Semiconductor Quantized Voltage Source

Abstract

We realize and investigate an all-semiconductor quantized voltage source which generates quantized
output voltages V_out = f (h / e) linked only to two fundamental constants, the electron’s charge e and
Planck’s constant h, and to an applied excitation frequency f. The device is based on an integrated
quantized circuit of a single-electron pump operated at pumping frequency f and a quantum Hall device
monolithically integrated in series. Robust output voltages up to several �V are generated, which are
expected to be scalable by orders of magnitude using present technology. The device might open a new
route towards the closure of the quantum metrology triangle.

Full text

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Semiconductor Quantized Voltage Source
F. Hohls, A.C. Welker, Ch. Leicht, L. Fricke, B. Kaestner, P. Mirovsky, A. Mu ¨ller, K. Pierz,
U. Siegner, and H.W. Schumacher*
Physikalisch-Technische Bundesanstalt, Bundesallee 100, 38116 Braunschweig, Germany
(Received 17 June 2011; published 31 July 2012)
We realize and investigate an all-semiconductor quantized voltage source which generates quantized
output voltages Vout¼ fðh=eÞ linked only to two fundamental constants, the electron’s charge e and
Planck’s constant h, and to an applied excitation frequency f. The device is based on an integrated
quantized circuit of a single-electron pump operated at pumping frequency f and a quantum Hall device
monolithically integrated in series. Robust output voltages up to several ?V are generated, which are
expected to be scalable by orders of magnitude using present technology. The device might open a new
route towards the closure of the quantum metrology triangle.
DOI: 10.1103/PhysRevLett.109.056802PACS numbers: 73.23.?b, 73.43.Fj
The Josephson effect [1] in superconductors links a quan-
tized output voltage Vout¼ fðh=2eÞ to the two natural
constants, the electron’s charge e and Planck’s constant h,
andtoanappliedexcitationfrequencyf [2].Ithasimportant
applications in electrical quantum metrology where the
electrical units are directly derived with high precision
from the universal natural constants e and h [3]. Also,
semiconductorsareappliedinelectricalquantummetrology,
making use of the Hall resistance quantization in units of
h=e2[4]. Other electrical quantization effects occurring in
semiconductor devices are, e.g., one-dimensional conduc-
tance quantization [5,6], single-charge quantization [7],
and single-charge-based current quantization in units of ef
where f is the frequency of a suitable ac drive voltage [8].
However, despite the well-elaborated semiconductor device
technology, the generation of quantized voltages by a semi-
conductordevicehasnotbeenobtaineduptonow.Quantized
voltagegenerationinasemiconductorbyaneffectotherthan
the Josephson effect could provide an independent quantum
voltage standard to compare Josephson-based voltage quan-
tization to. It could thus allow an independent universality
test of voltage quantization, which is a cornerstone of elec-
trical quantum metrology.
Here, we report a semiconductor quantized voltage
source generating quantized voltages Vout¼ fðh=eÞ upon
application of an ac drive voltage with frequency f. The
deviceconsistsofanintegratedquantizedcircuit(IQC)ofa
nonadiabatic single-electron pump [9] and a quantum Hall
devicemonolithicallyintegratedinseries.Robustoperation
with output voltages of several ?V corresponding to oper-
atingfrequenciesofafewGHzisdemonstrated.Theoutput
voltage is expected to be scalable by orders of magnitude
using present technology. A direct comparison with metro-
logical precision of the output voltage of the IQC to a
Josephson voltage could allow an alternative approach to
theclosure ofthequantummetrologytriangle[10]andthus
an independent validation of the foundations of electrical
quantummetrology.Byusingeitherthefunctionalityofthe
complete IQC or the functionalities of its two components,
it could further serve as a universal electrical quantum
reference, allowing us to link the three most relevant elec-
trical units of voltage, current, and resistance to the funda-
mental constants e and h in a single device.
The quantum Hall effect (QHE) in two-dimensional
semiconductors quantizes the relation between the applied
currentI andthegeneratedHallvoltageVHbyVH¼ I1
where the filling factor ? is an integer determined by the
applied magnetic field and the carrier density of the Hall
bar. Its precision quantization in different material systems
like Si [4], GaAs [11], and graphene [12–14] provides one
ofthemoststringenttestsoftheuniversalityofthisrelation.
Likewise, generation of quantized currents I ¼ nef has
been obtained in different metallic [15–18] and semicon-
ducting systems [9,19–21]. As opposed to the purely quan-
tum mechanical Josephson effect and the QHE, quantized
current generation generally relies on ‘‘classical’’ electron
charge quantization in a nanostructure with large Coulomb
energy anda precise controlofthe time-dependentelectron
transfer probabilities. Up to now, components, which real-
ize these different concepts, have generally been viewed
as separate entities. However, by using devices based on
the same material system, their integration into an IQC
becomes a straightforward option to realize new quantized
output functionalities.
The IQC under investigation consists of a high current
nonadiabatic quantized charge pump [9] and a QHE resis-
tor in series, as illustrated in Fig. 1(a). Both devices
are fabricated from a modulation doped GaAs=AlGaAs
heterostructure containing a two-dimensional electron
gas (2DEG) situated 95 nm below the surface with carrier
concentration Ne¼ 2:1 ? 1015m?2and electron mobility
of 97 m2=Vs at 4.2 K. More details on the layer sequence
of the heterostructures and the fabrication process can be
found in the Supplemental Material [22]. Figure 1(b)
shows a scanning electron micrograph of a typical non-
adiabatic single-electron pump. It consists of a 900 nm
?
h
e2,
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wide etched constriction in the heterostructure crossed
by three narrow metallic top gates G1-G3with 100 nm
width and 250 nm center spacing. The 2DEG channel of
the pump and the Hall bar are defined by electron-beam
lithography and wet etching. The Hall bar has lateral
dimensions of 1000 ?m ? 600 ?m. Alloyed AuGeNi
contacts to the 2DEG and TiAu top gates are defined
by electron-beam lithography and lift-off.
Negative dc voltages V1and V2are applied to G1and
G2, respectively, to define two electrostatic barriers to the
source and the drain and a quantum dot (QD) in between.
G3is not used and is grounded. To induce pumping, an
additional sinusoidal ac gate voltage component Vacwith
drive frequency f and ac power Pacis applied to G1. The
principle of pumping is described with respect to Fig. 1(c),
where the potential landscape induced by G1and G2is
sketched during different phases of the pumping cycle. In
the initial phase, (i) the left barrier to the source is low
and highly transparent. The QD is large and is filled by
electrons from the source. Then, (ii) the left barrier rises
and becomes more opaque. At the same time, the QD
becomes more confined and the bound QD states are lifted
above the chemical potential ? of the source and the drain.
During this so-called decay cascade phase, the QD levels
of electrons with higher charging energy are still strongly
tunnel-coupled to the source. Hence, these excess electrons
tunnel back to the source with tunnel rates ? ? f much
higher than the pumping frequency f, resulting in a high
fidelity initialization of the dynamic QD with n electrons
[23]. In the sketch, this is shown for n ¼ 1. Finally,
(iii) due to further rise of the QD levels, the barrier to the
drain (right) becomes highly transparent with tunnelling
rate ? ? f and the previously captured n electrons are
reliably ejected to the drain (right) before the cycle starts
over again. Under continuous ac excitation, the device
generates a quantized current I ¼ nef, where n can be
controlled by the dc gate voltages V1and V2.
Figure 1(d) shows a typical operation curve of such a
pump. The datawere taken at a temperature of T¼350mK
in a perpendicular magnetic field of B¼4:2T. Mea-
surement parameters are V1¼ ?0:325 V, f ¼ 675 MHz,
and Pac¼ ?16 dBm. The pumped current (orange circles)
is plotted as function of V2. For these parameters, well-
defined and robust pumping plateaus appear for I ¼ nef
with n ¼ 1;2. With increasing V2(towards the right), the
height of the drain barrier decreases, the QD becomes
larger,and then > 1QDenergylevelsarelowered,leading
to reduced tunnel rates ? to the source. Hence, the
number n of electrons that are kept in the dot during the
initialization phase increases stepwise with V2, leading to
the observed current steps. Recent traceable precision
measurements [24,25] of similar devices have confirmed
current quantization of I ¼ ef within a measurement
uncertainty down to 1 ppm. Note that in these experiments
the uncertainty of 1 ppm was limited by the traceable
measurement setup [25] and not by the intrinsic uncertainty
ofthepump.Theoptimumtheoreticallimitoftheuncertainty
ofcurrentquantizationcanfurtherbeevaluatedbyfittingthe
measuredIðV2Þ curves tothe so-calleddecay cascade model
for QD initialization [23]. Note that the above precision
measurements [25] have revealed a very good agreement
of this model and the experimental data. Also, our data of
Fig. 1(d) can be well described by a fit to the decay cascade
model (black line). From the model parameters (see the
SupplementalMaterial[22]),aminimumcurrentuncertainty
of about 10?8is derived for the given operation parameters.
The given device should therefore be well suited for metro-
logical applications of an IQC.
Note that the current quantization of such pumps is
robust against variations of the applied dc and ac gate
voltages, against an applied bias voltage, and against
high magnetic fields [26,27]. The latter two properties
are very important for the IQC under investigation.
V2(V)
-0,30-0,27
0
1
2
3
exp
fit
I / (ef)
(a)
µ
µ
(i)
(ii)
(iii)
(c)
(d)
1 µm
G1G2G3
(b)
5010
0
30
Rxy
Rxx
Rxy(k )
B (T)
0
2
Rxx (k )
(e)
= 1
= 2
FIG. 1 (color online).
(gray): semiconductor 2DEG; raised outer parts (orange): ohmic
contacts; gates marked G1–G3 (yellow). (b) Scanning electron
microscope image of a pump. (c) Sketches of pumping operation
(i)–(iii). (d) Orange circles: pumped current I vs V2, f ¼
675 MHz.
V1¼ ?0:325 V,
T ¼ 350 mK. Black line: fit to decay cascade model. (e) Rxy
(upper orange line) and Rxx(lower black line) of Hall bar vs B
for I ¼ 1 ?A, T ¼ 350 mK.
(a) Device scheme. Light inner part
Pac¼ ?16 dBm,
B ¼ 4:2 T,
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When the pump drives a current through a resistive load,
the voltage drop across the load will result in an effective
bias across the pump. The bias robustness therefore en-
ables serial integration with a high resistive quantum Hall
device. Additionally, the magnetic field robustness enables
quantized current generation while the Hall device is
operated in the QHE regime.
Figure 1(e) shows the Hall resistance Rxyand the longi-
tudinal resistance Rxxas functions of the applied perpen-
dicular field B. The data are acquired at T ¼ 350 mK with
a measurement current of I ¼ 1 ?A. Rxyreveals the well-
known Hall quantization with quantization plateaus of
Rxy¼ ??1ðh=e2Þ at integer filling factors ? ¼ 1;2;3;....
The most robust resistance quantization occurs for ? ¼ 1
and 2 for vanishing Rxxaround 8.4 and 4.2 T, respectively.
Here, optimum operation of the quantized voltage source
can be expected.
When operated at these fields, the IQC consists of
a single-charge pump delivering a quantized current
I ¼ nef toaseriallyconnectedQHEdevice,whichconverts
I intoavoltage Vout¼n
bytwonaturalconstantseandh,bythetwointegersnand?,
and by the applied drive frequency f. It should thus reveal
a quantization relation similar to the Josephson relation,
however, based on completely different physics.
Figure 2 shows typical measurements of the quantized
voltage generation of two IQC devices. The device already
characterized as current source in Fig. 1(d) is now operated
as a voltage source in Fig. 2(a). Operation parameters
are V1¼ ?0:325 V, f ¼ 675 MHz, Pac¼ ?16 dBm,
B ¼ 8:4 T (? ¼ 1), and T ¼ 350 mK. In the figure, the
output voltage Voutis plotted vs V2. Similar to the case of
quantized current [Fig. 1(d)], now quantized voltage
plateaus Vout¼ nh
plateaus are also observed as functions of both dc voltages
V1and V2, as shown in the three-dimensional plot of
Fig. 2(b). Operation parameters are again f ¼ 675 MHz,
Pac¼ ?16 dBm, and B ¼ 8:4 T (? ¼ 1). The plot reveals
a well-defined voltage plateau [flat shaded (green) area] at
n ¼ 1. The generated quantized voltage is thus robust
against variation of V1and V2. The position and width of
the plateau as a function of V1and V2are determined by
the position and width of the quantized current plateau in
V1-V2space [28].
A test of the robustness of the voltage quantization
against variations of B and Pacis displayed in Figs. 2(c)
and 2(d). To test the influence of a variation of B within the
quantum Hall plateaus at ? ¼ 1;2, the average Voutwithin
the VoutðV2Þ plateau and its statistical uncertainty are de-
termined for different values of B. V1is in the center of the
two-dimensional voltage plateau [compare with Fig. 2(b)].
In (c) and (d), Voutis plotted (symbols) for f ¼ 320 MHz
and Pac¼ ?6 dBm in natural constants as a function of B
around (c) ? ¼ 1 and (d) ? ¼ 2. Also, Rxxis displayed in
the graph (solid black line, right scale). For vanishing Rxx,
?
h
ef. Voutshouldthen be determined
ef with n ¼ 1;2;3 are observed. Such
Voutremains well quantized within the statistical uncer-
tainty, as indicated by the error bars (compare with the
Supplemental Material [22]). However, in the plateau tran-
sitions indicated by a nonvanishing Rxx, deviations of Vout
from the quantized value become apparent. Note that the
data of Rxxand Voutof Figs. 2(c) and 2(d) have been taken
in different cooldown cycles. This might result in a slight
shift of the field position of the filling factors due to carrier
density variations.
Next, the dependence of the voltage quantization
on Pac(and hence Vac) is considered. The pump requires
a minimum threshold power beyond which the generated
current remains quantized [28]. This robustness against the
variation of Pacis well transferred to the IQC, as shown in
the inset to Figs. 2(c) and 2(d). Here, Voutis plotted vs Pac
for excitation powers of ?16... ? 10 dBm. The corre-
sponding Vac at G1 varies between ?100...200 mV
[28]. Parameters for the given data are f ¼ 1:1 GHz,
FIG. 2 (color online).
V2. (b) Voltage plateau as a function of V1and V2. Parameters in
(a),(b): ? ¼ 1, f ¼ 675 MHz. Data are taken in different cool-
down cycles. (c),(d) Vout(symbols) vs B within QHE plateaus at
? ¼ 1 (c) and 2 (d). Device A. f ¼ 320 MHz, Pac¼ ?6 dBm.
Colors mark different field sweep directions. Solid black lines:
Rxxin the plateau region. Inset: Voutvs Pac. f ¼ 1:1 GHz, ?¼1,
n ¼ 1 (black squares), n ¼ 2 (red circles). Data in (a), (b), and
the inset are taken on device B.
Quantized voltage operation. (a) Voutvs
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B ¼ 8:4 T (? ¼ 1), n ¼ 1 (black squares), and n ¼ 2 (red
circles). Again, V1and V2are in the center of the voltage
plateau. No significant dependence of Vouton Pacis ob-
served. Note that the statistical uncertainty is smaller than
the dot size in the graph. Hence, Voutcan be considered to
be robust over a broad range of Pac.
Finally, the linear dependence of Voutwith f is tested.
Figure 3 shows the quantized Voutvs f. Data are compiled
from two devices using two different measurement setups
(see the Supplemental Material [22]) during a number of
cooldowns. To collect the data, the devices were operated
insidetherobustoperationregionofV1,Vac,andB((Rxx¼0).
Then,tracesofVoutðV2Þ [comparewithFig.2(a)]weretaken.
TheaverageVoutintheflatcenterregionoftheplateauandits
statistical uncertainty were derived. Three groups of data are
foundwhichcanbeclassifiedaccordington and?andhence
according to the operation parameters of the two devices
forming the IQC. The star-shaped black data correspond to
Voutmeasuredforn ¼ ? ¼ 1; thediamond-shapedbluedata
correspond to n ¼ 1, ? ¼ 2; and the triangular red data
correspond to n ¼ 2, ? ¼ 1. All data sets are well described
by the theoretically expected curves indicated by the straight
lines. The data thus clearly evidence the quantization of the
generated voltage in units of h=e. To evaluate quantitatively
theagreementwiththetheoreticalprediction,wefittheslope
of the measured voltages of Fig. 3 for n ¼ ? ¼ 1. The result
of 4:1330ð12Þ ?V=GHz deviates less than 1w from the
expected value of 4:1356 ?V=GHz. Note that this deviation
is well within the systematic uncertainties of our measure-
ment setups (compare with the Supplemental Material [22]),
and hence no significant deviation of Voutfrom the theoreti-
cally predicted value is evident. Figure 3 furthermore shows
thatthedevicecangeneratequantizedvoltagesabove10 ?V
corresponding to operation frequencies of about 3 GHz.
Higheroutputvoltagescanbeexpectedforoptimizeddevices
and optimized experimental conditions. Here, e.g., nonsinu-
soidal drive waveforms should allow an increase of f and
hence I and Voutby a factor of 5 without significant loss of
accuracy [25]. Furthermore, Voutis expected to scale linearly
with the number of the integrated devices of the IQC. For
example, the pumped current could be increased by the
operation of a number of NPpumps in parallel [29–31],
thereby scaling up the output voltage by NP. Likewise, in-
stead of a single Hall device, a serial array of NHquantum
Hall devices could be used [32], resulting in an accordingly
scaled Vout. Combining both concepts and employing values
for the available technology of NP? 10 [29,31] and of
NH? 100 [32], scaled-up quantized output voltages of the
order of tens of mV seem to be possible and would enable
high precision fundamental experiments in electrical quan-
tum metrology [3].
The IQCs clearly demonstrate a new semiconductor
device functionality of robust quantized voltage generation
linked to the two fundamental constants h and e. Based on
that, it enables an alternative approach to the closure of the
quantum metrology triangle [10] by direct high precision
comparison of the (scaled-up) output voltage of the IQC
to the output voltage of a Josephson-based quantum volt-
age standard [2]. The IQC concept could further be trans-
ferred to different materials like silicon [4,20] or graphene
[12–14], provided a suitable graphene-based quantized
charge pump is available.
This work has been supported by DFG, QUEST, and EU
EURAMET Grant No. 217257.
*Corresponding author
hans.w.schumacher@ptb.de
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