Pumping Heat wit Quantum Ratcets T. E. Humprey a H. Linke ab R. Newbury a a Scool of Pysics University of New Sout Wales UNSW Sydney 5 Australia b Pysics Department University of Oregon Eugene OR 9743-74 USA Email: l@pys.unsw.edu.au P: +6 9385 598 Fax: +6 9385 66 PACS codes: 5.6.+w 73.3.Ad 73.5.Fq 73.5.Lw Keywords: Non-linear mesoscopic quantum ratcet eat pump Abstract We describe ow adiabatically rocked quantum electron ratcets can act as eat pumps. In general ratcets may be described as non-equilibrium systems in wic directed particle motion is generated using spatial or temporal asymmetry. In a rocked ratcet wic may also be described as a non-linear rectifier an asymmetric potential is tilted symmetrically and periodically. Te potential deforms differently during eac alf-cycle producing a net current of particles wen eraged over a full period of rocking. Recently it was found tat in te quantum regime were tunnelling contributes to transport te net current may cange sign wit temperature. Here we sow tat a Landauer model of an experimental tunnelling ratcet [Linke et. al. Science 86 34 (999)] predicts te existence of a net eat current even wen te net particle current goes troug zero. We quantify tis eat current and define a coefficient of performance for te ratcet as a eat pump finding tat more eat is deposited in eac of te two electron reservoirs due to te process of rocking tan is pumped from one reservoir to te oter by te ratcet. Introduction In general a ratcet may be described as a non-equilibrium system in wic directed particle motion is generated troug te use of asymmetry [-3]. Often te non-equilibrium condition is acieved by varying an asymmetric potential wit time. In tis case te asymmetry defines a preferred direction wile te time variation provides te source of energy necessary to create a net current of particles [3]. One suc example is a rocked ratcet in wic an asymmetric potential is tilted symmetrically and periodically. Te potential deforms differently during eac alf cycle of tilting so tat a net current is produced wen te current is eraged over a full period of rocking [4]. Trougout te present paper we will only discuss socalled adiabatically rocked ratcets [5] in wic symmetric tilting occurs on time scales muc slower tan all oter time scales of te system. Suc a system is essentially a non-linear rectifier. In te classical regime a net current can only occur wen te eigt of te potential barrier depends upon te tilt direction. In te quantum regime owever were we reflection and tunnelling can occur not only te eigt but also te sape of te barrier becomes important. In particular narrow barriers will transmit more tunnelling particles tan wide barriers and smoot barriers wic cause less we reflection tan sarp barriers will allow more ig energy particles to pass over te barriers. It ten follows tat a cange in te sape of te asymmetric barrier in a quantum ratcet as a result of tilting is sufficient to produce a net current even wen te barrier eigt remains constant. Tis quantum net current as been found to cange direction wit temperature [56]. Te origin of te temperature dependent beiour is illustrated in Fig. wic sows an asymmetric energy barrier between two-dimensional electron reservoirs. A square-we voltage of amplitude V is applied to rock te potential. Wen tilted in one direction te asymmetric barrier deforms
Fig : A rocked electron ratcet. Te solid lines are an estimation of te confinement potential experienced by electrons as tey trerse te experimental ratcet we-guide sown in Fig (inset). Te Fermi distribution of electrons as a function of energy is indicated by te grey regions were ligter grey corresponds to an occupation probability of less tan one. Te boldness of te arrows is indicative of te relative strengts of te contributions of ig and low energy electrons to te current across te barrier under negative (a) and positive (b) voltages. Te dased lines indicate te spatial distribution of te assumed voltage drop over te barrier wic is scaled wit te local potential gradient of te barrier at zero voltage. to be ticker and smooter (Fig. a) suppressing tunnelling but also reducing we reflection of electrons so fouring transmission of electrons wit ig energies. Wen it is tilted in te oter direction owever te potential deforms to be tinner and sarper (Fig. b) enancing tunnelling but increasing we reflection tus fouring te transmission of low energy electrons. In tis way te two contributions to te net current tunnelling troug and excitation over te energy barrier flow on erage in opposite directions. By tuning te temperature rocking voltage or Fermi energy suc tat one of tese two contributions exceeds te oter te net current direction can be cosen. In te present paper we point out tat at parameter values were te contributions of te two components of te net particle current are equal and opposite (tat is were te net particle current goes troug zero) a net energy current still exists because te erage energy transported in eac direction is not te same. In te following we briefly describe te experimental results of [6] and introduce a Landauer model for tis experiment wic will allow us to quantify te eat current generated by te ratcet. Experimental quantum ratcet A scanning electron microscope (SEM) image of te experimental quantum ratcet device is sown in Fig. (inset). Te darker areas are trences wic were defined by sallow wet etcing and electron-beam litograpy in a two-dimensional electron gas (DEG) AlGaAs/GaAs eterostructure. Tis process created an asymmetric one-dimensional (D) we-guide connecting D electron reservoirs. Te crucial feature of te ratcet is te asymmetric point contact on te rigt wic can be adjusted in widt by applying a voltage to te DEG areas above and below te rigt point contact tat serve as side gates (marked SG in te SEM image). Te side gate voltage tunes te energy of te D we modes effectively creating an asymmetric energy barrier wic is experienced by te electrons as tey trerse te weguide. Te left point contact wic is not influenced by te side gates plays no significant role in determining te beiour of te device as a ratcet. Te dimensions of te device (~ µm) were muc smaller tan te lengt scales for elastic (6 µm) and inelastic (> µm) scattering at te temperatures and voltages used in te experiment (k B T and ev mev). A low-frequency square-we voltage of amplitude V was applied between te two electron reservoirs to adiabatically rock te device and te resulting net current eraged over many periods of rocking was measured using pase locking tecniques. Te direction of te net current was found to depend upon temperature rocking amplitude and te applied gate voltage. In Fig. we sow measurements of te net current versus te amplitude of te rocking voltage for various temperatures at constant side-gate voltage. For small voltages all tree curves display parabolic beiour as expected for a lowest order non-linear effect wic at te lowest two temperatures (.6 K and K) turns over to reverse direction at a rocking
Fig. : Main: Measured net current as a function of rocking amplitude at a number of temperatures as indicated. Reversals in te direction of te net current as a function of rocking amplitude and implicitly as a function of temperature are observed. Data taken from [6]. Inset: A scanning electron microscope image of te ratcet device (top view). Te dark regions are etced trences tat electrically deplete a two-dimensional electron gas located at te AlGaAs/GaAs interface beneat te surface forming a onedimensional we-guide. Due to quantum confinement inside te weguide an electron moving from left to rigt will experience an asymmetric potential barrier similar to tat sown in Fig.. Note te side gates (marked SG) wic are used to tune te eigt of te potential barrier wic is experienced by electrons moving toug te ratcet. Te left point contact does not play a significant role in te beiour of te device as a ratcet. voltage amplitude of V mv. Tese results may be interpreted by referring to Fig. wic illustrates te idea tat ig energy electrons and tunnelling electrons trel on erage in opposite directions. At low rocking voltage and temperature a positive net electrical current is measured (corresponding to a current of electrons from rigt to left in Fig. ) indicating tat tunnelling electrons dominate te net current. As eiter te temperature or voltage is increased te energy range of electrons wic contribute to transport widens resulting in a greater contribution from electrons wit energies iger tan te barrier leading to a negative net current. Wen te two contributions are equal in magnitude te net current undergoes a sign reversal. 3 Te Landauer model Te Landauer equation expresses te current flowing troug a mesoscopic device between two reservoirs as a function of te Fermi distribution of electrons in te reservoirs and of te energy dependent probability tat an electron will be transmitted troug te device [7]. It may be written as: I = e t ( ε V )[ f ( ε V ) f ( ε V )] R L dε () were f ( ε V ) = ε ± ev + exp k BT are te Fermi distributions in te left and rigt reservoirs (te upper/lower symbol in ± in all equations corresponds to te left (L) and rigt (R) reservoirs respectively). ε is te energy of te electrons for convenience cosen relative to te erage of te cemical potentials on te left and rigt sides µ =(µ L +µ R )/ (Fig. ). T is te temperature of te DEG V is te applied bias voltage and e=+.6-9 C. Lastly t(εv) is te probability tat electrons are transmitted across te barrier at a given bias voltage. Eqn. () assumes tat no inelastic scattering occurs inside te device. In addition we require te applied bias to be muc smaller tan te Fermi energy. Tis means tat te difference between te Fermi distributions will be negligible at low energies and allows us to use -µ = -.5(µ L +µ R ) as te lower limit of integration independent of te voltage sign. Non-linear effects wic form te basis of tis particular ratcet effect e been taken into account by solving te D Scrödinger equation to find t ( ε V ) for eac positive and negative bias voltage individually. In order to do tis te energy of te lowest mode of () 3
Fig. 3: Te bold curve (corresponding to te left vertical axis) is te difference between te transmission probabilities for +.5mV and.5mv tilting voltages as a function of electron energy. Te dotted and dased curves (corresponding to te rigt vertical axis) are te Fermi windows f (ε V =.5 mv) centred on an equilibrium Fermi energy µ =.7meV for temperatures of.3k and K respectively. As te temperature is canged from.3k to K note tat te integral of te product of f and t will make a transition from positive to negative leading to a reversal in te direction of te net particle current (Eq. 4). Small oscillations in t exist for ε > mev. te experimental we guide (Fig. inset) was estimated resulting in te energy barrier sown in Fig. (for more details see [6]). Te eigt of te barrier corresponds to te confinement energy for lowest mode electrons at te narrowest point in te constriction. To obtain te barrier sape at finite voltage an assumption about te spatial distribution of te voltage drop needs to be made. Arguing tat a smoot potential variation can be approximated by a series of infinitesimally small steps and tat a step-like potential cange may be assumed to cause a corresponding step-like voltage drop [8] we distribute te voltage drop in proportion to te local derivative of te barrier [9]. It is important to stress tat te qualitative quantum beiour of te ratcet does not depend upon te details of te voltage drop. Te present coice owever as te desirable side-effect tat te barrier eigt remains independent of te sign of te voltage resulting in te suppression of te classical contribution to te net current. In te present model we include only contributions to transport from te lowest we mode. Te contribution from iger modes is qualitatively similar and also negligibly small wen te Fermi energy is approximately equal to te eigt of te barrier. Te net current is defined as te time erage of te current over one period of rocking wit a square-we voltage of amplitude V : I net = [ I ( V ) + I( V )]. (3) Tis can also be written as: e I net = t ( V ) f ( ε V ) ε were f ( ε V ) f ( ε V ) f ( ε V ) dε (4) R L te Fermi window is te difference between te Fermi distributions on te rigt and left of te barrier and gives te range of electron energies wic will contribute to te current. Te Fermi window is centred on ε = and as a widt wic depends upon te t ε V t ε V t ε V also bias voltage and te temperature of te DEG (Fig 3). Te term ( ) ( ) ( ) sown in Fig. 3 is te difference between te transmission probabilities for an electron wit energy ε under positive (Fig. b) and negative (Fig. a) voltages. Electrons wit energies under te barrier eigt are more likely to flow from rigt to left wen te barrier becomes tinner (under positive voltage Fig. rigt) tan from left to rigt wen te barrier becomes ticker (under negative voltage Fig. left). Tis results in t being negative in tis energy range and ten positive for energies above te barrier eigt were above 4
situation is reversed. As f is adjusted (troug canging T V or Fermi energy µ ) to sample te t curve were it is negative rater tan positive te net current will cange sign from positive to negative. 4 Energy current Te eat cange associated wit te transfer of one electron to a reservoir wit cemical potential µ is given by []: Q = U µ (5) Te internal energy U associated wit te electron is taken wit respect to te same global zero as te cemical potential wic is assumed to be uncanged by te electron transfer. Te cange in eat in te two reservoirs upon transfer of one electron from te rigt to te left is ten given by Q L/R = ε + (µ L +µ R )/ - µ L/R = ε ± ev/. Note tat te eat removed from one reservoir by an electron crossing te barrier differs from te eat it adds to te oter reservoir by ev as a result of te kinetic energy acquired by te electron in te electric field driving te current. Te eat current entering te left and rigt reservoirs associated wit te particle current generated by a voltage V across te device is ten obtained from te equation for te electrical current (Eq. ). Tis is done by replacing te electron carge e by a factor of Q L/R inside te integral. Te eat current can ten be written as q = m ( ε ± ev ) t ( ε V ) f ( ε V ) dε (6) Te net eat current into te left and rigt reservoirs over a full cycle of square-we rocking net q = q V + q V is ten: ( )[ ( ) ( )] / = m q net L R [ ( ε ± ev ) t( ε + V ) f ( ε + V ) + ( ε ev ) t( ε V ) f ( ε V ) ] m dε (7) To obtain an intuitive understanding of te action of te ratcet as a eat pump at parameter values were te net particle current goes troug zero it is elpful to rewrite Eq. (7) as: q ev = m ε t fdε + τ fdε = m + Ω net E Here ( ε V ) = ( [ t( ε + V ) + t( ε V )]) τ is te erage transmission probability for an electron under positive and negative bias voltage. E = q L qr is te eat pumped from te left to te rigt sides of te device due to te energy sorting properties of te ratcet and can be non-zero only for asymmetric barriers. Ω = ( V )[ ( ) ( )] I + V + I V is te omic eating eraged over one cycle of rocking. E can be interpreted as te eat pumping power of te ratcet eraged over a period of rocking wile Ω = q L + qr is te electrical power input eraged over a period of rocking. We terefore define a coefficient of performance for te ratcet as a eat pump as: E qr ql χ ( T V ) = = (9) Ω q + q R L (8) 5
Coefficient of Performance (%) Fig. 4: Te eat-pumping coefficient of performance of te ratcet model sown in Fig. plotted as a function of rocking voltage and temperature. Eac point on te surface corresponds to a set of values of rocking voltage temperature and Fermi energy for wic te net particle current goes troug zero. Temperature (K) Bias Voltage (mev) It is important to note tat E will be trivially non-zero wen te net particle current is non-zero because eac electron carries eat. Tis definition of χ terefore only makes sense for parameter values were te net particle current is zero. To evaluate our model potential (Fig. ) in terms of a eat pump we e calculated χ for parameter sets were te net current goes troug zero (Fig. 4). As required te eat pumping power goes to zero for small bias and temperature corresponding to te linear response limit were by definition te ratcet cannot work. Te positive coefficient of performance indicates tat eat is always pumped from left to rigt for te range of parameters used in te calculation. Small oscillations in t at energies iger tan te barrier exist and placing f around tese would result in eat being pumped from rigt to left. Te fact tat χ is small means tat te total eat deposited in eac reservoir due to omic eating is muc larger tan te eat pumped from te left to te rigt sides of te device. Tis means tat despite te eat pumping action of te ratcet te internal energy of bot reservoirs increases but one reservoir is eated sligtly less tan te oter. Te experimental quantum ratcet may terefore be viewed as a poorly designed refrigerator were alf of te waste eat is deposited inside instead of outside te refrigerated region. For one reservoir to be cooled using te ratcet effect E would need to be larger tan Ω (χ > ) so tat more eat was pumped out of one reservoir tan was deposited tere as a result of omic eating. Te low coefficient of performance of te ratcet of [6] as a eat pump is a result of te fact tat te potential barrier studied ere transmits electrons wit a wide range of energies in bot rocking directions (all of wic contribute to eating) wile te ratio t/τ is less tan %. Tus omic eating of eac reservoir greatly exceeds te eat pumped from one side to te oter. Te eat pumping coefficient of performance of te ratcet would be enanced by designing a potential wic only transmitted electrons wit energies iger tan te equilibrium Fermi energy in one direction and only transmitted electrons wit energies lower tan equilibrium Fermi energy in te oter direction so tat t/τ. One way of acieving tis may be to employ resonant tunnelling as a means of energy filtering. Resonant tunnelling barriers e in fact been predicted to be able to cool a reservoir wen operated in DC mode []. An adaptation of tis idea to rocked ratcets is currently under investigation. Tis work was supported by te Australian Researc Council. 6
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