Coulomb blockade and single-electron devices

Transcription

1 oulomb blockade and sinle-electron devices One interestin limit of nanoelectronics are devices with impedances stronly influenced by the positions of sinle electrons. Imaine a transistor where the source-drain current is controlled by the presence or absence of a sinle electron on a controllin ate. The plan: oulomb blockade physics intro Sinle junctions Sinle-island, double-junction systems Sinle-island, double-junction + a ate: the sinle-electron transistor. Many islands and junctions: the electron turnstile oulomb blockade physics As we ve mentioned before, for any classical conductors coupled toether electrostatically, it s possible to define a capacitance that relates the chare on each conductor to the potential between the conductors: QiQ j Q E = i = ijv j j i j ij Two conductors with a mutual capacitance attached to opposite terminals of a battery with voltae V build up a chare Q = V (one +, one -). Work done in charin the capacitor: Q /. That chare does is a polarization chare and does not have to be quantized in units of e.

2 oulomb blockade physics apacitances: (assume embeddin dielectric ε) radius a α = 4πεε a + α , α l α 0 = a l d radius R = εε 0πR d 8εε R 0 d << R d >> R an make systems where charin enery is larer than k B T. Room temperature ~ 3 x 0-8 F. For vacuum, a ~ 8 nm. oulomb blockade physics oulomb blockade occurs when charin enery for movin a sinle electron throuh a system (e.. e / ) exceeds the available (thermal) enery. Ideally no current flows at T = 0 in a fully blockaded circuit. A blockaded circuit element has a definite number n electrons in its blockaded confiuration. Blockade may be lifted / broken if: A sufficiently lare bias is applied, so that ev exceeds the charin enery. An additional capacitively coupled electrode is used to offset the charin enery.

3 oulomb blockade: a sinle tunnel junction R >> h/4e models tunnelin. I Ideal version: chare accumulates and voltae across capacitor ramps up at rate iven by I/ until ev exceeds charin enery e /. At that point it is possible for one electron hops across the tunnel barrier, and the voltae drops back down. ( Q ± e) Q e E = = ( ± Q ) e / imae from P. Delsin thesis oulomb blockade: a sinle tunnel junction imae from P. Delsin thesis Result: voltae across current-biased sinle junction is expected to oscillate as shown, with a typical fruency f = I / e. In practice, this effect is nearly impossible to see: there is always stray capacitance in the leads that dwarfs the junction capacitance. The current source chares up the stray capacitance, and then the system acts like. 3

4 oulomb blockade: a sinle tunnel junction - voltae bias Voltae-biased junction. R >> h/4e V Here no current flows unless V exceeds the charin threshold, V c = e/. This is oulomb Blockade. models tunnelin. I V offset Actual capacitor chare not quantized, b/c it s a polarization chare (can vary continuously by slihtly shiftin all electrons in leads). V c V Shown here is typical IV curve for voltae biased sinle junction (blue). Hih temperature limit = red; zero temperature ideal case = reen. Sinle-island, double-junction One of the most commonly examined cases is that of an island connected to a voltae supply by two junctions: What happens to the current throuh this system as the voltae V a is varied? V n V, R, R V a 4

5 The oulomb staircase +n V V +n We know: Q = V V Q = V a = V + V, R, R For n electrons on the island, Q = Q Q = ne V a If the total capacitance of the island is = +, V = ( Va + ne) V = ( V a ne) Total electrostatic enery: Q E = Q + = s ( Va + Q ) The oulomb staircase Q Es = ( Va + ) When an electron hops off the island, the voltae source must transfer that chare as well as any polarization chare ruired to keep the total voltae drop ual to V a. Suppose an electron tunnels throuh off the island, so that Q = Q+e, and n = n -. The voltae across becomes V = V - e/, so a polarization chare must flow throuh the voltae source to compensate. Total work done to pass n chares throuh junction : W ( s n ) = neva Total work done to pass n chares throuh junction : W ( s n ) = nev a 5

6 The oulomb staircase Total enery of whole circuit after movin n electrons throuh junction and n throuh junction : E( n, n ) = E s = W s ( V a eva + Q ) + ( n + n ) Now, at T = 0, only tunnelin chanes that lower this enery are allowed. Tunnelin a particle throuh chanes the enery by: E ± ( Q ± e) = Q eva ± e e = ( en Va ) Similarly, tunnelin a particle throuh chanes the enery by: e e E ± = ± ( en + Va ) Upper sin = tunnelin electron off island The oulomb staircase Start with a neutral island (n = 0). ombinin the two expressions, E ±, e = eva ± For = =, the enery chane is positive (forbidden) unless V a > e =, e This is oulomb blockade. Suppose one electron has already tunneled onto the island. Then the (Fermi) enery of the island is raised by e /, and no chare can flow until V a crosses a new threshold, 3e/. 6

7 The oulomb staircase The result is the oulomb staircase: Only resolvable when there s a stron asymmetry in resistances, such as R >>R. Then, as soon as an electron tunnels out of junction, one from comes in to maintain n. Voltae drop across for ual capacitance case: Va ne V = + Size of current step: V R I = e R Ruirement on resistances to see these effects: e E t > h R h e ( ) > h R >> imae from Ferry The sinle-electron transistor: sinle-island, double-junction + ate an redo the same kind of electrostatic and stability analysis, but now include ate: Q = ( V V ) Q Q Q Q = ne + = + + Q p +n, R Buckley Prize (000): Fulton, Dolan, Kastner +n V V, R V V = (( + ) V V + ne Q ) a p V = ( V a + V ne + Q p ) V a 7

8 The sinle-electron transistor +n +n E ( ( ) s = Va V + V + Va + Q W s ( n ) = n eva + e( Va V ) Ws ( n = n eva + ev E ± = ) ( Q e) = e Q e ± ( en Q ± p e + ( [( + ) V V ] + ) V Q p accounts for offset chare on or near island. a a V ) E ± = ) ( Q ± e) = e e, R Q ( en Q V V, R ± p V a e V V Gate allows one to effectively vary the chare on the island! [ V + V ] a a V ) The sinle-electron transistor Aain, at T = 0, only transitions that lower the total enery are allowed. Define a new voltae: V ' V + Q / Forward tunnelin then ruires: [ en + ( + ) Va V '] e Backward tunnelin ruires: ± [ en V V '] p > a > Gate voltae can be used to tune system in and out of oulomb blockade! At the riht ate voltaes, the couplin to the ate can offset the enery cost that would otherwise exist for hoppin an electron on or off the island. The ate voltae ruired to chane the conductance is offset from its ideal value if there is excess chare coupled to the island (Q p ). e 8

9 The sinle-electron transistor Gate voltae can be used to tune system in and out of oulomb blockade! Diaram for = =, =. imae from Ferry Shaded reions: no conduction; definite number of chares on island. Points of intersection: system can lower its enery by chanin number of chares on island; conductance peak. Plots like this from experimental data are called diamond plots. Notice that modulatin ate chare by less than one electron can chane from blockaded to conductin conditions. Sinle-electron transistor Ralph et al., PRL 78, 4087 (997) For fixed V a, source-drain conductance is periodic in V, with peaks spaced by e/. Red lines = hih conductance near zero source-drain bias. imae from Ferry Manitude of conductance when not in blockade reime: G max = R + R 9

10 Sinle-electron transistor - finite T Blockade ets smeared out at finite temperature. In case of SET, oulomb blockade peaks (conductance as a function of ate voltae) take on a width in ate voltae (since k B T can make up for extra enery ruired): G G urve calculated for max cosh e kbt = e( / ) ( V V ).5k T B imae from Ferry The R-SET and other variations We ve been discussin -SETs - ate chare capacitively influences island chare. It s also possible to build R-SETs, in which the ate is actually connected (!) to the island via a lare (>> R, R ) resistor. For that case, island chare can truly be chaned by the ate. It s also possible, if done carefully, to substitute lare resistances for the tunnel junctions. This is called a resistive SET - see Krupenin et al., JAP 90, 4 (00): 0

11 The sinle-electron turnstile imae from Ferry More complicted structures can ive rich behavior. One can make a turnstile device that ideally lets throuh only one electron at a time. This particular approach dissipates sinificant power. The sinle-electron pump imae from Esteve This version is quasiadiabatic: By cyclin U and U out of phase with each other, once can pump electrons throuh the two islands with minimal power dissipation. Dominant errors in these approaches due to cotunnelin. an be minimized by increasin number of tunnel junctions.

12 Summary: lassical electrostatics plus slow chare relaxation (tunnelin?) plus (low temperatures or small sizes) can lead to oulomb blockade devices. Sinle junction can show B effects if arraned carefully. Double junction can exhibit oulomb staircase. Additional ates allow sinle-electron transistors, switchable from low (ideally zero) to hih conductance by movin a sinle electronic chare on or off the ate. More complicated arranements permit pumpin of electrons, ideally one at a time. Next time: Implementations of sinle-electron devices Sinle-electron loic: voltae loic Sinle-electron loic: chare loic Problems with industrial implementation

13 Sinle-electronics: the story so far For small enouh structures and low enouh temperatures, oulomb charin effects can determine the conductin properties of circuits. Two-terminal double-junction devices can show complicated (oulomb staircase) IV curves. Multiterminal devices can show transistor-like functionality, with substantial current switchin modulated by the positions of individual electrons. Main driver for possible applications: Need E c = e / >> k B T, so want smallest junctions possible. Sizes of thins: for E c = 50 k B T at 300 K, need ~ 0. af. Implementations of sinle electron devices Several different ways of makin SETs: Shadow-evaporation + oxidation of Al Most common approach, best suited for lare-scale fabrication of arrays. Oxidation of silicon ompatibility / ease of interation with Si hemically-aided approaches Trapped nanoparticles; AFM contactin; molecular devices.

14 Shadow evaporation and oxidation of Al Takes advantae of ease of rowth of thin, hih-quality Al O 3 for tunnel barriers. Uses eometry to make smallest possible junctions. Double-layer resist for e-beam lithoraphy leads to overhan. source island drain Evaporate at an anle. Then oxidize for controlled lenth of time. Evaporate straiht down - forms first junction. Then oxidize for controlled lenth of time. Third evaporation forms second junction.. Shadow evaporation and oxidation of Al Lateral size of junction overlap determined by resist thicknesses (well-controlled) and anles (also well-controlled). By tiltin in different directions, can make complicated structures: imae from PTB, Germany imae from KTH, Sweden

15 Shadow evaporation and oxidation of Al d Josephson junction array imae from Mooij, Delft, Netherlands Inverter from two coupled SETs imae from Mooij, Delft, Netherlands SET on tip of drawn lass fiber imae from Bell Labs Local oxidation imaes from Matsumoto et al., APL (996). Recall tunnelin transistor: local electrochemical oxidation (anodization) used to convert continuous Ti strip into island + insulatin tunnel barriers. Painstakin fabrication, but payoff is SET with some roomtemperature functionality. 3

16 Oxidation of Silicon imae from Steve hou, Princeton Oxidation of Silicon imaes from Steve hou, Princeton Even for an island as small as this, ettin room temperature oscillations is a real challene. 4

17 Trapped nanoparticles imaes from Dekker, Delft One approach: use chemical fabrication to make nanoparticles for use as SET islands. Trap particles between lithoraphically created electrodes. Electrostatic trappin in above nanoparticle drawn to reion of hih local electric field as source/drain are biased. Trapped nanoparticles imaes from Park et al., APL 75, 30 (999). Alternative: Start with continuous metal electrodes on top of insulated metallic substrate, to be used as a ate. Dust surface to decorate with nanoparticles, such as chemically synthesized dse nanocrystals. Break into separate source-drain electrodes by electromiration, and sometimes nanocrystal ends up ideally positioned to act as island. 5

18 Nanoparticles on surfaces an decorate surface with tethered nanoparticles, in this case Au colloid. Insulatin layer = selfassembled monolayer of thiolterminated alkane chains. Scanned probe microscope tip as drain electrode: oulomb staircase. imaes from Andres et al., Science 7, 33 (996). Nanoparticles on surfaces Special tip can incorporate ate, too, for SET action. imaes from Gurevich et al., APL 76, 384 (000). 6

19 lear technoloical challenes: Reliable fabrication of sub-0 nm structures with little or no variation for room temperature oulomb blockade physics. Tunin of environmental characteristics such as stray capacitance. Reliable (self-controllin?) tunnel junctions. ontrol of sinle electronic offset chares: individual chared defects can have effects identical to random offset voltaes on ates! Incentives: Dense interation. Possible ultralow power operation (reversible?). Ultimate limits of switchin technoloy. Technoloy possibilities Voltae-based loic: Very similar to typical MOS loic: hih voltae = loic hih; low voltae = loic low. Nonmonotonic drain current as function of ate voltae opens up desins not possible with reular MOSFET devices. hare-based loic: Since SETs can sense presence/absence of sinle electronic chares, use positions of chares to represent data / loic values. Some novel architectures possible, since chare positions can cause alter voltaes capacitively, which can then move chares: e.. Quantum ellular Automata. 7

20 Voltae-based loic: It is possible, with SETs, to achieve voltae ain. That is, the output voltae modulation of a circuit can be reater than the input voltae modulation: Vout KV V G input voltae output voltae imaes from Zimmerli et al., APL 6, 86 (99). Voltae-based loic: ircuit desiners can treat SET-based loic ates like conventional loic ates (keepin current desins) and leave the details up to the hardware folks. Good point: SET characteristics (oscillatory response to V G ) mean that one SET can sometimes replace more than one reular MOSFET. p-type n-type MOS inverter: transistors of opposite types SET inverter: SET + resistive load. 8

21 Voltae-based loic: Problems: Analysis shows that, for acceptable device performance, either temperatures must be very low, or devices must be extremely small. e k B T ~ 0.0 This can be mitiated slihtly (~factor of 3) by usin arrays of tunnel junctions, but at cost of lower packin density and more challenin fabrication. Off -state leakae leads to power consumption problems comparable to hihly-scaled MOS. hare-based loic: Try to use positions of individual chares to represent s and 0 s. Have switchin action depend on chare state of device (necessary for loic). Potential advantae: does not ruire substantial current flow for operation. Power consumption could therefore be advantaeous. Tricky problem: sinle chare errors now become very important. 9

22 hare-based loic: One approach analyzed in detail = SET parametron. Has series of islands with middle islands asymmetrically coupled. lock sinal pushes electrons alon chain. Summary of problems: Temperature rane of operation places reat restrictions on device sizes. Such small devices are likely to suffer tremendous variability. Further, (especially in semiconductor devices) quantum level spacin effects (which superpose on the charin eneries and alter the heihts of the blockade conductance peaks) are likely to be sinificant. Backround chares are also a serious problem. Some chance that this may be self-correctin at sufficiently tiny scales, but not at all clear. haracteristic resistances (~ 0 kω) mean that for realistic capacitances these devices are unlikely to operate at hih speeds. 0

23 Pronosis: Sinle electron devices are unlikely to be the technoloy that replaces MOS at the few nm scale, unless there is a major breakthrouh in fabrication, performace, or architectures. SEDs more likely to find applications in niche areas: Nonvolatile memory Electrometry Metroloy standards Next time: Applications of sinle electron devices apacitance standard Electrometers - the scannin SET Thermometry Intro to the rf-set

24 Applications of sinle electron devices urrent and capacitance standards Electrometers - the scannin SET Thermometry Intro to the rf-set Metroloy We ve established that, for a variety of reasons, SEDs are unlikely to become mass-fabricated substitutes / replacements for MOSFETs in consumer technoloy. What are they ood for? Niche applications like metroloy. Gadets that are challenin to fabricate and ruire moderately extreme conditions to operate are common in precision metroloy: Atomic clocks (UHV, temperature control) Quantum Hall resistance standard (low T, hih B)

25 urrent and capacitance standards Primary standards for electrical units exist for voltae and resistance: Voltae = Josephson effect Superconductor-insulator-superconductor junction. Applied ac current at fruency f produces dc voltae steps of size nhf / e. onstant: GHz/V Resistance = inteer quantum Hall effect d system, hih manetic field. Hall resistance is quantized to fractions of h/ne. onstant: Ω urrent and capacitance standards A primary current standard would allow testin the metroloy trianle : Quantum Hall R H = h / ne I sinle-electron tunnelin I = ef V f Josephson effect V = nhf / e These quantities had better all be consistent with one another, or there s new physics somewhere. Effectively would provide another check on the value of the fine structure constant.

26 urrent standard: SET pump imae from Zorin presentation urrent standard: SET pump Recall the basic idea here. yclin voltaes around one of the triple points in the diaram at the lower riht should transfer a sinle electron (on averae) throuh the pump, riht to left. (Thus a unidirectional current from left to riht.) imae from Esteve 3

27 urrent standard: SET pump Actual experimental data on - ate pump arees with this. urrent is nonzero only in vicinity of triple points. Three junction two island pump errors in experiment: ~ %. Error sources: thermal hoppin missed tunnelin events cotunnelin imae from Zorin presentation, PTB urrent standard: SET pump One approach to dealin with these errors: more junctions! State-of-the-art: NIST seven-junction six-ate pump. Really desined for low fruency work - electron countin rather than current standard. Errors with this set up: ~.5 x 0-8. M.W. Keller et al., Science 85, 706 (999) Much better, but very challenin. 4

28 apacitance standard Even slow pumpin of electrons can be very useful. apacitance standard: Make a capacitor. Pump a precisely known number of electrons onto the capacitor (e.. with the 7-junction pump). Measure the capacitor voltae precisely. Q = V ives the capacitance. M.W. Keller et al., Science 85, 706 (999) apacitance standard Here is the measured voltae chane after movin electrons on and off the capacitor repeatedly. The reproducibility is very ood, and the NIST team measured their little cryoenic capacitor to be = pf. Biest problem: lon term stability due to motion of offset chares in Al O 3 tunnel barriers. an be improved by workin with Si SETs instead. 5

29 Electrometry urrent flow throuh SET can be modulated between maximum and minimum values by movin a sinle electron off and on the ate: One possible eneralization: have the island be a moveable probe! As the island capacitively couples to test objects, its polarization chare Q p chanes. This shifts the G vs. V G plot at riht. From our SET analysis, V 0 ' V0 + Q p / G G max cosh 0 e( / ) ( V V ).5k T B imae from Ferry Electrometry Basic idea: Bias ate voltae to point where G vs. V G is most rapidly varyin. Apply a small source-drain voltae to measure G. Then chane the chare distribution near the island. Small chanes in V 0 lead to lare chanes in measured source-drain current. Sensitivity limits possible: < 0-5 e/hz / 6

30 Scannin SET electrometer One adaptation of this is to place the island on a movable tip, and scan it over a surface. Result: the SET scannin electrometer (SETSE). Imae from Bell Labs Scannin SET electrometer SETSE is easily sensitive enouh to see surface chare fluctuations caused by individual dopant atoms in semiconductor. Problems: slow fraile painful to fabricate ruires quite low T. Imae from Bell Labs 7

31 oulomb blockade thermometry Remember that a sinle junction (or for that matter, an array of junctions), when voltae biased, leads to an IV curve that looks like: I di/dv V offset V c V V The theory has been done for what this zero-bias anomaly looks like as a function of temperature for a d array of tunnel junctions. oulomb blockade thermometry slide from Pekola et al., NRS, Grenoble 8

32 oulomb blockade thermometry an improve reliability by havin parallel d arrays. d arrays also work, with essentially identical function for ZBA width (thouh hih temperature corrections are different). Work from 30 K down to < 0 mk, with basically no B dependence and comparatively low power dissipation! imae from Bersten et al., halmers, Sweden Intro to rf-set Our analysis of SETs has all been done at dc. Is it possible to use SET at hih fruencies, also? Yes. One approach: Reflection rf-set Reflection rf-set Schoelkopf et al., Science 80, 38 (998). 9

33 Intro to rf-set Basic idea: rf impedance of SET also chanes periodically in ate voltae. Make resonator, send in known rf power, and monitor chanes in reflected power as impedance mismatch is varied by SET impedance chanes. Since rf carrier can be at a hih fruency (.7 GHz), it should be possible to see rapid amplitude modulations (MHz). Sensitivity can still be incredibly hih. Schoelkopf et al., Science 80, 38 (998). Usin rf-set electrometer, Prof. Rimber can watch, in real time, as individual electrons tunnel on and off a quantum dot! Summary: While SETs are unlikely to replace reular FETs for a number of reasons, they can find excellent utility in niche applications, particularly metroloy. Sinle electron pumps can be used for fundamental physics tests + definin current and capacitance standards. SET electrometers can be incredibly sensitive, allowin unprecedented precision measurements of surface potentials and chares, possibly even in real time. oulomb blockade devices can also be used as precision thermometers. 0

34 Next time: Monday: Prof. Rimber: The rf-set and real-time observations of sinle-chare motion in a quantum dot (!) Wednesday: Prof. olvin: interfaces and thermodynamic stability of nanoparticles Friday: Molecular electronics and sinle-electron transistors