# Misfit layer compounds as ultra-tunable field effect transistors: from
charge transfer control to emergent superconductivity

Ludovica Zullo  Department of Physics, University of Trento, Via Sommarive 14,
38123 Povo, Italy<EMAIL_ADDRESS>Giovanni Marini  Graphene Labs,
Fondazione Istituto Italiano di Tecnologia, Via Morego, I-16163 Genova, Italy
Tristan Cren  Sorbonne Université, CNRS, Institut des Nanosciences de Paris,
UMR7588, F-75252 Paris, France  Matteo Calandra  Department of Physics,
University of Trento, Via Sommarive 14, 38123 Povo, Italy
<EMAIL_ADDRESS>
###### Abstract

Misfit layer compounds are heterostructures composed of rocksalt units stacked
with few layers transition metal dichalcogenides. They host Ising
superconductivity, charge density waves and good thermoelectricity. The design
of misfits emergent properties is, however, hindered by the lack of a global
understanding of the electronic transfer among the constituents. Here, by
performing first principles calculations, we unveil the mechanism controlling
the charge transfer and demonstrate that rocksalt units are always donor and
dichalcogenides acceptors. We show that misfits behave as a periodic
arrangement of ultra-tunable field effect transistors where a charging as
large as $\approx 6\times 10^{14}$ e-cm-2 can be reached and controlled
efficiently by the La-Pb alloying in the rocksalt. Finally, we identify a
strategy to design emergent superconductivity and demonstrate its
applicability in (LaSe)1.27(SnSe2)2. Our work paves the way to the design
synthesis of misfit compounds with tailored physical properties.

Sorbonne Université, CNRS, Institut des Nanosciences de Paris, UMR7588,
F-75252 Paris, France Sorbonne Université, CNRS, Institut des Nanosciences de
Paris, UMR7588, F-75252 Paris, France Graphene Labs, Fondazione Istituto
Italiano di Tecnologia, Via Morego, I-16163 Genova, Italy

The capability of inducing a controlled and tunable number of carriers in few
layer systems has been pivotal for the success of 2D materials 1. However, in
metallic few layers 2D dichalcogenides such as NbSe2, the largest carrier
doping that can be achieved via field effect gating are of the order of
$n_{e}\approx 3\times 10^{14}$ e- cm-2 2, corresponding to a Fermi level shift
of the order of $0.1$ eV, too small to drastically change the physical
properties.

Recently 3, it has been shown that overcoming this limit is possible in the
misfit layer compound (MLC) (LaSe)1.14 (NbSe2)2, an heterostructure composed
of periodically alternating rocksalt monocalchogenide units (RS) and few
layers transition metal dichalcogenides (TMDs) 4, 5. In this system, a massive
electron transfer from the LaSe RS to the NbSe2 TMD occurs, leading to a rigid
Fermi level shift as large as $+0.55$ eV. It is, however, unclear if the
electron doping in misfits can be in some way controlled by any physical
parameter and, more important, how general this mechanism to dope few layer
TMDs is.

MLCs have been known for a long time and their structures as a function of the
RS and TMD composition have been thoroughly investigated 4, 5. However, the
exploration of physical properties such as Ising superconductivity 6, 7, 8, 9,
10, 11 charge density waves (CDW) 12, 13, 14, 15, 16 or topological effects 17
are quite recent. The research in the field has lead to remarkable results but
it has mostly proceeded by isolated discoveries and trial and error chemical
synthesis, while general rules to understand what happens when assembling
different RS and TMDs are missing. The need of a global picture becomes
evident when considering that (i) many ternary alloys composed of
monochalcogenides can be assembled with practically any few layer
dichalchogenide, (ii) the thickness of the dichalcogenide layers can be chosen
at will. This makes a lot of possible combinations and leads to many
unanswered questions. For example, how does the charge transfer occur in these
structures ? Are the TMD layers acceptors or donors ? How can the charge
transfer be tuned ? To what extent the electronic structure of the TMD is
affected when inserted in the heterostructure ? Most important, what are the
emergent properties of the misfit, i.e. properties of the MLC that are absent
in the pristine constituents ? How can we design misfit properties from the
knowledge of their building blocks ?

In this work we answer these questions by performing extensive first
principles electronic structure calculations of MLCs. We identify the
fundamental mechanism ruling charge transfer and demonstrate how the charge
injection into the TMD layers can be efficiently controlled by chemical
alloying in the rocksalt unit. Most important, we show that superconductivity
can emerge in MLCs formed by assembling non-superconducting RS and TMDs.
Finally, we demonstrate that misfit layer compounds can be assimilated to
ultra-tunable field effect transistor with an unequaled charging of the TMD
layers. Our work paves the way to extensive experimental synthesis and
development of these promising systems.

Figure 1: a) Bulk structure of (LaSe)1.18(TiSe2)2. The field-effect modeling
scheme is depicted for the case of the most common TiSe2 terminated surface
(left) and for bulk TiSe2 (right). b) Sketch of the unit cell of
(LaSe)1.18(TiSe2)2 misfit layer compound $({\bf a},{\bf b})$ compared with the
ones of a TiSe2 monolayer $({\bf a}_{1},{\bf b}_{1})$ and of a LaSe unit
$({\bf a}_{2},{\bf b}_{2})$.

The chemical formula of MLCs is (RQ)1+δ(TX2)m, where (TX2)m is a $m-$layers
TMD and RQ is a rocksalt monochalcogenide unit (often referred to as Q-layer)
4, 5. Ternary alloys of two monochalcogenides within a single RS Q-layer (e.g.
LaxSr1-xS) have also been synthesized 18 leading to MLCs having chemical
formulas of the kind (RxM1-xQ)1+δ(TX2)m. As a prototypical example of the MLCs
crystal structure we consider (LaSe)1.18(TiSe2)2, shown in Fig. 1 (a) and (b).
Each TiSe2 and LaSe sublattice has its own set of cell parameters. Compared to
bulk 1TTiSe2, the lattice of the TiSe2 bilayer in the MLC is not perfectly
hexagonal and is slightly expanded along one direction. As a consequence, the
TiSe2 sublattice is described by a centered orthorhombic cell with in-plane
lattice vectors ${\bf a}_{1}\approx 3.6$ Å and ${\bf b}_{1}\approx 6$ Å. The
LaSe sublattice has also an orthorhombic symmetry but with similar in-plane
lattice parameters ${\bf a}_{2}\approx{\bf b}_{2}\approx 6$ Å. Both systems
have the same ${\bf b}$ vectors (${\bf b}_{1}\approx{\bf b}_{2}$) so that the
material is commensurate along this direction. The ratio between the norms of
the ${\bf a}_{1}$ and ${\bf a}_{2}$ vectors sharing the same direction is an
irrational number (see tables in Fig. 1 and 2 in Supplemental Material) making
the MLC incommensurate in the ${\bf a}$ direction. The mismatch ratio
$a_{2}/a_{1}=x/y$ is usually in the range $\sim 1.6-1.8$ and sets the
parameter $\delta$ in the chemical formula through the relation
$1+\delta=2\times(a_{1}/a_{2})$. In this work we adopt the convention of using
the value of $\delta$ as obtained from the lattice parameters a1 and a2 of the
pristine RS and TMD before assembling them in a MLC structure, as reported in
the tables in Figs. 1-2 in the Supplemental Material. The commensurate
approximant of each MLCs considered in the current work is reported in Fig. 3
in the Supplemental Material.

Figure 2: Calculated band-alignement of isolated Q-layer rocksalt structures
(red) and single layer transition metal dichalcogenides (blue). Dark (light)
bars represents the position of EF/valence band maximum (EF/conduction band
minimum) for metal/insulators. White spaces in the bars represent gaps in the
single particle spectrum. The zero of the energy is set to the vacuum level.

The RS layers have a strong intralayer bonding. A strong bonding also forms
among the RS and TMDs layers. On the contrary, Van der Waals bonding occurs
among the closer TMD layers. After cleavage, for $m>1$, the surface of the
sample is a perfect TMD layer (a single layer in the $m=2$ case considered in
this work 3). In the $m=1$ case , i.e. a single layer TMD sandwiched among RS
Q-layers, the bonding along the $z$ axis is always strong. As a result, the
cleavage occurs in-between the RS and TMD bonding and the surface is still a
TMD single layer, however it is often less clean and presents several steps
and defects 6, 9. In all cases, there is a substantial experimental evidence 3
that ARPES and STS/STM measurements mostly sample the terminating TMD layer
without accessing the bulk of the structure. On the contrary, Raman, transport
and superconducting measurements probe bulk properties of the crystal.

Figure 3: Band unfolding onto the NbSe2 single layer Brillouin zone for the
NbSe2 misfit series for different rocksalt Q-layers having comparable
mismatching ratio close to $7/4$. The band structure for the isolated single
layer NbSe2 (red line) is superimposed and aligned to the Nb d-band in the
misfit. The blue dashed line corresponds to the Fermi level EF of the misfit
compound, while the red one to the Fermi level of the isolated NbSe2 layer. In
the last two panels the dashed red line is superimposed to the dashed blue
one. Spin orbit coupling is neglected in all the calculations.

In order to gain insight on the charge transfer among the RS and TMD layers in
the MLC and its relevance for the electronic structure measurements (ARPES),
we perform extensive calculations of the work functions of $8$ isolated
rocksalt Q-layers and $12$ isolated TMDs single layers. The choice of
considering TMD single layers is motivated by (i) the fact that we consider
MLC with $m=2$ having a single layer TMD as terminating surface and (ii) by
the fact that the work functions of bilayers TMDs is fairly close to the one
of single layers 19. Thus, we expect that our results will also hold for the
surface and the bulk and for the $m=1$ case. Calculations are performed with
the quantum ESPRESSO 20 package and we use the PBE exchange and correlation
functional 21 (see SI for more technical details). Results are shown in Fig.
2.

The key quantities ruling the charge transfer in these systems are the work
function difference among RS and TMDs and the consequent band alignment, the
lattice mismatching ratio a${}_{2}/$a1 and, finally, the degree of
hybridization when the two subsystems are in contact. As shown in Fig. 2, the
TMDs globally possess substantially larger work functions than the RS
compounds. As the work function is the energy required to transfer an electron
from the Fermi level to the vacuum level, RS are always donor and TMDs always
acceptors. The net amount of charge transfer depends, however, not only on the
work function difference but also on the mutual concentration of the RS and
TMD that is related to the mismatching ratio. To explain this more clearly,
each RS can transfer a given amount of charge to the TMDs layer, if the
mismatching ratio is close to one. However, if the mismatching ratio
increases, the relative concentration of RS atoms per TMD cell decreases, and
so does the charge transfer. By looking at Fig. 3 in SI, it is clear that the
mismatching ratio varies mostly due to the change in the TMD lattice
parameter.

In order to demonstrate this global picture we perform explicit calculations
for several misfit surfaces terminated by a single layer NbSe2 but having
different RS Q-layers as building blocks and sharing comparable mismatching
ratios very close to $7/4$ (these compounds all belongs to the ninth column in
the table in Fig. 3 in the Supplemental Material). As it can be seen in Fig.
3, the behaviour of the (LaSe)1.15 (NbSe2)2, (BiSe)1.14(NbSe2)2,
(PbSe)1.14(NbSe2)2 and (SnSe)1.16(NbSe2)2 serie is almost completely
characterized by the work function differences. Indeed as
W(LaSe)$<$W(SnSe)$<$W(PbSe), the charge transfer decreases by progressively
decreasing the difference W(NbSe2)-W(RS), as expected. The work function of
BiSe is slightly larger than the one of SnSe, however BiSe seems to transfer
few more electrons than SnSe. We attribute this to the metallic character of
BiSe and the consequent stronger hybridization occurring between BiSe and
NbSe2, resulting in a substantial band deformation of the pristine NbSe2, as
shown in Fig. 3.

Figure 4: a) Band unfolding onto the single layer TiSe2 Brillouin zone of the
misfit compound (LaxPb1-xSe)1.18(TiSe2)2 for $x=1.0,0.34,0.0$. The band
structure for the isolated single layer TiSe2 (red line) is superimposed and
aligned to the bottom of the Ti d-band in the misfit. The blue dashed line
corresponds to the Fermi level EF of the misfit compound, while the red one to
the Fermi level of the isolated TiSe2 layer (in the lowest panel they
coincide). b) Lattice deformation of the TiSe2 layers generated by the partial
substitution of Pb atoms in (LaxPb1-xSe)1.18(TiSe2)2. The magnified portion
shows a bond length alternation in the TiSe2 lattice with two different
distances $d_{1}$ (red) and $d_{2}$ (green).

Finally we point out that the NbSe2 electronic structure in going from
(PbSe)1.14(NbSe2)2 to (LaSe)1.15 (NbSe2)2 is n-doped rigidly, i.e. the charge
transfer simply induces a Fermi level upshift. From this analysis two
questions arise: how general is this rigid doping effect and how can it be
used to effectively tune the doping ? We now show that it is possible to
engineer the misfit in such a way that the doping level is rigidly adjustable
through appropriate alloying of the RS Q-layer.

For this reason we consider MLCs having the following stoichiometry
(LaxPb1-xSe)1.18 (TiSe2) as a function of $x$. We point out that similar
substitutions (La$\leftrightarrow$Sr) have already been achieved in sulfur-
based MLC 18. A comparison between this system and the previous results for
the NbSe2 series will allow us to draw conclusions that are less dependent on
the chosen TMD.

From the previous reasoning and from Fig. 2, we expect that the La
concentration ($x$) allows to tune the carrier concentration in the TiSe2
layers with $x=1$ ($x=0$) corresponding to the highest (lowest) n-doping. In
Fig.[4] we show the calculated band structure of the full
(LaxPb1-xSe)1.18(TiSe2)2 misfit for $x=1.0,0.34,0.0$. We also plot (red
continuous line) the electronic structure for an isolated single layer. The
position of the bottom of the Ti d-band of the isolated single layer is
aligned to the corresponding band in the misfit. As it can be seen, by
increasing $x$ the doping is increased. Most important, the Ti d-band displays
no deformation upon doping. At the highest doping level ($x=1$, corresponding
to a charge transfer of $0.53$ electrons per Ti, which is $n_{e}\sim 5\times
10^{14}$ e- cm-2) two parabolic La bands cross the Fermi level along the
$\Gamma$K direction. These bands disappear by decreasing $x$ (see SI for
calculation at additional values of $x$). Remarkably, the electronic structure
of (PbSe)1.18(TiSe2)2 is almost indistinguishable from the one of the isolated
TiSe2 layer.

Despite this similarity in the electronic structure, we find that
(PbSe)1.18(TiSe2)2 does not display a $2\times 2$ CDW as it happens in the
case of the supported TiSe2 single layer 22, 23, 24. This result is in
agreement with resistivity data on this MLC 7 where no CDW was detected. We
attribute the suppression of the CDW to the strong bonding between TiSe2 and
the the RS Q-layer. We find that in (LaxPb1-xSe)1.18(TiSe2)2, for $x\neq 0,1$,
the Ti-Ti distances are modulated by the presence of Pb atoms in the host LaSe
lattice (i.e. the Ti-Ti distance becomes shorter if the Ti atoms are close to
a Pb atom). The reason is mostly sterical as the La atomic radius is larger
than the one of Pb, therefore Pb atoms are more strongly bounded to the RS
layer and a consequent deformation of the LaSe rocksalt host occurs (as shown
in Fig. 4 (b)) followed by a modulation of the Ti-Ti distances. We verified
that even starting from $2\times 2$ distorted TiSe2 layers in the misfit, the
structural optimization suppresses the CDW and leads to other distortion
patterns that essentially follow the Pb atoms superstructure. Our analysis
shows that altering the chemical composition of the rocksalt has a double
effect: on the one hand, it allows to precisely tune the rigid doping of the
TMD, on the other hand it suppresses the $2\times 2$ CDW of the TiSe2 bilayer
and introduces an additional modulation related to the alternation of La and
Pb.

After achieving a complete knowledge of the charge transfer in MLC, we now
demonstrate how to design a misfit superconductor starting from its
constituents. In particular we show that non-superconducting pristine RS and
TMD compounds can lead to a superconductor via charge transfer control
(emergent superconductivity).

We consider the layered indirect gap semiconductor 1TSnSe2 that can be
exfoliated and synthesized in single layer form 25. The electronic structure
of a single layer SnSe2 is shown in Fig. 5 (red line). The conduction band is
formed by an isolated band with a Van Hove singularity point at K. A maximum
in the density of states occurs at the energy corresponding to the band
flattening. If the Fermi level is tuned at the inflection point, this would be
beneficial for superconductivity. However, this involves a $\approx 1.4$ eV
Fermi level shift corresponding to a charge transfer of $0.77$ electrons
($\approx 6\times 10^{14}$e-cm-2), unreachable even in a ionic-liquid based
field effect transistor. However, as previously shown, this electron doping
level could be reached in the misfit (LaxPb1-xSe)1.27(SnSe2)2. In order to
confirm this hypothesis, we perform first principles calculations for this MLC
as a function of $x$ (see Fig. 8 in SI ). We find that the La-Pb alloying
allows a perfect control of the doping level due to the large work function
difference between LaSe and SnSe2 and an insulator-to-metal transition occurs
in SnSe2. At $x=1.00$ the Fermi level perfectly matches the inflection point,
as shown in Fig. 5. It is worth noting that at this high La concentration,
some LaSe bands cross the Fermi level close to the K point and along
$\Gamma$K, however their contribution to the total density of states is
marginal. In Fig. 5 we also compare the MLC surface electronic structure with
the one of an isolated layer (red line). As it can be seen, there is a
substantial band distortion with respect to the isolated single layer. A
better description of the surface electronic structure is obtained by
replacing the LaSe layer with a uniformly positive charged potential barrier,
as in a single gate field effect transistor setup by using the method
developed in Ref. 26. The electronic structure of an isolated SnSe2 layer
under this approximation is the green line in Fig. 5, in perfect agreement
with the complete calculation of the MLC surface electronic structure both for
what concerns the band bending at the Fermi level (some deviations are seen in
the empty states close to zone center) and for the position of the valence
band top. We attribute the band-bending occurring at the K high-symmetry point
to a modification of the intralayer spacing between Sn and Se in SnSe2 due to
the charging of the monolayer (a table with intralayer spacing comparisons can
be found in Fig. 7 of SI).

This result shows that it is possible via Pb/La alloying in the RS layers to
set the Fermi level at the Van Hove singularity. Furthermore, it shows that
the LaSe Q-layer can be assimilated to a capacitor plate in a Field Effect
Transistor (FET) (see Fig. 1(a)). This remains true even for the SnSe2
bilayers in the bulk of the sample, i.e. the full MLC can be assimilated to
several field-effect transistors stacked periodically along the z-axis of the
MLC, as shown in Fig. 1.

Figure 5: Panel a): band unfolding of (LaSe)1.27(SnSe2)2 misfit supercell onto
the hexagonal primitive Brillouin Zone (BZ) of single layer SnSe2 (the zero
energy is set to the Fermi level, dashed green line) . The superimposed solid
lines are the band structure of an isolated single layer $SnSe_{2}$ (red), and
of a single layer SnSe2 doped in a single FET setup as in (LaSe)1.27(SnSe2)2
by $0.7$ electrons per Sn atoms (green), respectively. Darker regions in the
colormap represent the most relevant projection of the misfit eigenvalues of
the band structure in the SnSe2 first BZ (band unfolding). In the adiacent
panel we plot the total DOS per SnSe2 formula unit of (LaSe)1.27(SnSe2)2
(yellow) and the projected density of states over atomic orbitals of the LaSe
layers (purple) and of the SnSe2 layers (blue), respectively. The green line
is the DOS of a single layer SnSe2 doped in a single FET setup of $0.7$
electrons per Sn atoms. Panel b): dynamical properties and electron-phonon
coupling of (LaSe)1.27(SnSe2)2 modeled by a bilayer SnSe2 in a double FET
setup. The phonon dispersion is shown in the first panel while, in the
adjacent panel, the Eliashberg function $\alpha^{2}F(\omega)$ (filled blue
curve) and the total electron-phonon coupling $\lambda(\omega)$ (red) are
depicted.

As superconductivity is a bulk property, we must simulate the complete 3D
crystal. The calculation of the vibrational properties and electron-phonon
coupling for the complete MLC is, however, a very cumbersome task due to the
large number of atoms. We then proceed differently, namely we consider a SnSe2
bilayer in a field effect configuration as in Fig. 1 with a $+0.7$ charge on
each of the two plates (double gate configuration). In order to prevent the
ions from moving too close to the gate electrodes, a potential barrier is
placed before the gates, and the total charge of the system is maintained
equal to zero 26. Additional details on these calculations can be found in the
SI. We have verified that this approach gives geometries for the SnSe2 bilayer
in excellent agreement with the complete MLC structural optimization.
Furthermore the electronic density of states of the MLC and that of the
monolayer in double gate configuration are practically indistinguishable, as
shown in Fig. 5.

We then calculate the phonon dispersion ($\omega_{\mathbf{q}\nu}$) and the
electron-phonon coupling $\lambda_{\mathbf{q}\nu}$ for each mode $\nu$ of
phonon crystal momentum $\mathbf{q}$ in double gate geometry. From these
quantities we obtain the Eliashberg function
$\alpha^{2}F(\omega)=\frac{1}{2N_{q}}\sum_{\mathbf{q}\nu}\lambda_{\mathbf{q}\nu}\omega_{\mathbf{q}\nu}\delta(\omega-\omega_{\mathbf{q}\nu})$
and the average electron-phonon coupling
$\lambda=\frac{1}{N_{q}}\sum_{\mathbf{q}\nu}\lambda_{\mathbf{q}\nu}=0.6$,
$N_{q}$ being the number of points in the phonon momentum grid used to
calculate the average (we used a 96$\times$96$\times$1 $\mathbf{q}$-grid, see
the SI). These quantities are plotted in Fig. 5 (b). Approximately $30\%$ of
the coupling arises from the Einstein optical modes at $\approx 45-50$ cm-1,
while the rest of the coupling is uniformly distributed throughout the other
modes. The phonon density of states (not shown) is very similar to the
Eliashberg function.

We calculate the superconducting critical temperature by solving the
anisotropic Migdal-Eliashberg equations 27, as implemented in the EPIq
software 28, 29, and by assuming $\mu^{*}=0.1$, obtaining a superconducting
critical temperature of $T_{c}=3.5$ K (see SI for details on Migdal-Eliashberg
calculations). This result matches well with the $T_{c}=4.8$ K detected in
ultrathin Li-intercalated SnSe2 via field effect gating and demonstrates that
superconductivity can emerge in MLC from pristine components that are not
superconducting.

In conclusion, by performing extensive first principles electronic structure
calculations on misfit layer compounds we unveiled the mechanism ruling charge
transfer in these systems. In particular, due to their large work functions,
we showed that TMDs are always acceptors while rocksalts are always donors.
The electron density that can be injected in the TMD layers can be as high as
6$\times$1014 e-cm-2, sensibly larger than in ordinary field-effect
transistors.

We have shown that the charging of the TMD layers can be efficiently
controlled via the La$\leftrightarrow$Pb substitution. Most interesting, by
replacing each RS Q-layer with a charged plate and a barrier, we have shown
that the surface of the MLC behaves as a single gated field-effect transistor
while the bulk can be seen as a periodic arrangement of double-gated field
effect transistor.

Finally and most important, we have shown that from the knowledge of the RS
and TMD constituents it is possible to infer the amount of charge transfer to
the TMD layers in the MLC and to predict the physical properties of the
heterostructure. As a practical demonstration, we showed that emergent
superconductivity occurs in (LaSe)1.27(SnSe2)2 via a 1.4 eV Fermi level shift
induced by the presence RS Q-layers in the misfit. The methodology developed
in this work paves the way to the synthesis and design of misfit compounds
with tailored physical properties.

##  1 Acknowledgements

We acknowledge EuroHPC for awarding us access to the LUMI supercomputer (grant
number $465000468$). We acknowledge support from the European Union’s Horizon
2020 research and innovation programme Graphene Flagship under grant agreement
No $881603$.

##  2 Supporting Information

Contains:

  * •

I. Geometrical Details of MLCs.

  * •

II. Technical details.

  * •

III. Band Alignment Calculation.

  * •

IV. Band unfolding method applied to (LaxPb1-xSe)1.18(TiSe2)2.

  * •

V. Doping-induced Superconductivity.

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200 Gbps444200 G transceivers and switch ports are estimated as 2$\times$ 100G cost. |  198 [12] |  815 [35] |  374 [24] |  100 [43] |  520 [41] |  25 [112]
Table 2: Cost of network components.

Table 2 lists the cost of network components we use in Section 5.2, namely
NICs, transceivers, fibers, electrical switches, patch panels, and optical
switches. The cost of transceivers, NICs, and electrical switch ports is based
on the lowest available prices in official retailer websites [28, 18]. Note
that for 200 Gbps, we use more 100 Gbps ports and fibers, because they were
less expensive than high-end 200 Gbps and 400 Gbps components, or their price
was not available. To estimate the cost of electrical switch ports, we
consider Edgecore bare metal switches with L3 switching and maximum number of
ports to amortize the per port cost. The cost of NICs is taken from the
Mellanox ConnectX series, and we consider two 2-port NICs as one 4-port NIC.
We obtain the cost of the patch panel, OCS, and 1$\times$2 optical switch
directly from their suppliers, Telescent [43] and Polatis [41] (with 40%
discount). The cost of transceivers matches that reported in Sirius [53].

To compute the network cost of Fat-tree and Ideal Switch, we consider number
of nodes in a full bisection bandwidth Fat-tree. For example, a standard $k=8$
Fat-tree has 80 switches with 64 ports, or 640 switch ports in total, in
addition to 1 NIC per host and one transceiver per NIC and switch port. A
TopoOpt system of 128 nodes with degree $d$ uses $128\times d$ NICs and
transceivers, but $128\times 2\times d$ patch panel ports because of the look-
ahead design. Note that the cost of optical components stays constant as link
bandwidth increases, an inherent advantage of optics. Following prior work, we
estimate the cost of fiber optics cables as 30 cents per meter [68] and select
each fiber’s length from a uniform distribution between 0 and 1000 meters
[148]. We calculate the cost of TopoOpt based on $2d$ patch panels and
1$\times$2 switches at each link to support its look-ahead design (§C). OCS-
reconfig’s cost is based on $d$ OCSs connected to all servers in a flat
topology.

##  Appendix H Impact of Server Degree on TopoOpt’s Performance

Figure 27: Dedicated cluster of 128 servers ($d$ = 8).

Figure 27 shows the same setting as Figure 11 except that each server has a
degree of eight ($d=8$). The results show a similar trend: even though per
server bandwidth has increased, the behavior of different network
architectures remains consistent.

Figure 28: Impact of server degree ($d$) on performance.

Next we do a sensitivity analysis of impact of server degree $d$ on TopoOpt’s
performance. Specifically, we vary the degree of each server in TopoOpt for
two link bandwidths: 40 Gbps and 100 Gbps. Figure 28 shows the trend for
different DNN models. Both DLRM and CANDLE are network-heavy; therefore, they
benefit more from the additional bandwidth obtained by increasing $d$.
CANDLE’s improvement is almost linear as degree goes up, as the strategy is
closer to data parallel and the amount of bandwidth available to AllReduce
operation increases linearly as well. In the case of DLRM, we observe a super-
liner scaling when $B$ = 100 Gbp because DLRM has one-to-many and many-to-one
MP transfers which require a low hop count in the topology. As we increase
$d$, TopologyFinder is able to find network topologies with much lower
diameter, consequently benefiting the performance by both increasing bandwidth
and reducing hop-count for MP transfers. Finally, BERT is mostly compute bound
at higher bandwidth; hence, increasing the server degree and bandwidth per
node has marginal impact on its iteration time.

##  Appendix I Enabling Host-based Forwarding in RDMA

Figure 29: Host-based RDMA forwarding to create a logical RDMA connection
between end hosts.

To support a multihop TopoOpt interconnect using host-based forwarding, we
enable RDMA RoCEv2 forwarding on all our HP NICs. RoCEv2 is an implementation
of RDMA on top of UDP/IP protocol, by utilizing a particular UDP port (4791)
and encapsulating an InfiniBand (IB) data packet. Hence, each RoCEv2 packet
can be routed with its source and destination IP addresses. However, host-
based forwarding is challenging in RDMA protocol, as the packet processing and
memory access are offloaded to the NIC, and the host does not have access to
individual packets. More precisely, if a packet’s IP destination IP address
does not match the NIC’s IP address, the RDMA engine silently drops the
packet.

To address this issue, we collaborated with engineers from Marvell, the
provider of the firmware and driver for our HP NICs. The solution that came
out of our collaboration does not require proprietary software or firmware,
and is applicable to commodity NICs with the same ASIC. We will release our
scripts publicly. At a high-level, we use a feature called NPAR, or network
partitioning. It allows us to split each 25 Gbps physical interface into two
logical interfaces in the hardware level: $if_{1}$ and $if_{2}$, as shown in
the right-most port of server A in Figure 29. $if_{1}$ is a normal RDMA
interface, where the RDMA engine of the NIC bypasses the kernel, and it has an
IP address. This enables the upper layer software to consider $if_{1}$ as a
normal RDMA interface. However, $if_{2}$ does not have an IP address and RDMA
is disabled. $if_{2}$ has a different MAC address from $if_{1}$, and we use
this address to split the traffic across $if_{1}$ and $if_{2}$. The traffic
that needs to be forwarded uses the MAC address of $if_{2}$ and hence is
delivered to the host networking stack instead of NIC’s RDMA engine.

Furthermore, we establish a set of iproute, arp, and tc flower rules in Linux
to enable the proper forwarding of packets. If two servers are directly
connected, such as the third port of server A and the second port of Server B
in Figure 29, we only need to indicate the outgoing interface on each of these
servers. RDMA engines will handle the communication. However, for the
connection between server A and D, we set the iproute and arp tables on server
A and server D to dictate which port the traffic should go out, as well as the
proper MAC address of the next server in the forwarding chain. In this case,
the packets are delivered to the kernel. Then, on servers B and C, we set the
tc flower rules to forward the packets to the next server with the proper MAC
address. In these tc flower rules, we look-up the final destination IP and
assert the routing that was computed by our algorithm.

Walk-through of an example of a packet going from server A to server D. In
Figure 29, the RDMA engine of server A assumes server D is connected on the
third port. It uses the kernel’s routing tables for the destination MAC
address, which is set to the MAC address of $if_{2}$ of the second port on
server B. Therefore, a packet which starts as an RDMA packet of server A is
treated as an Ethernet packet when it arrives at server B, and goes to server
B’s kernel. In the kernel, based on the packet’s final destination IP of
server D, server B redirects the packet to the fourth port, with destination
MAC address set to $if_{2}$ of server C. In this connection, the packet is
treated as a normal Ethernet packet. Finally, on server C, the kernel rewrites
the destination MAC address to that of $if_{1}$ on the third port of server D,
and redirects it to that port. In this connection, the outgoing Ethernet
packet is considered an RDMA packet because of the destination MAC address.
For the reverse connection from server D to A, the same process happens in
reverse, to support a bidirectional connection.

With these forwarding rules, we construct logical RDMA connections between all
pairs of servers. Upper layer communication libraries such as NCCL requires
all-to-all connectivity, and they will utilize these connections. We also
modify NCCL to be topology-aware, as certain pairs of servers are only
connected through specific ports.

Compared to native point-to-point RDMA, this approach takes a performance
penalty. Our experiments indicate the overhead is negligible when the amount
of forwarded traffic is small. Our NICs currently support TCP forwarding
offload. With firmware and driver modifications or future versions of the
NICs, they will also support RDMA forwarding offload. This will further reduce
the overhead of our approach.