Market) et les montées en cadence (Time-to-Volume), le DFM (Design for Manufacturing) a été intégré dès 1980 dans les outils de conception en tant que stratégie d'atteinte des objectifs de rendement. DFM est ainsi devenu un standard industriel pour estimer le rendement et la fabricabilité dès la conception. Cet article propose une méthodologie DFM pour inclure les résultats des mesures géométriques en corrélation avec les résultats des tests électriques, capturant les capabilités process liées aux performances des produits. L'étude de cas sur la modélisation d'interconnexion est réalisée chez un industriel de haut rang et une méthodologie destinée à aligner rapidement les modèles sur la cible des spécifications est proposée. Le BPR (business process reengineering) et IDEF0 sont utilisés pour définir les workflows qui accompagne la méthodologie proposée ; un modèle de données est ainsi mis en oeuvre via un outil destiné aux ingénieurs R&D.

. INTRODUCTION

L'industrie des semi-conducteurs (SMI) est caractérisée par une fluctuation cyclique de la demande et la loi de Moore qui prédit le doublement du nombre de composants par unité de surface tous les 18/24 mois (TR-IBS, USA). En 2001/2002, l'industrie a été confrontée à une chute de 31% des volumes suivie par 64% de croissance entre 2003 et 2007, elle même suivie d'une chute de 22% en 2009. Malgré ces chutes cycliques, les investissements en R&D n'ont cessé d'augmenter et devraient atteindre 18% du total des revenus d'ici 2012. Les efforts en R&D sont axés sur le développement de nouvelles plateformes technologiques incluant des méthodes de fabricabilité dans le but de développer la compétitivité. Le DFM a été adopté dans les années 80 comme une stratégie d'amélioration du rendement, mais maintenant le DFM est devenu un outil clé pour la compétitivité.

Le DFM est défini comme la capacité d'évaluer le rendement et la fabricabilité d'un produit dès les premiers stades de la conception (Hermann, 2004), conduisant à une meilleure maîtrise des couts et donc de la compétitivité [START_REF] Raina | What is DFM & DFY and Why Should I Care?[END_REF]. Les concepts du DFM sont classés en [START_REF] Mehrabi | Trends and perspectives in flexible and reconfigurable manufacturing systems[END_REF] "DFM produit" (design de produits manufacturables pour des processus de production définis) et "DFM processus" (processus de développement minimisant les modifications et itérations avec prises en compte précoces des contraintes). Les premières recherches sur le DFM, basées sur des estimations approximatives, visaient à assurer la fabricabilité mais laissaient l'obtention du rendement aux contrôles en ligne de la production (APC/AEC). Cette politique a porté ses fruits jusqu'aux technologies 250mm, mais avec le 300mm, la complexité croissante de la topologie des circuits et la diminution de longueur d'onde lors de la lithographie ont entraîné de nombreux rebuts et des pertes de rendement [START_REF] Cliff | DFM -An Industry Paradigm Shift[END_REF]. "Du point de vue d'un concepteur, les choses deviennent plus difficiles parce que les capabilités-machine issues de la fabrication sont tellement serrées qu'ils ont du mal à trouver les bonnes pratiques de conception" [START_REF] Peters | DFM Worlds Collide, Then Cooperate[END_REF].

Le concept de DFM a émergé accompagné d'un foisonnement de termes tels que DFY, DFV et DFT etc, regroupés sous le label DFX visant à la maîtrise des coût, qualité, rendement et délai de mise sur le marché [START_REF] Anderson | Design for Manufacturability & Concurrent Engineering; How to Design for Low Cost, Design in High Quality, Design for Lean Manufacture, and Design Quickly for Fast Production[END_REF]. Pour (Kuo et al., 2001), les outils DFX visent à apporter aux concepteurs, l'information sur la prévisibilité des éléments de cycle de vie. Ces outils s'appliquent directement sur des modèles de CAO et fournissent des conseils sur l'analyse des coûts, l'amélioration de performances du produit (qualitative, quantitative et binaire). Pour [START_REF] Seino | Manufacturing technology management to accelerate "design for manufacturability[END_REF] les raisons qui peuvent expliquer la mise en oeuvre infructueuse du DFM sont le manque de sensibilisation sur l'importance de DFM par les concepteurs de produits et l'analyse erronée des variations lors de la production par les différents d'ingénieurs/managers. L'industrie des semi-conducteurs [START_REF] Mcgregor | The Common Platform Technology: A New Model for Semiconductor Manufacturing[END_REF] a commencé avec une très forte intégration économique verticale comme un dispositif intégré (IDM) qui conçoit, fabrique et commercialise les équipements et les produits microélectroniques ; mais au milieu des années 70, la fabrication d'équipements et la conception/production des wafers se sont séparés avec des finalités industrielles distinctes. A la fin des années 1980, l'idée d'une industrie sans fabrication (fabless) a émergé afin de concentrer les ressources capitalistiques sur l'innovation de nouveaux modèles plutôt que sur des équipements de fabrication à obsolescence rapide.

Pour rester compétitives, [START_REF] Morinaga | Development of a Platform for Collaborative Engineering data flow between design and manufacturing[END_REF], les SMI ont besoin de mettre les produits sur le marché rapidement avec un rendement de production maitrisé, alors que les durées de vie des produits sont de plus en plus courtes et qu'on observe un déclin rapide des prix de vente. L'évolution des SMI poussée par la compétitivité soutenue s'appuie sur les plates-formes technologiques, l'équipement et les matériaux. Le DFM joue un rôle important dans le développement technologique et l'amélioration continue de processus de fabrication. Les coûts de maîtrise d'une technologie sont en constante augmentation : pour la plateforme 32 nm, un cout de 7 milliards de dollars est escompté (Web 2008). Chaque nouvelle technologie devrait voir un doublement de densité de transistors accompagné d'une capacité de montée en cadence rapidement avec une diversité de produits (axés sur les règles DFM). INTEL company a autorisé l'évolution des règles de conception et l'étude a montré que la quasitotalité des règles de conception a été modifiée avant la production de série afin d'assurer le rendement le plus élevé; par conséquent, au cours du prototypage, les règles de conception (DFM) doivent pouvoir changer si nous voulons une rapide montée en cadence des volumes de production.

Sur la base des faits énoncés ci dessus, il est bien évident que le DFM a pris une place importante à l'articulation de la conception et de la production en tant qu'outil méthodologique clé. Pour partager les coûts de R&D, l'industrie microélectronique constitue des "jointventure" pour le développement des plateformes technologiques ; ce Commençons par la description de la plaquette (Fig. 1) : elle est fabriquée à partir de silicium et est divisée en lignes horizontales et verticales qui sont connues sous le nom de lignes de découpe. Ces lignes ont un double objectif au cours du processus de fabrication, d'abord elles contiennent des structures de test (PCS: structures de contrôle, PCM: moniteurs de contrôle de processus et PMB: boîte de surveillance de processus) et sont utilisées lors des mesures électriques à différentes étapes du flux de production. À la fin du processus de production, la plaquette est découpée le long des lignes de découpe pour séparer les puces. Ces puces sont emballées et expédiées chez le client. Le "notch" est une coupe dans la plaquette utilisée pour identifier la position de référence au cours du processus. Les formes rectangulaires résultant de l'intersection des lignes horizontales et verticales constituent ce qui peut être un circuit complet (SoC: System on Chip) ou un composant. Afin de mesurer les variations d'épaisseur, des structures de test peuvent être placées dans la plaque à différentes positions comme le montre la figure 1. Il faut noter que les circuits réels ne sont pas testés jusqu'à ce que le test final soit effectué sur le produit. Les tests en ligne sont représentatifs du produit et sont utilisés pour surveiller et mesurer des mesures électriques et géométriques au cours du processus de fabrication.

Fig.1 description des tranches de 300 mm de diamètre

Le processus de fabrication est divisé en étapes amont ou front-end (FE) et étapes aval ou back end (BE). Le Front End est constitué par la fabrication de transistors sur une surface de silicium et 1-9 couches de métal dans une plaquette de silicium pour l'interconnexion. Au cours de chaque étape, les dimensions géométriques sont mesurées et contrôlées à travers des cartes de contrôle pour assurer le meilleur rendement paramétrique. Le premier test électrique (PT) est effectué pour mesurer les performances électriques (R, C et RC). L'objectif principal de l'analyse de la variance des caractéristiques dans le Back End (BE) est d'analyser de localisation en localisation, de plaquette en plaquette et de lot en lot, la corrélation spatiale et de classer les variations significatives, systématiques ou aléatoires. Notons que les plaquettes R&D ne comprennent pas de structures de test : aussi des domaines géométriques sont ils identifiés HG (en haut à gauche), HD (en haut à droite), C (Centre), CD (centre droit), BD (en bas à droite) et BG (en bas à gauche). La direction de ces éléments est essentielle pour la détermination du rendement fonctionnel de la plaquette.

Pour conserver une surface de la puce aussi constante que possible, les dimensions des lignes et des portes diminuent d'un facteur 2 tous les 2 ans augmentant ainsi la densité de circuits sur une plaquette. Parce que les périphériques sont connectés par le biais des interconnexions, l'augmentation de la densité augmente la longueur des interconnexions.

Les interconnexions sont des pistes de métal déposé ; elles présentent donc une résistance électrique et sont des causes de retard du signal. De même les interconnexions métalliques sont séparées par un diélectrique ayant un potentiel de condensateurs, de sorte que ces grandeurs parasites (résistance/capacité) doivent être modélisées sous peine de risque d'échec du produit final. Les ingénieurs R&D sont confrontés au défi d'adapter les spécificités locales d'interconnexion avec celles connues au sein d'une alliance d'industriels (DFM à l'échelle d'une filière industrielle).

PLATEFORME TECHNOLOGIQUE 'FONCTIONS ET USAGES'

Le concept de plate-forme technologique dédiée à une taille de gravure (22, 32 nm par exemple), et l'utilisation d'une bibliothèque standardisée de cellules fonctionnelles constituent les clés de la compétitivité. Nous présentons, en formalisme UML, les "cas d'utilisation" destinés à la mise en place des concepts informationnels du DFM étendu présenté dans cet article. Cette analyse a servi de base pour le déploiement logiciel qui a permis de valider la pertinence de l'approche chez un grand industriel du semi-conducteur (Fig. 2).

Les équipes directement impliquées dans ce processus sont l'intégration des procédés, la R&D, la modélisation de dispositifs et d'interconnexion ; elles focalisent leurs efforts vers le développement/amélioration de la plateforme technologique (32nm, 22nm, 40nm) qui sert de base pour les nouveaux modèles.

Les modèles de fonctions (devices) et d'interconnexion définissent l'interaction entre la conception et la fabrication, mais ils sont élaborés en utilisant des simulations de CAO et sont validées sur la base des résultats de tests électriques (dits Parametric Tests ou PT). Les principales étapes dans le Nos efforts se sont concentrés sur la modélisation des interconnexions et ont identifié trois principales sous-fonctions en vue d'une extension de la méthode DFM ; la méthode nécessite les résultats de mesure électriques (metal) et géométriques (processus) sur les lignes de découpe. Chaque mesure doit être identifiée par sa localisation (x, y) afin de permettre les corrélations par localisation. Jusqu'à présent, les ingénieurs R&D ont porté les corrélations sur les résultats des tests électriques, car ils ne pouvaient pas établir une cartographie basée sur les localisations (x, y) entre les résultats des mesures électriques et géométriques. Au cours des mesures électriques, la position de l'encoche (fig. 1) est orientée pour aligner la structure de test dans le sens horizontal et vertical avec le testeur. Les coordonnées lors des mesures électriques entraînent les coordonnées sont "normalisées" avant d'être transférées à la base de données à des fins de corrélation PT-PT.

PROCESSUS DE MODELISATION DES INTERCONNECTIONS

Les modèles d'interconnexion sont des équations physiques complexes qui prennent en entrée les caractéristiques des lignes métalliques, comme la largeur, l'épaisseur, la nature du matériau, le sens de dépôt, la rugosité ... et le contexte pour calculer la résistance et la capacité résultantes. Le processus générique de modélisation des interconnexions est présenté cidessous (Fig. 3 Voici les principales observations formulées lors de l'analyse:

• l'extraction de données avec des filtres prédéfinis est une pratique courante, d'où des données antérieures à l'extraction qui sont déjà normalisées et aucune corrélation significative ne peut conduire à la source des variations : les résultats peuvent être trompeurs

• les ingénieurs travaillent avec des feuilles Excel de 8 Mo et plus, taille qui non seulement exigent des ressources supplémentaires, mais pour lesquelles il est très difficile de traiter, d'harmoniser, de prétraiter et de dégager des analyses précises.

• l'erreur humaine est hautement probable dans ces traitements, ce qui conduit à des interprétations erronées.

• plusieurs outils sont utilisés pour établir une analyse des flux acceptée comme il est défini dans les produits appelés «flux de référence de calcul", où les étapes de conception et de simulation ainsi que les outils sont décrits sous forme de flux "en or". Cette stratégie peut s'avérer très trompeuse dans le cas de grande quantité des données à analyser et interpréter.

• la dispersion sur les caractéristiques fonctionnelles des circuits ne sera pas réduite par le fait que de bonnes corrélations existent et que les données électriques s'avèrent se conformer à des distributions "normales".

CORRELATIONS INTER-LOCALISATIONS

Aucune amélioration importante ne peut être envisagée sans que des corrélations inter-localisations sur les mesures en ligne et les tests électriques sur la plaquette soient faites, parce que les résultats électriques sont directement influencés par les caractéristiques géométriques relevées sur la plaquette.

Observons cette question par un exemple (Fig. 4). Il a été admis que les résultats électriques dans la base de données sont affectés à des coordonnées x, y, qui sont normalisés par rapport à l'orientation de la plaquette pendant les essais. Le centre de la plaquette avec l'encoche en position «haut» est pris comme référence 0,0 et dans toutes les autres zones, les coordonnées sont calculées sur la base de ce point de référence. Un index de localisation variant de 1 à n est affecté à chaque localisation par déplacement de bas en haut et de gauche à droite, par exemple loc1 a x, y (-5, -1) et loc63 à (4,2). Dans la base de données en ligne l'index de localisation n'est pas présent mais la référence est toujours 0,0 au centre. Il n'est donc pas possible d'associer des coordonnées aux numéros d'index de localisation. vérification, l'utilisateur est invité à importer les données de feuilles Excel (résistance, capa et les mesures géométriques) dans la base multidimensionnelle, tout en ayant la possibilité d'appliquer les filtres LSL-USL (limite de spécifications) ou LVL-UVL (limite de validation). Après traitement, le résultat final (analyse de variance) est sauvegardé dans des fichiers avec d'éventuels graphiques au choix de utilisateur. Le résultat met en évidence la déviation et la corrélation avec les mesures géométriques : cette étape est particulièrement critique parce que les limites de chaque paramètre doivent être définies in fine. Si les limites n'englobent pas les valeurs cible, des itérations sont enclenchées. L'applicatif fournit une option à l'utilisateur de choisir les algorithmes de calcul programmé. Les diagrammes de Box-Whisker confirment que capa et résistance calculés avec le filtre LSL-USL sont biaisés; néanmoins il n'y a pas de variation dans le calcul lorsque les fonctions programmées sont utilisées. Une réponse possible peut être qu'on a calculé seulement des statistique descriptives alors qu'avec une statistique prédictive, l'impact serait plus important. Nous avons pu observer que les ingénieurs R & D utilisent plusieurs outils statistiques en conformité avec la stratégie utilisée pour le flux de conception de référence. La référence de calcul pour chaque technologie est définie par l'alliance et comprend, modèles, des outils, des séquences de faible utilisation de chaque outil dans les flux de conception et les critères d'acceptation. Les multiples combinaisons des outils statistiques pour l'analyse des données sont sources de confusions : chaque outil est une boîte noire et la mise en oeuvre interne de l'algorithme est cachée. Ce fait prend plus d'importance lorsque des filtres sans causes assignables sont appliqués et que des analyses sont effectuées sans veiller à ce que les paramètres sont distribués normalement. Les ingénieurs R&D sont la ressource clé d'une organisation et la compétitivité est directement issue de l'innovation conduite par ces ingénieurs ; il est certainement préférable de ne pas disperser leur capital productif en leur confiant l'extraction de données et les tâches d'analyse, mais de focaliser leurs compétences pour leur permettre de trouver les causes profondes afin de mieux comprendre les mécanismes d'obtention des meilleurs rendements.

Fig. 2

 2 Fig.2 Les "cas d'utilisation" de la plateforme technologique développement d'une plate-forme technologique commencent par la définition des valeurs cibles sur les principaux paramètres critiques. Une liste de paramètres physiques est élaborée en conformité avec le manuel des règles et bonnes pratiques de conception (dit Design Rules Manual DRM) ; les valeurs sont simulées et itérées jusqu'à la cohérence du modèle avec les spécifications du design. L'objectif est d'identifier la séquence des interconnections pour chaque couche de métal en utilisant les connaissances existantes qui comprennent le DRM précédent (manuel des règles de conception), l'expertise humaine et des données issues de la production. Pour chaque nouveau produit, chaque couche de métal doit être simulée dans le simulateur SPICE ainsi que' l'empilement des couches pour être enfin testé et comparé par rapport aux résultats d'essais (capacité, résistance). La dernière étape est la comparaison des résultats obtenus par prototypage dans la 'fab' avec ceux de simulation. Les modèles finaux sont alors traduits en 'kit de conception' intégré aux outils de CAO. Les équipes de développement se concentrent sur l'identification des variations systématiques en production afin d'enrichir le 'kit DFM' par les contraintes induites sur la conception.

Fig. 3

 3 Fig.3 Processus de modélisation des interconnectionsLa cible de résistance et de capacité est fournie par l'équipe de Process Integration (PI). La liste des valeurs cible est définie en utilisant le DRM, l'expertise humaine et un modèle type qui est simulé et validé par rapport aux limites des spécifications. Si des divergences apparaissent, la liste est itérée en étroite collaboration avec l'équipe d'intégration processus jusqu'à ce que les résultats de simulation soient dans les spécifications. Cette étape suit la définition du modèle de "bord de plaque" qui sont validées par l'objectif du modèle de simulation. La liste des paramètres de définition est la sortie du processus (P0) à partir de simulations CAD. Une fois le modèle approuvé, une plaquette d'essai est préparée pour vérification physique dans la ligne de production afin d'évaluer la production en terme de qualité et volume. Lors de l'évaluation de ces règles sur les prototypes physiques, des règles DFM ou contraintes supplémentaires pourraient être identifiés et ajoutés au manuel DRM. La dernière étape est la documentation où le DRM pour le modèle est figé et les modèles numériques sont envoyés pour être ajoutés à l'ensemble pour la conception des kits. Après cette étape, bien que les modèles puissent encore être améliorés avec la dispersion, les concepteurs ne permettent généralement pas le changement dans les règles du manuel de conception ou dans les modèles de bords de plaque.

Fig. 4

 4 Fig.4 Problème de corrélation inter-localisations 2.4 NORMALISATION DES CORRELATIONS INTER-LOCALISATIONS Afin de permettre d'améliorer les corrélations entre étapes successives du process, une classe Mask a été créée dans le modèle de données (Fig. 5) ; cette classe permet d'établir un lien pour la cartographie des coordonnées entre tests paramétriques PT et bases de données en ligne. Sur la plaquette de gauche de la Fig.5, nous disposons de données PT avec des coordonnées x, y normalisées tandis que sur la plaquette droite les données en ligne ne sont identifiées qu'à l'aide de l'indice de localisation. Un utilitaire défini comme une méthode de la classe Mask et Mask_Sequence permet de corréler les localisations des différentes mesures géométriques et électriques effectuées sur des machines différentes.

Fig. 5 Fig. 6

 56 Fig.5 Solution de corrélation inter-localisations 3. PROPOSITION D'EXTENSION DE LA METHODOLOGIE DFMLes principaux constats mis en évidence lors de l'analyse des processus de la section 2-2 montre l'intérêt d'une méthode basée sur un DFM étendu et s'appuyant sur un triangle Données-Méthode-Statistique (DMS), qui est décrit ci dessous en IDEF0. Tel que discuté dans les sections 2.1 et 2.2, le s valeurs moyennes sont calibrées sur la base des résultats de simulation validés par les résultats des tests électriques. La corrélation PT-PT avec la localisation sur la plaquette est possible ; les résultats des mesures électriques sont pris en charge par la normalisation des coordonnées x, y, mais la corrélation PT-ligne avec une localisation est très demandée par les ingénieurs R&D. En l'absence de coordonnées x, y pour les mesures en ligne, il n'était pas possible de la fournir. Cet article présente un modèle de données qui recueille la localisation des coordonnées x, y pour le masque associé avec le produit pour des mesures en ligne. L'outil d'analyse de variance Back End Off Line (BEOL) PT-Inline est conçu pour faciliter aux ingénieurs R&D l'analyse des variations PT, mais aussi la cartographie PT-ligne. Le modèle IDEF0 de la méthodologie proposée est présenté sur la (Fig.6):

Fig. 8

 8 Fig.8 Diagramme de Classes pour l'outil d'analyse de variance

Fig. 9

 9 Fig.9 Outil d'analyse de variance BEOL Le flux de travaux usuel pour l'outil développé est présenté sur la Fig.10. On commence par analyser les variances de résistance R, de capacité C et enfin du produit R*C suivi par le calcul des statistiques et l'élaboration du graphique. L'étape suivante est d'appliquer la normalisation aux coordonnées des localisations afin de faire la corrélation de localisation à localisation. L'option de réaliser une modélisation prédictive est implémentée dans l'outil KLA-ACE, un outil très connu dans l'industrie microélectronique pour les analyses statistiques avancées.

Fig. 10

 10 Fig.10 Workflow pour l'outil d'analyse de variance Comme nous disposons de 4 422 mesures, on pourrait, en application du théorème central limite facilement faire l'hypothèse que les données suivent une loi normale. On propose que pour tous les lots R&D, on ajoute une plaquette de référence dans les étapes des mesures qui valide l'équipement et aussi les mesures faites. Cet article est centré sur l'industrie microélectronique qui est caractérisée par les changements

  changement stratégique permet de réduire le temps de déploiement et le coût d'une plate-forme technologique, mais met sous pression la production pour adopter les modèles issus du DFM délivrés par une alliance. Dans cet article, nous proposons une méthodologie étendue basée sur DFM pour le développement d'interconnexions pour identifier, estimer et corriger les sources de variation.

	Cet article est divisé en 4 parties. La première présente une synthèse
	des concepts de DFM, des difficultés rencontrées par l'industrie et de
	mettre en évidence la nécessité d'une approche rigoureuse pour le
	modèle DFM. La seconde partie présente le processus de
	modélisation et l'approche utilisée par l'industrie pour aligner /
	développer des modèles de référence. La partie 3 propose une
	méthodologie DFM étendue en ajoutant un contrôle supplémentaire
	des corrélations entre tests géométriques et électriques. Elle est
	soutenue par un diagramme de classes UML du système et est mise en
	oeuvre en tant qu'outil pour les ingénieurs R&D. Enfin la partie 4
	conclut l'article en discutant les résultats obtenus et les perspectives.
	1	MODÉLISATION DES INTERCONNECTIONS
	Schématiquement, les circuits électroniques sont constitués de
	transistors fabriqués sur chaque plaquette de silicium et reliés
	par un réseau de câblage (couches d'interconnexion en métal).
	La taille standard des plaquettes est aujourd'hui de 300nm de
	diamètre, une taille couramment utilisée aujourd'hui avec la
	technologie	CMOS	(Complementary	Metal	Oxide
	Semiconductor) ; cette technologie nécessite plus de 400
	opérations et environ 8 semaines de traitement.