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the transport properties of single electron transistors ( sets ) have been extensively studied theoretically and experimentally.@xmath0 the main structure of sets consists of a single quantum dot ( qd ) and three electrodes ( source , drain and gate ) .
the manipulation of sets is based on coulomb blockade effect arising from the particle interactions of qd . from the practical point of view , it is important for sets to operate at room temperature .
therefore , the size of qd is required to be less than @xmath1 for silicon ( si ) or germanium ( ge ) semiconductor qds . in such size range of qd , the quantum confinement effects and charging energies of qds are larger than the thermal energy of room temperature .
recently , the coulomb oscillation and staircase features of tunneling current of room temperature sets have been reported by several groups , where si or ge qds are embedded into @xmath2 matrix@xmath3 .
even though it is difficult to align a single qd of nanometer with electrodes in the fabrication of individual sets , several methods were used to solve this difficulty .
nevertheless , those methods still can not precisely control the barrier thickness , which significantly influences the tunneling time for electrons to access qds .
further understanding for the location of qds , the implementation technology of sets using thermal oxidation method@xmath4 can be improved .
the measurement of transient current can provide above information . in this study
we apply the current formula of ref .
[ 2 ] to examine the transient current of the set with asymmetrical tunneling rates at room temperature .
it is found that the transient current exhibits exponential decay when electrons are injected into the qds from the left electrode and @xmath5 , where @xmath6 represents the tunneling rate for electrons from the left ( right ) electrode to the qd , on the other hand the transient current exhibits the exponential growth as @xmath7 .
in addition to theoretical studies of transient current , we report the experimental measurement of transient current of single hole transistor ( sht ) .
the transient current of a set consisted of a single qd and three electrodes was theoretically derived by jauho , wingreen and meir.@xmath8 when applied voltage is insufficient to overcome the charging energies arising from electron - electron repulsion interactions in the qd , the time - dependent tunneling current through the ground state of qd can be expressed as ( ref.[2 ] ) @xmath9,\ ] ] where @xmath10 and @xmath11 according to eq .
( 1 ) , there are four components for the net current from the left electrode to the right electrode .
the current of eq .
( 2 ) , @xmath12 , denotes the current flowing out from the qd to the left ( right ) electrode .
this current results from the time - dependent electron occupation number of qd , which is given by @xmath13 @xmath14 denotes the current flowing into the qd from the left ( right ) electrode .
obviously , @xmath14 is determined by the time - dependent spectrum function @xmath15 where @xmath16 means taking the imaginary part of @xmath17 .
notations @xmath18 and @xmath19 denote , respectively , the electron charge and plank s constant . for the simplicity
, we assume that the tunneling rates @xmath6 are bias and energy independent .
@xmath20 and @xmath21 denote , respectively , the tunneling rates from the left and right electrodes to the qd .
@xmath22 is the fermi distribution function of the left ( right ) electrode .
the chemical potential difference is related by the applied voltage @xmath23 . from the results of eqs .
( 3 ) and ( 4 ) , the time - dependent spectrum function plays a crucial role in the determination of time - dependent tunneling current .
the system studied here considers that the step - like modulation is added into the gate electrode , the expression of @xmath17 is obtained as    @xmath24\\ \nonumber & \cdot & exp^{i(\epsilon-(e_1-e v_g)+i\frac{\gamma}{2 } ) t},\end{aligned}\ ] ]    where @xmath25 is the ground state energy level of qd .
@xmath26 . in the absence of gate voltage
@xmath27 the @xmath28 becomes time - independent retarded green s function @xmath29 .
the resonant energy level @xmath25 is shifted to @xmath30 , that is @xmath31 , when the system goes into steady state . in the transient process
the density of states of qd depends on time .
consequently , @xmath32 displays time - dependent behavior in the transient process .
due to the complicate spectrum function , the transient current lacks analytic form . to numerically calculate @xmath32
, we set the fermi energy level of electrodes and the ground state energy level @xmath33 and @xmath34 , respectively .
therefore , the energy levels of qd are empty at zero temperature under zero bias .
first of all , we plot the time - dependent tunneling current through a symmetric double - barrier tunneling structure with @xmath35 and @xmath36 in response to a step - like modulation for different applied voltages at zero temperature in fig . 1 : solid line @xmath37 , dashed line @xmath38 and dotted line ( @xmath39 ) .
the tunneling current is zero at @xmath40 since the fermi level of electrodes is below the resonant state @xmath25 .
when the gate voltage is added into the system , electrons from the left electrode are injected into the new resonant energy level @xmath41 .
consequently , the tunneling current jumps instantly and finally reaches the steady state .
current also displays an interesting ringing behavior , which was pointed out in ref.[2 ] .
the oscillations of the curve do not maintain constant frequency in the transient process .
in addition , the frequency of oscillations is increased , when the separation between the fermi energy of left electrode and the resonant level @xmath25 is decreased . due to symmetry tunneling rates ,
the behavior of @xmath32 can be understood by the analysis of @xmath42,where    @xmath43\\ \nonumber & \cdot & cos((\epsilon - e_1+ev_g)t)exp^{-\frac{\gamma}{2}t}\},\end{aligned}\ ] ]    and @xmath44\cdot sin((\epsilon - e_1+ev_g)t)exp^{-\frac{\gamma}{2}t}\}.\end{aligned}\ ] ] we plot @xmath45 ( dashed line ) and @xmath46 ( dotted line ) in fig .
2 . @xmath45 exhibits the behavior of @xmath47 , but @xmath46 exhibits the oscillation . owing to the fermi energy of right electrode away from @xmath41 , we see small contribution from the @xmath48 . from the results of fig .
2 , the oscillation of the solid curve shown in fig . 2
is consisted of two frequencies @xmath49 and @xmath50 , which are , respectively , given by @xmath46 and @xmath48 .
( 7 ) reveals that the frequency of oscillation depends on not only @xmath51 , but also @xmath27 . to demonstrate it
, we show @xmath32 for different applied gate voltages and @xmath52 in fig .
3 : solid line ( @xmath53 ) , dashed line ( @xmath54 ) , dotted line ( @xmath55 ) and dot - dashed line ( @xmath56 ) .
we see that the frequency of oscillation can be tuned by the magnitude of applied gate voltage . for @xmath56 ,
the fermi energy level is just aligned with @xmath41 , the oscillatory behavior of tunneling current almost vanishes .
although the charge density of qd is not be directly measured , we plot @xmath57 in fig .
4 , where the curves correspond to those of fig .
it is very clear that qd is empty before the gate voltage turns on .
when the system reaches the steady state , the value of @xmath57 is fractional due to the result of opened system .
it is unexpect that @xmath57 does not exhibit very manifest oscillation feature as well as @xmath32 .
because the width of tunneling barrier could not be precisely controlled in the fabrication of sets , we attempt to study the tunneling current for the case of asymmetrical tunneling rates .
we show @xmath32 for different tunneling rate ratio in fig . 5 : the curves from the bottom to the top ( for @xmath58 )
correspond to @xmath59 and @xmath60 , respectively .
meanwhile @xmath61 .
note that other parameters used in fig .
5 are the same as those of the solid line of fig .
1 . it is worthy noting that the maximum steady current occurs for the symmetry tunneling rates .
this is consistent with other theoretical reports@xmath62 .
in addition , the steady current of @xmath63 ( @xmath64 ) is the same as that of @xmath65 ( @xmath66 ) .
this indicates that the formula given by eq .
( 1 ) maintains the asymmetry of @xmath67 and @xmath68 .
it is not avoided for high temperature sets to study the temperature effect on the transient current .
figure 6 shows @xmath32 with symmetry tunneling rates for different temperatures at @xmath52 and @xmath53 . the ringing behavior of @xmath32 is swept out by temperatures . in addition , the magnitude of @xmath32 is suppressed with increasing temperature since the right electrode provides current @xmath69 into the qd , which is opposite to @xmath70 . consequently , the net current becomes small with increasing temperature .
we also show the charge density @xmath57 in fig . 7 :
solid line ( @xmath71 ) and dashed line ( @xmath72 ) .
we see that @xmath73 approaches the same value for @xmath74 and @xmath71 in the steady state . to understand this feature , we plot @xmath75 and @xmath76 , which are defined as the charge density provided from the left and right electrodes , respectively . in the steady state ( or a large time ) , @xmath75 declines with increasing temperature , on the other hand @xmath76 increases with increasing temperature .
we see that they compensate in each other .
next , we study the transient current for the case of different tunneling rate ratio @xmath77 at room temperature @xmath71 . comparing with the results at zero temperature ,
the oscillation structure of @xmath32 at room temperature is more complicate due to the enhancement of @xmath48 .
for @xmath63 and @xmath65 , the former exhibits the exponential decay , the latter exhibits the exponential growth .
this indicates that the location of qd exists a considerable effect on the transient transport properties of sets .
even though the above theoretical analysis is for sets , we attempt to report the experimental measurement of the transient current of ge sht .
for multivalley conduction band of germanium semiconductors , the intervalley interaction effect can not be ignored . however , such effect is not included in eq .
the more realistic system described by eq .
( 1 ) is a ge sht , where valence band is a single valley . in figure 9
we show the transient current of ge sht for two different applied voltages @xmath78 and @xmath52 at room temperature and @xmath79 .
the detailed fabrication process of ge shts and measurement technique of transient current will be discussed in elsewhere@xmath80 .
the solid lines are the experimental curves .
the dashed lines are the fitting curves . according to fitting curves ,
the transient currents display exponential growth feature .
based on previous theoretical analysis , the ge sht has the characteristic of @xmath81 .
two fitting curves also provide the bias - dependent tunneling times @xmath82 and @xmath83 for the applied bias @xmath78 and @xmath52 .
this very long tunneling time is arising from a fact of very high barrier of @xmath2 which the ge qd is embedded into .
besides , we observe the small oscillation structures on the exponential growth curves .
the oscillation structures vanish for sufficient long time .
this indicates that this oscillation structures are not the measurement error .
owing to the lack of information about the detailed size and shape of qd , there are some difficulties to do the detailed comparison between the theoretical calculation and experimental measurement .
in this study the transient current of a set is investigated by using the formula derived by jauho , wingreen and meir.@xmath8 the tunneling rate ratio exists a considerable influence on the transient current of sets .
the tunneling current of a single ge sht at room temperature displays the oscillation structures in the transient process .
this feature is attributed to time - dependent density of states of qd in a switching on the gate voltage .
1 . transient current through the symmetry barrier for different applied voltages at zero temperature and @xmath53 : solid line ( @xmath52 ) , dashed line ( @xmath84 ) and dotted line ( @xmath78 ) , where @xmath85 and @xmath86 .
transient current through the symmetry barrier for @xmath53 , @xmath52 and zero temperature .
solid line is duplicated from that of fig .
dashed line and dotted line denote @xmath87 and @xmath88 .
dash - dotted line represents @xmath89 .
transient current through the symmetry barrier for different applied gate voltages at zero temperature and @xmath52 : solid line ( @xmath53 ) , dashed line ( @xmath54 ) , dotted line ( @xmath55 ) and dash - dotted line ( @xmath56 ) , where @xmath90 and @xmath86 .
transient current for different tunneling rate ratio at zero temperature , @xmath52 and @xmath91 : the curves from the bottom to the top ( for @xmath92 ) correspond , respectively , @xmath93 and @xmath60 .
meanwhile , @xmath61 . here @xmath90 and @xmath86 .
transient current through the symmetry barrier for different temperatures at @xmath52 and @xmath53 : solid line ( @xmath94 ) , dashed line ( @xmath95 ) , dotted line ( @xmath96 ) and dash - dotted line ( @xmath71 ) , where @xmath90 and @xmath86 .        fig
transient current of ge single hole transistor for different applied voltages at room temperature and @xmath97 .
the curves from the bottom to the top correspond to @xmath78 and @xmath98 , respectively .
the transient tunneling current of single electron transistors ( sets ) is theoretically investigated . the time - dependent current formula given by jauho , wingreen and meir [ phys . 
 rev . 
 b 50 , 5528 ( 1994 ) ] is applied to study the temperature effect on the transient current through a single quantum dot embedded into asymmetry barrier . 
 it is found that the tunneling rate ratio significantly influences the feature of transient current . 
 finally , the oscillation structures on the exponential growth transient current of single hole transistors composed of germanium quantum dots is analyzed .    
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