1 Scalability of S uperconductor Electronics Limitations Imposed by AC Clock and Flux Bias
2025-04-28
0
0
2.73MB
18 页
10玖币
侵权投诉
1
Scalability of Superconductor Electronics:
Limitations Imposed by AC Clock and Flux Bias
Transformers
Sergey K. Tolpygo, Senior Member, IEEE
Abstract—Flux transformers are the necessary component of all
superconductor digital integrated circuits utilizing ac power for
logic cell excitation and clocking, and flux biasing, e.g., Adiabatic
Quantum Flux Parametron (AQFP), Reciprocal Quantum Logic
(RQL), superconducting sensor arrays, qubits, etc. On average,
one transformer is required per one Josephson junction. We
consider limitations to the integration scale (device number
density) imposed by the critical current of the ac power
transmission lines (primary of the transformers) and cross
coupling between the adjacent transformers. The former sets the
minimum linewidth and the mutual coupling length in the
transformer, whereas the latter sets the minimum spacing between
the transformers. Decreasing linewidth of superconducting (Nb)
wires increases kinetic inductance of the transformer’s secondary,
decreasing its length and mutual coupling to the primary. This
limits the minimum size. As a result, there is a minimum linewidth
~100 nm which determines the maximum achievable scale of
integration. Using AQFP circuits as an example, we calculate
dependences of the AQFP number density on linewidth for various
types of transformers and inductors available in the SFQ5ee
fabrication process developed at MIT Lincoln Laboratory and
estimate the maximum circuit density as a few million AQFPs per
cm2. We propose an advanced fabrication process for a 10x
increase in the density of AQFP and other ac-powered circuits. In
this process, inductors are formed from a patterned bilayer of a
geometrical inductance material (Nb) deposited over a layer of
high kinetic inductance material (e.g., NbN). Individual pattering
of the bilayer layers allows to create stripline inductors in a wide
range of inductances, from the low values typical to Nb striplines
to the high values typical for NbN thin films, and preserve
sufficient mutual coupling in stripline transformers with
extremely low crosstalk.
Index Terms— AQFP, crosstalk, inductance, kinetic
inductance, microstrip, mutual inductance, NbN, RQL, RSFQ,
SFQ circuits, stripline, superconductor electronics,
superconducting flux transformer, superconductor integrated
circuit
I. INTRODUCTION
UPERCONDUCTOR digital electronics easily beats
CMOS and prospective beyond CMOS technologies
in such important performance metrics as energy
This material is based upon work supported by the Under Secretary of Defense
for Research and Engineering under Air Force Contract No. FA8702-15-D-0001.
Sergey K. Tolpygo is with Lincoln Laboratory, Massachusetts Institute of
Technology, Lexington, MA 02421 (e-mail: sergey.tolpygo@ll.mit.edu).
dissipation and processing speed. Superconductor single flux
quantum (SFQ) electronics [1] holds the record in clock speed
of simple circuits of about 770 GHz [2] and in its slower,
adiabatic implementations can operate with energy per bit near
the Landauer’s thermodynamic limit ln 2 [3]-[6].
However, these performance advantages so far have not
benefited any large-scale computational system because
integration scale of superconductor digital circuits is currently
three to four orders of magnitude lower than of the CMOS
circuits. For instance, the largest demonstrated circuits in
superconductor electronics have about one million Josephson
junctions [7], [8] whereas the modern CMOS circuits have over
50 billion transistors, a 50,000× difference [9].
Due to the recent progress in fabrication technology of
niobium-based superconductor integrated circuits, the
minimum feature size was reduced to 120 nm [10], [11]. This
allowed for an increase in the circuit density to about 1.5∙107
Nb/Al-AlOx/Nb Josephson junctions (JJs) per square
centimeter [10], [12], about ten-fold increase from the previous
level [7]. For a comparison, the present density of CMOS
circuits is 1000x higher, about 1.4∙1010 transistors per cm−2 [9].
The largest demonstrated density of superconductor Josephson
junction-based random access memory (JRAM) is 1 Mbit cm−2
[13]. For a comparison, RAM technology based on spin-
transfer torque magnetic RAM, the so-called STT-MRAM,
operating at room temperature has a 1000x higher density [14],
although it uses devices similar to multilayered sandwich-type
Josephson junctions.
In [15], the author argued that superconductor digital
electronics is fundamentally less scalable than semiconductor
electronics because of the fundamental difference in
information encoding. Indeed, in superconductor electronics
information is encoded, stored, and transferred by magnetic
flux quanta created by superconducting currents circulating in
Color versions of one or more of the figures in this article are available
online at http://ieeexplore.ieee.org
S
2
closed superconducting loops (inductors) interrupted by
Josephson junctions. In semiconductor electronics, information
is encoded by a stationary electric charge on the gates of field-
effect transistors. Obviously, localized charge on a capacitor
occupies less space than the moving charge – superconducting
loop current. Hence, charge-based devices can always be made
smaller and their circuits made denser and more scaled-up than
flux-based devices and circuits.
The goal of this work is to establish fundamental limits on
the scalability, i.e., the maximum circuit density, of ac-powered
superconductor digital electronics, imposed by two basic
components of all superconductor integrated circuits: inductors
and transformers. Limitations imposed by sandwich-type
(trilayer) Josephson junctions like Nb/AlOx-Al/Nb were
discussed in [15].
A problem of superconducting transformers has emerged
with advancement of ac powered and ac clocked
superconductor logic solutions instead of the dc-powered RSFQ
logic [1]. Starting from the original Parametric Quantron (PQ)
[3], [16], [17] and going to its analogs, renames and
incarnations − DC Flux Parametron [18], [19], Quantum Flux
Parametron (QFP) [20], [21], and Adiabatic Quantum Flux
Parametron (AQFP) [22], [23] − all logic solutions based on
parametric devices require a multiphase ac excitation. These ac
signals are inductively coupled (via transformers) to the devices
in order to modulate their Josephson inductance (critical current
of Josephson junctions) and produce a change in the logic state
and a parametrically amplified output current upon applying a
weak input current . The only exception is a dc-powered
nSQUID logic whose devices (nSQUIDs) use transformers to
create a large negative mutual inductance between the SQUID
arms [4], [24].
Another example is Reciprocal Quantum Logic [25], [26]
which requires four-phase ac power delivered via transformers
to propagate positive and negative single flux quantum (SFQ)
pulses in the same direction along Josephson transmission lines
(JTLs) and provide energy to and synchronization of RQL
gates.
In addition to the power and clock delivery, in all types of
superconductor logic and memory circuits, superconducting
qubit circuits, superconducting sensor arrays, etc.
superconducting flux transformers (mutual inductors) are used
to provide flux biasing, signal inverting (NOT function), and
coupling between cells.
As an example, we will consider what is now known as
AQFP cell shown in Fig. 1a. It is identical to QFP and PQ cells,
and may only slightly differ in parameter values. The AQFP
logic requires, at least, one ac transformer per Josephson
junction. Therefore, the circuit density (device number density)
cannot be higher than the density of the transformers. A very
similar analysis to the offered below can be easily done for RQL
gates and their JTLs, which typically require one transformer
per two junctions [25]-[27], and for any other ac-powered logic,
memory, and quantum circuits.
II. AC EXCITATION AND DC FLUX TRANSFORMERS:
PHYSICAL LIMITATIONS
A. Main Features of ac Excitation Transformers
The typical use of transformers in superconductor electronics
is to provide a dc flux bias and/or ac flux excitation to logic
cells with amplitude
; typically, = 2. Very often, e.g., in
AQFP, both dc flux bias =2
and ac excitation are
required. They can be applied either via the same or two
separate transformers. For simplicity, we consider only the
former case, requiring = 1, because using two transformers
increases the total transformer area by about 2x.
In order to provide the required dc bias and ac excitation, the
mutual running length of the transformer primary wire and the
(a)
(b)
Fig.
1. (a)
Schematics of PQ, QFP and AQFP cells. It consists of two
identical RF SQUIDs connected in parallel. In the adiabatic regime of
operation, the typical parameters of AQFPs are
2
= 0.2,
2
= 1.6, and =50 µA is the critical current of junctions J1 [34].
(b) A sketch of a planar transformer between the primary inductor
, a part
of the ac
-power transmission line, and the QFP cell inductors which
are
either
a microstrip (one ground plane) or stripline (two ground planes) inductor
s
with length
, laying in the same or adjacent plane as the
. The mutual
running length of the inductors
and , which determines their total
mutual
inductance
, is . The full QFP consists of two connected half-
cells (RF
SQUIDs). Also shown is the second
row of QFPs. Their inputs are
inductively
coupled to
other QFPs via the output inductor . Spacing
between the
QFP
rows
(or the ac power lines) determines their cross coupling − ac excitation
in QFP#
2 produced by the power line of the QFP#1 in the same column,
and
vice versa.
Spacing
determines cross coupling between the output
transformers
,
of QFPs in the same row. For layout examples, see [37].
Inductor
needs to be place perpendicular to inductors and
in order to
minimize direct coupling of the ac excitation to the output.
3
transformer secondary wire should be
=(), (1)
where = + is the amplitude of the total excitation
current that is fed into the , and is the mutual inductance
per unit length between the wires of the primary and secondary.
On the other hand, the transformer secondary, is always a
part of the logic cell and its inductance is determined by the cell
design parameter
= 2
, (2)
where is the Josephson junction critical current of junction
J1. Hence, length of the inductor is given by
=
, (3)
where is the inductance per unit length of the secondary.
Obviously, the transformer with the smallest area in this
simple two-wire configuration can only be formed if or
if
. (4)
Multiturn transformers with > are possible but will not be
considered here because they always occupy much larger area
and, hence, restrict the circuit density more than the simplest
parallel-wire transformer shown in Fig. 1b.
The mutual inductance of any two conductors is smaller than
the magnetic part of the self-inductance of the conductors
=()
, (5)
= +, (6)
=
, (7)
where is kinetic inductance per unit length of a wire with
thickness and width , is the magnetic field penetration
depth; 1 is the coupling coefficient between the magnetic
inductances of the primary, and the secondary, . If the
primary and secondary are formed by inductors of the same type
with equal magnetic inductance per unit length, = =
, the required primary excitation current from (4) is
1 +
. (8)
This current is significantly larger than the and must
increase with decreasing the cross section of the inductors, i.e.,
with increasing the scale of integration.
On the other hand, must be smaller than some maximum
current related to the critical current, Ic of superconducting
(Nb) wires forming the transformer,
==, (9)
where < 1 is a safety factor and is the superconductor
critical current density. Hence, the minimum possible inductor
cross sectional area = can be found from the condition
1 +
, (10)
which is not a quadratic equation because of a nontrivial
dependence of and on and .
In superconductors, the fundamental limit to the is the
Ginzburg-Landau critical current density
given by
=
, (11)
where is the thermodynamic critical magnetic field of the
superconductor and is the magnetic field penetration depth
[28]. Using the microscopic theory [29], Bardeen expressed this
depairing critical current density as
(
)
(1
)
, (12)
where and are the thermodynamic critical field and the
energy gap at zero temperature = 0, respectively; is the film
resistivity in the normal state [30].
For niobium, () = (1 )
, = 9.1 K and
=0.155 T, giving = 0.122 T at 4.2 K. The measured
value of the penetration depth in our Nb films is =90 nm.
Therefore,
(4.2 K)= 0.59 A/µm2 for our Nb films at 4.2 K.
The measured critical current density in Nb wires with
200 nm is a factor of two lower than this theoretical value,
about 0.37 A/µm2 [10], [31]. Since may decrease further with
reducing the wire dimensions, e.g., due to increasing Nb
contamination and resistivity, it would be practical to take
= 0.25 A/µm2, a 33% lower value than the measured
critical current density, in order to have some safety margin.
To proceed further with (10) and find the dependence of the
transformer area on the linewidth, we need to specify the type
of the transformer in order to determine and . Before going
to numerical results, as a simple illustration, we consider a
transformer formed by two parallel microstrips laying in the
same plane.
For superconducting microstrips with rectangular cross
section, magnetic part of the inductance per unit length is given
in [32] by
=
ln 1 +
.(), (13)
where is the dielectric thickness between the microstrip and
the ground plane, and is magnetic permeability of the
dielectric, hereafter assumed to be 1. From the fabrication
process practicality, we are interested in wires with nearly
square cross sections and deeply scaled features with
,2(+). In this case, the minimum cross section area
is achieved at the maximum possible coupling = 1 in (10)
and given by solution of the equation
1 +
((
)
.). (14)
To solve (14), let us use the typical parameters of ac
excitation and flux bias transformers in AQFPs as an example:
= 50 µA, = 1, 0.2 [22], [33]. In the widely used
4
fabrication processes SFQ5ee [34], transformers using Nb
microstrips M6aM4 with = 615 nm (signal trace on the layer
M6 above the ground plane M4) are the most convenient
because they are the closest to the layer of Josephson junctions.
Then, solving (14) gives the minimum cross section area
= 1.34∙10−2 µm2. Hence, the typical minimum linewidth
and the film thickness are ()
116 nm. At smaller cross
sections, the required transformer cannot be made because the
excitation current required to induce flux amplitude in the
transformer secondary would exceed the critical current of the
transformer primary wire. In reality, the coupling coefficient is
always less than 1. For a more realistic = 0.5,
= 2.18∙10−2 µm2 and ()
=148 nm.
The minimum linewidth following from the minimum cross
section area) determines the minimum possible coupling length
between the transformer primary and secondary, i.e., the
minimum possible size of the ac-powered cell along the ac
power transmission line. Existence of the minimum linewidth
and the minimum cell size is the first limit on scalability of ac-
powered superconductor electronics (AQFP, RQL, etc.) caused
by the finite superconducting critical current of the primary
wire in the ac and dc flux-biasing transformers. This limit can
be reached in a 90-nm technology node (90-nm linewidth).
Further reduction of the linewidth would not significantly
increase the density of superconductor integrated circuits using
ac powering of logic gates. This is our first conclusion.
In the following sections we will consider other limitations
on the cell height (in the y-direction perpendicular to the ac
power transmission line) and width. For specificity, we will use
the typical parameters of AQFPs, whereas the same arguments
and estimates apply to other types of ac-powered
superconductor logics.
B. Inductor and the Output Coupling Transformer
The length of the inductor in AQFP, is given by
=
=
, (15)
Here is inductance per unit length of inductor which may
differ from the per length inductance of the excitation
transformer secondary . The typical AQFP parameters are
= 50 µA and = 1.6, giving =10.53 pH. At
~1 pH/µm, is about 10 µm.
Inductor needs to be placed perpendicular to the
inductors and , forming a T-shape, in order to minimize
direct coupling of the ac excitation to the output. The aspect
ratio of this “T” in the typical AQFP cell (width to length ratio)
is 2
=1:4 because the ratio of the optimal parameters
=
is 1.6:0.2=8:1. The minimum possible area of the typical AQFP
is = 2=
if the area occupied by the
junctions , the transformer primary, and vias between the
inductors can be neglected. The AQFP maximum number
density is then
=
=
, (16)
where < 1 is the area filling factor.
Using the AQFP parameters in Fig. 1 and the maximum
expected value of magnetic inductances =~1 pH/µm,
we get < 3.6∙106 cm−2. This is our second conclusion: the
number density of AQFPs using Nb inductors is limited from
above by three to four million AQFPs per cm2, corresponding
to about 1M cm−2 AQFP majority gates composed of three
(MAJ3) AQFPs.
This estimate does not account for possible limitations
imposed by the supercurrent-carrying capacity of the ac
transformer and for a possibility of using inductors with higher
values. Accounting for these factors does not change the
order of magnitude in the AQFP density estimate, but requires
a more detailed analysis.
The superconducting material and cross section of wires for
inductors and are selected such that both currents
(17a) and (17b) are smaller than the critical current of the
corresponding wires. For Nb wires with = 0.25 A/µm2, the
minimum cross section of is = 8∙10−4 µm2, which is an
order of magnitude smaller than following from (14) and
corresponds to wire dimensions ( )
~30 nm. This is the
ultimate limit to reduction of the linewidth of Nb wires.
Increasing values significantly above 1 pH/µm is
possible by using kinetic inductance (7). For instance, a 40-nm
thick Mo2N kinetic inductor in the SFQ5ee process [35] has
8
pH/µm (with is in micrometers) [36], which is
>>1 pH/µm at < 1 µm. However, a short strip of kinetic
inductor can be used only if is galvanically coupled to the
next AQFP (so-called directly coupled AQFP [38]). If inductive
(transformer) coupling to the next gate is required, the mutual
inductance between the short strip of a kinetic inductor and
the output inductor , which is proportional to the length of
, is going to be small. Careful optimization is needed in this
case, as will be discussed in Sec. III.
Consider inductive coupling with mutual inductance
between the and the output inductor connected to the
next AQFP or forming a part of the buffer which sums up the
output currents of three AQFPs comprising the majority
(MAJ3) gate [37]. Parametrically amplified input current
creates current
= 2
200 µA (17a)
in the , which in turn creates current
=
2 ()
(17b)
in the output inductor. The maximum value of can be
determined from the condition 10 µA, which is
the current required to drive the next AQFP. This gives
=19.63, (18a)
=.
. (18b)
摘要：
展开>>
收起<<
1ScalabilityofSuperconductorElectronics:LimitationsImposedbyACClockandFluxBiasTransformersSergeyK.Tolpygo,SeniorMember,IEEEAbstract—Fluxtransformersarethenecessarycomponentofallsuperconductordigitalintegratedcircuitsutilizingacpowerforlogiccellexcitationandclocking,andfluxbiasing,e.g.,AdiabaticQuant...
声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！
相关推荐
-
【词汇变形总汇】2025高考词汇变形总汇 - 教师版VIP免费
2024-12-06 3 -
【超简37页】新课标高考英语考纲3500词汇VIP免费
2024-12-06 4 -
《高考英语3500词详解》（WORD版）VIP免费
2024-12-06 13 -
《高考英语3500词详解》VIP免费
2024-12-06 11 -
高中英语-[教师版]80天通关高考3500词汇VIP免费
2024-12-06 12 -
高中人教选修7课文逐句翻译VIP免费
2024-12-06 7 -
高中人教选修7课文原文及翻译VIP免费
2024-12-06 13 -
高中人教必修4课文逐句翻译VIP免费
2024-12-06 7 -
高中人教必修4课文原文及翻译VIP免费
2024-12-06 13 -
高考英语核心高频688词汇VIP免费
2024-12-06 10
分类：图书资源
价格：10玖币
属性：18 页
大小：2.73MB
格式：PDF
时间：2025-04-28
作者详情
相关内容
-
3-专题三 牛顿运动定律 2-教师专用试题
分类：中学教育
时间：2025-04-07
标签：无
格式：DOCX
价格：5.9 玖币
-
2-专题二 相互作用 2-教师专用试题
分类：中学教育
时间：2025-04-07
标签：无
格式：DOCX
价格：5.9 玖币
-
6-专题六 机械能 2-教师专用试题
分类：中学教育
时间：2025-04-07
标签：无
格式：DOCX
价格：5.9 玖币
-
4-专题四 曲线运动 2-教师专用试题
分类：中学教育
时间：2025-04-08
标签：无
格式：DOCX
价格：5.9 玖币
-
5-专题五 万有引力与航天 2-教师专用试题
分类：中学教育
时间：2025-04-08
标签：无
格式：DOCX
价格：5.9 玖币
渝公网安备50010702506394
