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Temporally correlated transport and shot noise suppression in a ballistic quantum point contact
surface science ELSEVIER
Surface Saence 361/362 (1996) 726-729
Temporally correlated transport and shot noise suppression in a ballistic quantum point contact M. R e z n i k o v *, M. H e i b l u m , H a d a s S h t r i k m a n , D. M a h a l u Braun Centerfor Submicron Research. Department of Condensed Matter Physics. Wetzmarmln~l,'~te of Sck,,nce. Rthovot 76100. Israel Received 16 June 1995;~ t e d
for publicauon 19 September 1995
Almract
Wido-band thot home, a~xSatrd with DC current flow through a quantum point contact (QPC), is me,~ur~ m the microwave frequency ranse 8-18 GH2. As the number of conducting channeis m the QPC change•, the noise power oscillates, with • value of almost zero at the conductance plateaux. Coati,tent with e=ssling theoriea, the noise pea.ks depend linearly on the DC ~ t _ Surpritin~ly, however, in the pinch-off region, where QPC is expected to behave as • classical injector, we find tm'ong noise suppremion, poufibly mediated by the Coulomb interaction in the QPC r~giorL
Keyword~: Heterojunctions~ Quantum effects; Surface electrical transport
1. Introduction
It is well k n o w n that the c o n d u c t a n c e of a Q P C , e.g. a s m o o t h one-dimensional ( I D ) c o n s t r i c t i o n made in a two-dimensional electron gas, is quantized in units of 2e2/h. This quantization is a direct result of almost perfect transmission ( T ~ . I ) t h r o u g h the 1D channels confined in the Q P C . Aside from that quantization, it was predicted by Khlns [ 1 ] and Lesovik [ 2 ] that when T = I , another remarkable effect takes place: a total suppression of shot noise in the Q P C . The shot noise, originating from the granularity of the electrons and their stochastic injection, is characterized by averaged squared current fluctuations. In classical systems such as vacuum diode, where electron emission can be considered as a truly stochastic " C o ~ n d i n g author. Fax: + 972 8 9344128; o-mail: ~ . w e i , ~ - - - a c . ~ 0039-6028/96/$1500 Copyright O 1996 ~ PII S0039-6028 (96)00520- I
(Poisson-like) process, the average of the squared current fluctuations di, measured in a frequency range dr, is given by the classical shot noise expression S ( v ) d v = < ( d i ) ~ > , , = 2 e l d v . S(v)is the white (frequency-independent) spectral density and 1 the average current. In a Q P C , the theoretically obtained low-frequency spectral density of the current fluctuations, at zero temperature 0, for energy independent T~s is [ 1 - 3 ]
2e2
S(v=O)=2e ~
,,
VDs ~.
T~(1-TI), ksO
|=l
(1) where V ~ is the voltage across the Q P C , N is the n u m b e r of occupied channels in the Q P C , and Tt is the transmission coefficient through the channel i. Since (2ea/'n)V~T, = It is the ith channel current, a noise suppression of ( l - T 3 , relative to the classical noise for the /th c h a n n e l is expected. In an experiment where the T~s axe continuously
Science B.V. All rights reserved
M. Reznikov et al./Surface Science 361/362 (1996) 726-729
varied, the shot noise is expected to have a maximum (half of the classical shot noise due to one channel) whenever the transmission of the uppermost conducting channel is 1/2. Note that the frequency dependence of S(v) was shown (neglecting Coulomb interactions between electrons) to decrease linearly with v with S(v~toer)=0 for hv~toer=eVm (i_e. for V m = l m V , v~toer~ 250 GHz) [4]. . : Several attempts to measure shot noise, in a Q P C have been published [5,6]. However, these attempts were restricted to low frequencies (v< 100 kHz) where I/f noise or fluctuations and instabilities of the conductance are dominant. Moreover, the noise power was found to have a square dependence on the D C current, and not the expected linear dependence. A linear dependence, however, was observed for noise measured in diffusive mesoscopic conductors [7]. First some order of magnitude estimates. injected DC current It. and the applied voltage, VDs should be as small as possible in order (i), to prevent electron heating, and (ii) to prevent injection into higher 1D channels. However, the applied voltage Vm should be greater than KsO/e in Order to make Eq: (3)"applicable. For example, Vre =' 1 mV leads, according to Eq. (3), to a peak noise S(0)=6.2x 10-Z7 As Hz -1. At high enough frequencies, this shot-noise signal has to compete with the noise of the amplifiers. Even a cold amplifier has an equivalent noise temperature of 40 K at its 50 ~ input impedance, leading to a spectral density of unwanted noise kBO/50=lO -23 Az Hz -1, more than three orders of magnitude higher than the shot noise we are searching for. Consequenfly,in order to improve the signal-to-noise ratio (S/N) we modulate the DC current through the QPC at a low frequency f (<1 kHz). and measure the amplified noise synchrononsly with f using a lock-in technique. It turns out that the wide band, measurements improve the SIN ratio relative' to. the ratio of the spectral densities given above by a factor of (Av/Af) 1/2, where Af is the low-frequency bandwidth (determined by the time constant of the lock-in amplifier). We thus expect that measuring the s h o t noise synchronously, in a band v = 8-18GHz (dv=10GI-Iz) with .ztf
727
improve the SIN ratio by a factor of (1-3)x 105, leading to an acceptable S/Nratio of ~ 102.
2. Experimental" Our QPC is induced electrostatically in the plane of a 2DEG, embedded in a GaAs-A1GaAs heterostructttre only 33 nm below the surface. The 2DEG has an area electron density of 4.6 x 1011 cm -2 and a low-temperature mobility of 5 x l0 s cme/V.s. The QPC is formed by direct electronbeam Writing and metal gate (TiAu) deposition, with a 100 nm gap between the two gates (shown schematically in Fig. 1). The voltage Vo applied to the gates. ~ia a low-pass-filter with a n upper frequency o f 1MHz, controls the number of 1D conducting channels in the QPC. The DC voltage drop across the sample was monitored during the measurement with a lock-in amplifier and was kept constant via a feedback loop (taking into account series contact resistance). Similarly, the DC current was kept constant by monitoring the DC voltage drop on a series 1M_~ resistor. Note that the highfrequency path to ground is provided via the capacitor Ct, effectively maintaining a constant voltage on the sample. The high-frequency current fluctuations are amplified in the 8-18 GI-Iz band, and are finally applied to a high-frequency diode and a load capacitor C~. The low-frequency output Vo~t(f)°c(zli) 2 > ioom is converted, after a calibration of the system gain and the diode sensitivity, to the spectral density at the input S[A 2 H z - l ] .
3. 'Results and discussion ' Typical results for the D C conductance G and the noise signal Vout(f),,measured as a function of gate voltage Vo at T = 1.5 K are shown in Fig. 2. The linear conductance:is quantized in units of 2e2/h, after subtracting the series resistance of the ohmic contacts. The noise signal SocVout(f) is measured for different injection voltages, Vm(f)~-(VDs/2)(l'+cos2~ft), imposed on the QPC. As predicted by Eq. (3), keeping Vm conStant allows the, injected current It.(f) to change as the conductance of the QPC varies, leading to a noise dependence proportional to T(1-T).
728
M. Reznikov et aL /Surface Science 361/362 (1996) 726-729
r"
.
.
' /
/ '1
.
.
.
.
1 1
'
1Ma
.
.
.
-
il
I il . / ' .z..',4f,"
I~
-
1 1
\
\ ~
Cold
Warm
Amplifier ,..
I
High Freq.
Amplifier Diode ) (
I
Fig. 1. Experimental set-up. The voltage Vo controls the number of 1D conducting channels in the QPC. The current, modulated at low frequency f, is provided to the QPC via a variable current source, with a voltage Vm(f) appearing across the QPC. A high f~xtuency path to ground is provided via a capacitor C~. 10 0
S
8
,~ ....... %s/my.
~/d>,,\-~ ....... .,,~,.k..\\k~.
-1
2 .
.J. ........
0
O-
I+-ff2. . . .
I -t.2
-1/2
rs-.~Y2~Vos_--.0,~,D.~ .t.1 -t.o .~
.............
Gate Voltage, V G ,IV] Fig. 2. Noise spectral density S(v) and normalized linear conductance G, measured with current 10 nA versus gate voltage Vo. The noise is measured for Vm =0.5, 1, 1.5..... 4 inV. The lines through the mLxima and minima of the peaks are guides to the eye. Pmitions of Ti = 1/2, where the noise m n ~ m n Bre thcoretically expected to be, are marked with arrows. Inset: dependence of the peak height (same scale as in main figure) on injection voltage Vm. 0 , 0 and <> mark the first three conducting ID channels. The straight dashed line is the predicted behavior.
Indeed the measured noise signal, as a function of Vo and for different Vm, exhibits strong oscillations. The magnitude of the peaks (each peak relative to the average of two adjacent minima) grows almost linearly with the current, as seen in the inset of Fig. 2. This linear dependence of the
spectral density also agrees quantitavely with Eq. (3) and is a crucial fact in substantiating the origin of the noise. Even though the magnitude of the peaks agrees rather well with the predicted noise maxima (being (1/4)2e(2e2/h)Vvs) the peak positions (and respective minima) shift from the predicted positions at Vo(T~=I/2) (and T~--1 for the minima) to Vo (higher Tis), a shift that increases with VDS. This shift is not yet understood. As a possible explanation one may consider injection into the upper channel of the QPC mediated by the finite temperature and injection voltage, thus leading to added noise from this channel and to an apparent shift in the position of the peaks. However, the fact that the measured noise signal for relatively small injection voltages drops to almost zero at the minima, implies that Tl=l, Tl+l ~ 0 at the minima, making the above hypothesis questionable. Another possible explanation for the shift is the finite temperature and the fact that TIS energy-dependent (T= HE)) [8,9]. Taking this into account we find that the calculated noise peaks shift toward smaller ~ , contrary to our data. 4. Conclusion
We performed detailed investigations of the noise in the pinch-off region of a QPC, namely, when
M. Reznikov et a£/Surface Science 361/362 (1996) 726--729
all carriers are depleted from the QPC and transport is via tunneling or thermally assisted emission. As the gate voltage becomes more negative and the resistance of the QPC increases, the current and its fluctuations tend to zero. We thus performed noise measurements at a constant current I, expecting to obtain the behavior S--2eI(1-T~), which is nearly classical when T~<<1. Contrary to this, we find that the noise tends to saturate and become independent of the DC current above 100 hA, which is contradictory to the expected classical behavior. Although the voltage Vm across the PC in this r e , m e could be tens of mV this should not be a problem, for the linear r e , m e is not a condition for classical shot noise to occur. Since in the pinch-off r e , me the estimated dwell time in the QPC is similar to eli
729
(the impinging time interval) we suggest that Coulomb repulsion between the electrons leads to temporally correlated transport in the pinched QPC. and thus to noise suppression.
References El"] V.A. Khlus, Soy. Phys. JETP 66 (1987) 1243. 1"2"1G.B. Lesovik, JETP Lett. 49 (1989) 592. I-3"1M. Buttiker, Phys. Rev. Lett. 65 (1990) 2901. E4"1 G.B. Le~vik and L.S. Levitov, Phys. Rev. Lett. 72 (1994) 538. 1-5"1Y.P. Li et aL, AppL Phys. ~ 57 (1990) 774. 1-6"1F. Liefrink, J i Dijkhuis and H. van Houten, Semicond. S~. TechnoL 9 (1994) 2178, and references there~ [7] F. Liefrink et aL Phys. Rev. B 49 (1994) 14066. 1-8] IL Landaner and TIL Martin, Physica B 175 (1991) 167. 1-9"1G~B. Lesovik and R. L o o e ~ Z. Phys. B 91 (1993) 531.
Surface Saence 361/362 (1996) 726-729
Temporally correlated transport and shot noise suppression in a ballistic quantum point contact M. R e z n i k o v *, M. H e i b l u m , H a d a s S h t r i k m a n , D. M a h a l u Braun Centerfor Submicron Research. Department of Condensed Matter Physics. Wetzmarmln~l,'~te of Sck,,nce. Rthovot 76100. Israel Received 16 June 1995;~ t e d
for publicauon 19 September 1995
Almract
Wido-band thot home, a~xSatrd with DC current flow through a quantum point contact (QPC), is me,~ur~ m the microwave frequency ranse 8-18 GH2. As the number of conducting channeis m the QPC change•, the noise power oscillates, with • value of almost zero at the conductance plateaux. Coati,tent with e=ssling theoriea, the noise pea.ks depend linearly on the DC ~ t _ Surpritin~ly, however, in the pinch-off region, where QPC is expected to behave as • classical injector, we find tm'ong noise suppremion, poufibly mediated by the Coulomb interaction in the QPC r~giorL
Keyword~: Heterojunctions~ Quantum effects; Surface electrical transport
1. Introduction
It is well k n o w n that the c o n d u c t a n c e of a Q P C , e.g. a s m o o t h one-dimensional ( I D ) c o n s t r i c t i o n made in a two-dimensional electron gas, is quantized in units of 2e2/h. This quantization is a direct result of almost perfect transmission ( T ~ . I ) t h r o u g h the 1D channels confined in the Q P C . Aside from that quantization, it was predicted by Khlns [ 1 ] and Lesovik [ 2 ] that when T = I , another remarkable effect takes place: a total suppression of shot noise in the Q P C . The shot noise, originating from the granularity of the electrons and their stochastic injection, is characterized by averaged squared current fluctuations. In classical systems such as vacuum diode, where electron emission can be considered as a truly stochastic " C o ~ n d i n g author. Fax: + 972 8 9344128; o-mail: ~ . w e i , ~ - - - a c . ~ 0039-6028/96/$1500 Copyright O 1996 ~ PII S0039-6028 (96)00520- I
(Poisson-like) process, the average of the squared current fluctuations di, measured in a frequency range dr, is given by the classical shot noise expression S ( v ) d v = < ( d i ) ~ > , , = 2 e l d v . S(v)is the white (frequency-independent) spectral density and 1 the average current. In a Q P C , the theoretically obtained low-frequency spectral density of the current fluctuations, at zero temperature 0, for energy independent T~s is [ 1 - 3 ]
2e2
S(v=O)=2e ~
,,
VDs ~.
T~(1-TI), ksO
|=l
(1) where V ~ is the voltage across the Q P C , N is the n u m b e r of occupied channels in the Q P C , and Tt is the transmission coefficient through the channel i. Since (2ea/'n)V~T, = It is the ith channel current, a noise suppression of ( l - T 3 , relative to the classical noise for the /th c h a n n e l is expected. In an experiment where the T~s axe continuously
Science B.V. All rights reserved
M. Reznikov et al./Surface Science 361/362 (1996) 726-729
varied, the shot noise is expected to have a maximum (half of the classical shot noise due to one channel) whenever the transmission of the uppermost conducting channel is 1/2. Note that the frequency dependence of S(v) was shown (neglecting Coulomb interactions between electrons) to decrease linearly with v with S(v~toer)=0 for hv~toer=eVm (i_e. for V m = l m V , v~toer~ 250 GHz) [4]. . : Several attempts to measure shot noise, in a Q P C have been published [5,6]. However, these attempts were restricted to low frequencies (v< 100 kHz) where I/f noise or fluctuations and instabilities of the conductance are dominant. Moreover, the noise power was found to have a square dependence on the D C current, and not the expected linear dependence. A linear dependence, however, was observed for noise measured in diffusive mesoscopic conductors [7]. First some order of magnitude estimates. injected DC current It. and the applied voltage, VDs should be as small as possible in order (i), to prevent electron heating, and (ii) to prevent injection into higher 1D channels. However, the applied voltage Vm should be greater than KsO/e in Order to make Eq: (3)"applicable. For example, Vre =' 1 mV leads, according to Eq. (3), to a peak noise S(0)=6.2x 10-Z7 As Hz -1. At high enough frequencies, this shot-noise signal has to compete with the noise of the amplifiers. Even a cold amplifier has an equivalent noise temperature of 40 K at its 50 ~ input impedance, leading to a spectral density of unwanted noise kBO/50=lO -23 Az Hz -1, more than three orders of magnitude higher than the shot noise we are searching for. Consequenfly,in order to improve the signal-to-noise ratio (S/N) we modulate the DC current through the QPC at a low frequency f (<1 kHz). and measure the amplified noise synchrononsly with f using a lock-in technique. It turns out that the wide band, measurements improve the SIN ratio relative' to. the ratio of the spectral densities given above by a factor of (Av/Af) 1/2, where Af is the low-frequency bandwidth (determined by the time constant of the lock-in amplifier). We thus expect that measuring the s h o t noise synchronously, in a band v = 8-18GHz (dv=10GI-Iz) with .ztf
727
improve the SIN ratio by a factor of (1-3)x 105, leading to an acceptable S/Nratio of ~ 102.
2. Experimental" Our QPC is induced electrostatically in the plane of a 2DEG, embedded in a GaAs-A1GaAs heterostructttre only 33 nm below the surface. The 2DEG has an area electron density of 4.6 x 1011 cm -2 and a low-temperature mobility of 5 x l0 s cme/V.s. The QPC is formed by direct electronbeam Writing and metal gate (TiAu) deposition, with a 100 nm gap between the two gates (shown schematically in Fig. 1). The voltage Vo applied to the gates. ~ia a low-pass-filter with a n upper frequency o f 1MHz, controls the number of 1D conducting channels in the QPC. The DC voltage drop across the sample was monitored during the measurement with a lock-in amplifier and was kept constant via a feedback loop (taking into account series contact resistance). Similarly, the DC current was kept constant by monitoring the DC voltage drop on a series 1M_~ resistor. Note that the highfrequency path to ground is provided via the capacitor Ct, effectively maintaining a constant voltage on the sample. The high-frequency current fluctuations are amplified in the 8-18 GI-Iz band, and are finally applied to a high-frequency diode and a load capacitor C~. The low-frequency output Vo~t(f)°c(zli) 2 > ioom is converted, after a calibration of the system gain and the diode sensitivity, to the spectral density at the input S[A 2 H z - l ] .
3. 'Results and discussion ' Typical results for the D C conductance G and the noise signal Vout(f),,measured as a function of gate voltage Vo at T = 1.5 K are shown in Fig. 2. The linear conductance:is quantized in units of 2e2/h, after subtracting the series resistance of the ohmic contacts. The noise signal SocVout(f) is measured for different injection voltages, Vm(f)~-(VDs/2)(l'+cos2~ft), imposed on the QPC. As predicted by Eq. (3), keeping Vm conStant allows the, injected current It.(f) to change as the conductance of the QPC varies, leading to a noise dependence proportional to T(1-T).
728
M. Reznikov et aL /Surface Science 361/362 (1996) 726-729
r"
.
.
' /
/ '1
.
.
.
.
1 1
'
1Ma
.
.
.
-
il
I il . / ' .z..',4f,"
I~
-
1 1
\
\ ~
Cold
Warm
Amplifier ,..
I
High Freq.
Amplifier Diode ) (
I
Fig. 1. Experimental set-up. The voltage Vo controls the number of 1D conducting channels in the QPC. The current, modulated at low frequency f, is provided to the QPC via a variable current source, with a voltage Vm(f) appearing across the QPC. A high f~xtuency path to ground is provided via a capacitor C~. 10 0
S
8
,~ ....... %s/my.
~/d>,,\-~ ....... .,,~,.k..\\k~.
-1
2 .
.J. ........
0
O-
I+-ff2. . . .
I -t.2
-1/2
rs-.~Y2~Vos_--.0,~,D.~ .t.1 -t.o .~
.............
Gate Voltage, V G ,IV] Fig. 2. Noise spectral density S(v) and normalized linear conductance G, measured with current 10 nA versus gate voltage Vo. The noise is measured for Vm =0.5, 1, 1.5..... 4 inV. The lines through the mLxima and minima of the peaks are guides to the eye. Pmitions of Ti = 1/2, where the noise m n ~ m n Bre thcoretically expected to be, are marked with arrows. Inset: dependence of the peak height (same scale as in main figure) on injection voltage Vm. 0 , 0 and <> mark the first three conducting ID channels. The straight dashed line is the predicted behavior.
Indeed the measured noise signal, as a function of Vo and for different Vm, exhibits strong oscillations. The magnitude of the peaks (each peak relative to the average of two adjacent minima) grows almost linearly with the current, as seen in the inset of Fig. 2. This linear dependence of the
spectral density also agrees quantitavely with Eq. (3) and is a crucial fact in substantiating the origin of the noise. Even though the magnitude of the peaks agrees rather well with the predicted noise maxima (being (1/4)2e(2e2/h)Vvs) the peak positions (and respective minima) shift from the predicted positions at Vo(T~=I/2) (and T~--1 for the minima) to Vo (higher Tis), a shift that increases with VDS. This shift is not yet understood. As a possible explanation one may consider injection into the upper channel of the QPC mediated by the finite temperature and injection voltage, thus leading to added noise from this channel and to an apparent shift in the position of the peaks. However, the fact that the measured noise signal for relatively small injection voltages drops to almost zero at the minima, implies that Tl=l, Tl+l ~ 0 at the minima, making the above hypothesis questionable. Another possible explanation for the shift is the finite temperature and the fact that TIS energy-dependent (T= HE)) [8,9]. Taking this into account we find that the calculated noise peaks shift toward smaller ~ , contrary to our data. 4. Conclusion
We performed detailed investigations of the noise in the pinch-off region of a QPC, namely, when
M. Reznikov et a£/Surface Science 361/362 (1996) 726--729
all carriers are depleted from the QPC and transport is via tunneling or thermally assisted emission. As the gate voltage becomes more negative and the resistance of the QPC increases, the current and its fluctuations tend to zero. We thus performed noise measurements at a constant current I, expecting to obtain the behavior S--2eI(1-T~), which is nearly classical when T~<<1. Contrary to this, we find that the noise tends to saturate and become independent of the DC current above 100 hA, which is contradictory to the expected classical behavior. Although the voltage Vm across the PC in this r e , m e could be tens of mV this should not be a problem, for the linear r e , m e is not a condition for classical shot noise to occur. Since in the pinch-off r e , me the estimated dwell time in the QPC is similar to eli
729
(the impinging time interval) we suggest that Coulomb repulsion between the electrons leads to temporally correlated transport in the pinched QPC. and thus to noise suppression.
References El"] V.A. Khlus, Soy. Phys. JETP 66 (1987) 1243. 1"2"1G.B. Lesovik, JETP Lett. 49 (1989) 592. I-3"1M. Buttiker, Phys. Rev. Lett. 65 (1990) 2901. E4"1 G.B. Le~vik and L.S. Levitov, Phys. Rev. Lett. 72 (1994) 538. 1-5"1Y.P. Li et aL, AppL Phys. ~ 57 (1990) 774. 1-6"1F. Liefrink, J i Dijkhuis and H. van Houten, Semicond. S~. TechnoL 9 (1994) 2178, and references there~ [7] F. Liefrink et aL Phys. Rev. B 49 (1994) 14066. 1-8] IL Landaner and TIL Martin, Physica B 175 (1991) 167. 1-9"1G~B. Lesovik and R. L o o e ~ Z. Phys. B 91 (1993) 531.