Shot noise control in an AC-driven quantum point contact

Physica B: Condensed Matter 560 (2019) 215–219

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Shot noise control in an AC-driven quantum point contact

T

X.K. Yue Laboratory of Mesoscopic and Low Dimensional Physics, College of Physical Science and Technology, Sichuan University, Chengdu, 610064, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: Quantum point contact Shot-noise Time-dependent transport

We propose a scheme to manipulate the shot noise spectrum via the gate voltage Vg(t) of the quantum point contact. It has been known that shot noise can be used as a hallmark of electron temperature in the reservoirs. We show that, by choosing the pulse shape of Vg(t) properly, the zero-frequency shot noise can be tuned so that it is the same with the DC shot noise of two electron reservoirs at different temperatures. The voltage pulse width is rather fixed and cannot be changed efficiently. We provide a way to construct an amplitude-modulated pulse train to increase the signal strength of shot noise, which is useful in the study of shot noise experimentally.

1. Introduction In earlier quantum point contacts (QPC) experiments, the quantized conductance is mainly concerned, which has been seen as the hallmark of the ideal quantum channel. Besides, it is found that current noise spectrum can provide more detailed information of the quantum conS ductor [1]. For example, at zero temperature, the Fano factor F = 2eI0 , 0 which is the ratio between the zero-frequency noise S0 and the current I0, can be used as a quantity to describe the statistic property of the electron transport. A suppression of the Fano factor below the classical limit F = 1 has been demonstrated [2,3], which can be attributed to the Pauli blocking of electrons in quantum conductors [4,5]. In quantum dot systems, an enhancement of the Fano factor (F > 1) has also been observed, which is related to the Coulomb interaction and the shape of the density of states [6]. When a finite temperature is concerned, the current noise spectrum can also be used as a tool to detect the energy distribution of the electrons. Electronic thermometers based on Johnson-Nyquist noise and/or shot noise have been fabricated and widely used in various experiments [7,8]. In recent years, it has been shown that the current noise can be manipulated via the time-dependent bias voltage Vds, as illustrated in Fig. 1. By properly tuning the profile of Vds in the time-domain, the creation of electron-hole pairs, which are related to the bidirectional single-charge transfer process [9], can be suppressed. This leads to minimal excitation states in the Fermi sea of the quantum conductor, which has been predicted by Levitov et al. theoretically [10–12] and justified in a series of experiments by the group of D.C. Glattli [13,14]. Moreover, it has also been demonstrated that one can use the bias voltage Vds to realize an electron source with tunable temperature [15]. As shown in Fig. 1, the energy distribution of the emitted electrons follows the Fermi distribution

fD (ω) =

1 , (1 + exp[(ℏω − μ)/(kB TD)])

(1)

with kB being the Boltzmann constant and TD being the electron temperature. Such an electron source shows a special shot noise spectrum, which is illustrated by the red dots in Fig. 1(b). It is the same with the DC shot noise of two electron reservoirs at different temperatures TS and TD, as indicated by the blue dotted curve. In this paper, we propose that, by using gate voltage pulses, shown in Fig. 1(c), the same shot noise spectrum as described above can be also realized, which is indicated in Fig. 1(d). For typical values TS = 10 mK and TD = 100 mK, this gives a realistic single pulse width Δt ≈ 0.2 ns. The paper is organized as follows. In Sec. 2, we derive the shot noise of electron current through a time-dependent QPC. In Sec. 3, we show how to tune the transmissivity to get the desired shot noise. In Sec. 4, by modeling the potential barrier of QPC as a delta function, we show how to realize the above-mentioned manipulating via gate voltage Vg(t). We summarize in Sec. 5. 2. Shot noise of a time-dependent QPC A QPC can be modeled as a one-dimensional quantum wire connecting two reservoirs S and D, as shown in Fig. 1. It is biased with a DC voltage Vds and the zero of the energy is set so that the chemical potential μ = 0. A negative voltage Vg(t) is applied to the gate electrodes on the top of the conductor to form an elastic scattering region with tunable transmissivity. It is worth mentioning that the mean dwell time of an electron in the nanostructure is much smaller than the inverse of driving frequency so the adiabatic approximation holds in most experiments [16]. In the adiabatic regime, the transmissivity D(t) of the

E-mail address: [email protected]. https://doi.org/10.1016/j.physb.2019.02.018 Received 18 May 2018; Received in revised form 20 December 2018; Accepted 11 February 2019 Available online 16 February 2019 0921-4526/ © 2019 Published by Elsevier B.V.

Physica B: Condensed Matter 560 (2019) 215–219

X.K. Yue

BˆL (t ) = ∫ e−iEt /ℏ bˆL,E dE ,

(5)

where the operator âL,E satisfies: ⟨â†L,EâL,E′⟩ = fL (E ) δ (E − E′) and ⟨…⟩ represents the thermal expectation. The scattering states at the two sides of QPC are connected by the time-dependent scattering matrix [1,17]

⎛ BˆL (t ) ⎞ = ⎜ ⎟ ˆ ⎝ BR (t ) ⎠

ÂL (t ) ⎞ S (t ) ⎜⎛ ⎟, Â ⎝ R (t ) ⎠

D (t ) ⎛−i 1 − D (t ) ⎞ = ⎜ ⎟, D ( t ) − i 1 − D ( t ) ⎝ ⎠

S (t )

(6)

The Fermi operator ÂR(t) and BˆR (t ) represent the left- and right-moving mode, respectively, in the right lead. The current operator is constructed as † † ĵL (t ) = e /(2π ℏ)[ÂL (t )ÂL (t ) − BˆL (t ) BˆL (t )]. Then the zero-frequency t t noise within the time interval [− 20 , 20 ] can be given as

S¯ =

1 t0

t0

t0

∫− 2t0 dt ∫− 2t0 dt ′ [⟨ĵL (t ) ĵL (t ′) ⟩ − ⟨ĵL (t ) ⟩⟨ĵL (t ′) ⟩] 2

2

S¯eq + S¯shot,

=

(7)

with S¯eq being the equilibrium noise term and S¯shot being the nonequilibrium or shot noise term which depends on the bias voltage Vds. The shot noise tuned by D(t) can be written as

S¯shot =

e2 2π

¯

D (1 − D) ∫ dω {[1 − fS (ω)] fV (ω) + fS (ω)[1 − fV (ω)]},

(8)

with dω′ Π (ω′ 2π V

fV (ω) = ∫ Fig. 1. Schematic of the quantum point contact (QPC) (top figure) and comparison of the main results in Ref. [15] and this paper. Two-dimensional electron gas (grey part) connects electron reservoirs S and D. At the centre region, gate voltage Vg is applied to form a scatter. TS and fS (TD and fD) are the temperature and energy distribution, respectively, of incoming (outgoing) electrons. ÂL(ÂR) represents electrons ejected from left (right) reservoir. (a) This driving voltage pulse is used in Ref. [15] to manipulate shot noise. The detected shot noise corresponding to Vds is shown in (b) (red dots). The blue dotted curve represents the DC shot noise with two reservoirs at different temperatures, while the orange dashed line represents the DC shot noise without manipulating. S0 is the zero-bias noise. The gate voltage pulses in (c) are used in this paper, and the corresponding shot noise is shown in figure (d). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

fS (ω) =

me∗ [ÂL (t + x / v F ) + BˆL (t − x / v F )], 2π ℏ2kF

,

D (1 − D)

(10)

1 , (1 + exp[(ℏω − μ)/(kB TS)])

(11)

and D (1 − D) is the mean value of D(t)[1 − D(t)] within the time int

¯

t

terval [− 20 , 20 ]. Note that the parameter D (1 − D) reflects the signal strength of shot noise spectrum. When DC transport is concerned, the two Fermi reservoirs have the same temperature TS and the above formula of noise is integrable ¯

3

eV eV S¯shot = D (1 − D) coth( ). 2π ℏ 2kB TS

(12)

As a function of voltage, we can fit the noise spectrum with temperature TS as fitting parameter. Thus the shot noise measurement provides us a simple method to detect electron temperature. However, if the two reservoirs' temperatures are different, Eq. (8) is not integrable and we can not get a simple analytic formula of noise spectrum. Numeric method must be needed to do the above fitting process and extract the electron temperatures.

(2)

3. Tuning distribution via transmissivity (3)

If fV(ω) given in Eq. (9) follows the Fermi distribution fD(ω) given in Eq. (1), then we'll get the desired zero-frequency shot noise. To do so, the integral kernel ΠV(ω) would satisfy

me∗

being the effective mass of electrons and kF being the Fermi with wave vector. ÂL(t) and BˆL (t ) can be expressed by the annihilation operators

ÂL (t ) = ∫ e−iEt /ℏ âL,E dE ,

¯

2

¯

with vF being the Fermi velocity. The sign ± corresponds to right- and left-going electrons, respectively. The field operator of the electron in the left lead can be written as

ˆ L (x , t ) = Ψ

D (t )[1 − D (t )] iωt e dt

∫− 2t0

where fS represents the Fermi distribution of electrons in reservoir S

scattering potential can be obtained by solving the static scattering problem. In most mesoscopic experiments, the relevant energy scales, such as electron temperature and the bias voltage Vds, are much small compared to the Fermi level. It is convenient to consider such electron system as dispersionless Fermi systems with the corresponding singleparticle Hamiltonian

H = −iℏ(± v F ) ∂x ,

(9) 2

t0

1 t0

ΠV (ω) =

− ω) fS (ω′),

ΠV (ω) = Πh (ω) =

(4)

with 216

∫ dte−iωt Π¯h (t ),

(13)

Physica B: Condensed Matter 560 (2019) 215–219

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¯ h (t ) . The red solid, green dashed and blue Fig. 2. Schematic of the function Π dotted curves represent the heating up from TS = 10 mK to TD = 100 mK, 200 mK and 300 mK, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 4. The same as Fig. 3, but with higher right electron reservoir temperature TR = 60 mk. The green dashed line in inset (b) corresponds to the noise spectrum in Fig. 3, which is presented here as a comparison. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

πk T

csch( ℏB D t ) ¯ h (t ) = TD Π . TS csch( πk B TS t ) ℏ

(14)

solid curve in the main panel, which is calculated from Eq. (9) and Eq. (10) numerically. One can see that it follows the Fermi distribution fD (green dashed curve) at the temperature TD = 100 mK. The blue dotted curve represents the energy distribution fS of the incoming electrons from reservoir S at the temperature TS = 10 mK. We show the shot noise in the inset (b) of Fig. 3. It is normalized to the zero-bias noise

Some typical profiles are illustrated in Fig. 2. Note that since ΠV(ω) is non-negative according to Eq. (10), TD must be higher than the original electron temperature TS. From Eq. (10), we get the relation between D(t) and Πh(ω) ¯

D (t )[1 − D (t )] = t 0 D (1 − D)

∫

Πh (ω) ei [θ (ω) − ωt ]

dω 2 , 2π

¯

S0 = e 2D (1 − D)kB TS/(2π 2ℏ) with x = eVds∕(kBTS) being the normalized voltage. The red dots represent the shot noise spectrum tuned by the transmissivity pulse, while the predicted shot noise is plotted as the blue dotted curve. One can see that they agree quite well. Another case should be concerned, where electron temperatures of two contacts are different. The Eq. (8) and Eq. (9) show that the desired energy distribution fV of outgoing electrons is only affected by the timedependent transmissivity D(t) and the temperature TS of the incoming electrons. The temperature of the right reservoir does not have influence on the manipulating process. However, it does impact the shot noise spectrum, which is shown in Fig. 4. We increase the temperature of right reservoir from TR = 10 mk to TR = 60 mk. It is obvious that the transmissivity curve D(t)(1 − D(t)) remains unchanged, compared to Fig. 3. The red dots in the inset (b) represent the shot noise with higher temperature TR. As a comparison, the green dashed line is presented here, which corresponds to the noise spectrum in Fig. 3. It should be emphasized that the time duration t0 is rather restricted. In principle, t0 should be large enough to include the main positive part of D(t)[1 − D(t)]. Beyond this time range, D(t)[1 − D(t)] approaches zero so does not contribute to shot noise. Finally, we need to choose a

(15)

where θ(ω) is an arbitrary phase factor. In the following discussion, we set θ(ω) to zero for simplicity. An example is shown in the inset (a) of Fig. 3, with t0 = 0.4 ns. It is a single pulse of bell shape. The corresponding fV is presented as the red

¯

proper value of D (1 − D) so that D(t) ranges from 0 to 1. In Fig. 5, we perform a case with short duration, t0 = 0.1 ns. In its main panel, we can see that the function fV(ω) does not follow the Fermi distribution fD(ω) anymore. As a consequence, the corresponding shot noise deviates from the predicted noise, as shown in the inset (b). Note that we can not find a proper way to reduce the duration t0 since the pulse width is fixed by temperatures TD and TS within our theory. We also show a case with long duration in Fig. 6. In this case, t0 is chosen to be 2 ns, which is five times larger than the case shown in the Fig. 3. It is long enough to include the main positive part of D(t)[1 − D (t)] and the function fV(ω) is almost distinguishable from fD(ω). We get the desired shot noise indeed. However, its signal is too weak due to the

Fig. 3. Comparison of the function fV(ω) with the Fermi distribution fD(ω) and fS(ω) (main panel). The blue dotted line represents the distribution of incoming electrons fS(ω), and corresponding temperature TS = 10 mK. The function fV(ω) are shown with the red solid curve. The green dashed curve represents the Fermi distribution at the temperature TD = 100 mK. The pulse D(t)[1 − D(t)] with duration t0 = 0.4 ns is shown in inset (a). The red dots in inset (b) is the corresponding shot noise Sshot∕S0 as a function of normalized DC bias voltage ¯

¯

x = eVds∕(kBTS), with S0 = e 2D (1 − D)kB TS/(π ℏ) and D (1 − D) = 0.02 . The blue dotted curve represents shot noise with two reservoirs at temperatures TS = 10 mK and TD = 100 mK, while the orange dashed line represents the shot

¯

small value of D (1 − D) , which is demonstrated in the inset (b). It would make the noise hard to be detected experimentally. To fix this problem, we propose a scheme to find an approximate solution with multi-pulse, which satisfies this relation

¯

noise without manipulating S¯ = D (1 − D)e 3V /(2π ℏ)coth[eV /(2kB TS)]. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 217

Physica B: Condensed Matter 560 (2019) 215–219

X.K. Yue

Fig. 5. The same as Fig. 3, but with the pulse duration t0 = 0.1 ns.

Fig. 8. The same as Fig. 3, but with multi-pulse transmissivity. N

ΠG (ω) =

∑ n =−N

¯

Δω t0 −Δω2t 2 e π

= 2D (1 − D) π N

× ∑n = 1 An cos(nω0 t ) +

A0 2

2

.

(17)

where Δω and ω0 represent the width of Gaussian functions and the interval between two adjacent sub-functions, respectively. To make it simplified, we assume that these sub-functions are well-separated, which means Δω should be small enough. The overlap of the subfunctions is negligible, so we can get the analytic form of ΠG (ω) and deduce Eq. (16) by substituting Eq. (17) into Eq. (15). The exponential term in Eq. (16) ensures that when t is beyond the time interval t t [− 20 , 20 ], the function D(t)[1 − D(t)] is small enough and does not contribute to shot noise. Thus Δω and t0 have an inverse relationship and t0 should be large enough. For example, in Fig. 8, we choose t0 = 2 ns, which is approximately twenty times larger than the full ¯ h (t ) . width at the half maximum (FWHM) of the profile Π In Eq. (9), the convolution of fS(ω) and Πh(ω) gives the function fV(ω), which follows the Fermi distribution fD(ω). In order to make the substitution ΠG(ω) have the same effect, we can initially set the value of An, through the relation

Fig. 6. The same as Fig. 3, but with the pulse duration t0 = 1 ns.

D (t )[1 − D (t )]

An ω + nω0 2 exp[−( ) ], π Δω Δω

(16)

ω nω0 + 0

∫nω − ω 2 0

This is a form of amplitude-modulated pulse train with amplitude tuned by the exponential term. Terms in the brackets constitute a periodic pulse signal. Parameters Δω, ω0 and An are introduced due to the approximation. In the following, we'll show the derivation of the above equation and explain how to get these parameters' value. As shown in the Fig. 7, a series of evenly spaced Gaussian functions are used to replace the original integral kernel Πh(ω)

0

ΠG (ω)dω = An ≈ Πh (nω0) ω0 .

2

(18)

Then, another requirement must be considered, that the integral of Πh(ω) equals 2π, which has a great influence on the shot noise, since it ensures fV(ε) = 1 at energy ε being much less than chemical potential. So after obtaining the rough values of An, we need to normalize them to satisfy

∑ An

= 2π .

n

(19)

Now, we can calculate the approximate integral kernel ΠG(ω) and D (t)[1 − D(t)]. An example is shown in Fig. 8 with t0 = 2 ns and 13 Gaussian functions being used to substitute the origin integral kernel. One can see that fV(ω)(red solid curve) follows the Fermi function fD(ω) with TD = 100 mK(green dashed line) quite well. The zero-frequency shot noise tuned by D(t)[1 − D(t)] (red dots) is shown in the inset (b) of Fig. 8. In the same figure, we also present the DC noise (green dashed line) with TS = 10 mK and TD = 100 mK as a comparison. Despite some small ripples of fV(ω), the shot noise spectrum still fits with the target noise very well. 4. Gate voltage on the QPC Fig. 7. The integral kernel ΠG(ω) with 13 Gaussian functions (red solid curve) being used to replace the original Πh(ω) (green dashed curve) from Eq. (13). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Experimentally, a gate voltage Vg(t) is applied to control the QPC's transmissivity. We model the scattering potential as a delta function

U (x , t ) = αeVg (t ) δ (x / a), 218

(20)

Physica B: Condensed Matter 560 (2019) 215–219

X.K. Yue

5. Summary To summarize, we have shown a realization of a special shot noise spectrum via the gate voltage Vg(t) of QPC. This spectrum is the same with the DC shot noise with two electron reservoirs at different temperatures, TS and TD respectively. For typical values TS = 10 mK and TD = 100 mK, this gives a realistic single pulse width Δt ≈ 0.2 ns, making it helpful in the study of DC noise with a modulating AC signal. Acknowledgments The authors gratefully thank Y. Yin for his kind help and acknowledge the support of the National Key Basic Research Program of China under Grant No. 2016YFF0200403, the Key Program of National Natural Science Foundation of China under Grant No. 11234009, and Young Scientists Fund of National Natural Science Foundation of China under Grant No. 11504248.

Fig. 9. The gate voltage curve corresponding to the transmissivity D(t) in Fig. 8.

with α = 0.1 describing the electrostatic coupling between gate electrodes and electron gas [18] and a = λF∕π being the cut-off length. The transmissivity D(t) is determined by the time-dependent Schrödinger equation

∂Ψ(x , t ) ℏ2 ∂2Ψ(x , t ) =− ∗ + U (x , t )Ψ(x , t ). iℏ ∂t 2m ∂x 2

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(21)

We express the electron wave function in the leads as

ΨL (x , t ) = C1 (t ) eik F x + C2 (t ) e−ik F x , x < 0, ΨR (x , t ) = C3 (t ) eik F x + C4 (t ) e−ik F x , x > 0.

(22)

The boundary conditions are

ΨL (x , t )|x = 0 = ΨR (x , t )|x = 0 ,

(23)

and

∂ΨR (x , t ) ∂x

− x = 0+

∂ΨL (x , t ) ∂x

= x = 0−

2m∗αaeVg (t ) ℏ2

Ψ(x , t )

. x=0

(24)

Substituting Eq. (22) to the boundary conditions, one obtains the relation between the gate voltage Vg(t) and the transmissivity D(t)

D (t ) =

C3 (t ) C1 (t )

2

1

= 1+

aαeVg (t ) ( v ℏ )2 F

. (25)

In Fig. 9, we show the voltage pulse corresponding to the case in Fig. 8.

219