Voltage Tunable Quantum Dots (VTQDs) for Realizing Quantum Superposition in Complementary-Metal-Oxide- Semiconductor (CMOS) structures

Sanatan Chattopadhyay

Quantum Materials, Devices and Digital Microfluidics Group (QMat & DigiMF)

Department of Electronic Science

University of Calcutta

Topics of discussion

• Few introductory statements (MOS devices).
• Semiconductor quantum structures.
• MOS devices and surface quantization.
• Concept of voltage tunable quantum dots (VTQDs).
• Fabrication of quantum dot devices.
• Quantum superposition in quantum- /nano- wire FETs.
• Summary.

Introduction

• Realization and implementation of quantum theory is perhaps the most valuable contributions in physics in the last 100 years.
• This invaluable discovery has enabled the creation of multiple branches of modern physics.
• The most fascinating essence of quantum theory is that it is implementable in almost all the material systems.
• The modern buzz of quantum computing, quantum information processing, quantum communication and quantum cryptography, is also the gift from quantum theory.

Introduction: the domain

• Quantum computing, quantum information processing, quantum communication and quantum cryptography are the invaluable gifts of the progress of quantum theory.
• The basic unit is qubit and such scalable Qubits operate only at very low temperature (~ mK).
• The emerging device schemes for qubit generation are different from the state-of-the-art classical bits, and these are not compatible with the CMOS technology.
• Presently 97% of the electronics market is dominated by CMOS technology.
• Continuous scaling of MOSFETs since 1970’s has resulted in the emergence of NW-/QW FETs.
• Currently, TSMC, Intel and Samsung are manufacturing NW/QW- FETs of 3 nm technology node.

MOSFET: basic unit of Systems

MOSFET: basic unit of Systems

Constraints:

• Power.
• Speed.
• Size.

• Data volume: Tera-bits (TB).
• Numerous functionalities.

Market

80% dominated by CMOS

97% covered by Si

SRAM

Physics of MOSFETs

Quantum Superposition Principle

Qubit

• Weightage of the two states, i.e., |α|² and |β|² are the probability to get either |0〉 or |1〉 on read-out.
• Weightage of the two states and the phase difference, i.e., φ, can be manipulated by gate operations.
• It can be represented by Bloch sphere.

|ψ_A〉 = α|0〉 + β|1〉

|ψ_A〉 = cos(θ/2)|0〉 + sin(θ/2)e^iφ|1〉

• Any quantum system is a superposition of two orthogonal states, i.e., |0〉 and |1〉, is a qubit.

Quantum Gate Operations

Basic Gates

NOT

σ_x =

0 1

1 0

|0〉 ↔ |1〉

Phase

σ_z =

1 0

0 -1

|0〉 → |0〉

|1〉 → -|1〉

Hadamard

H =

1 1

1 -1

|0〉 → |0〉 + |1〉

|1〉 → |0〉 - |1〉

Gate

Matrix

Result

Qubit Implementation

Device

Information

|0〉

|1〉

Challenge

Photonic

Polarization

Horizontal

Vertical

Scalability

Absorption/

scattering

No. of photons in Fock state

Vacuum

1-photon

Arrival time

Early

Late

Super-conducting

Josephson junction

Charge

One extra Cooper pair

No extra Cooper pair

Flux/

Current

Counter-clockwise

Clockwise

Phase/

Energy

1^st excited state

Ground state

QD

Up

Down

Dot spin

Double-QD

Electron on Right dot

Electron on Left dot

Electron localization

T ~ mK

Coherence time low

T ~ mK

T ~ mK

Advantage

Room Temp

operation

Fidelity 99.4%

No. of entangled qubit high

Semiconductor

Charge Based

Fidelity 99%

Trapped ion

Lower Hyperfine level

Energy level

Upper Hyperfine level

Fidelity 99.9%

Coherence time very high

Laser require-ment

Topology

Emerging (claimed to be braids)

Anyons

?

Coherence

Challenges

• The conceptual areas with relevant theories have progressed significantly.
• The area of quantum algorithm has also been developed.
• Data transmission with quantum communication protocol finds good progress.
• However, the technological implementation at room temperature has not been affordable yet.
• Very difficult to communicate with the people working in different sections of these areas.
• Should have an appropriate way to exchange ideas and their possible physical implementations.

Why Quantum Dots (QDs) ?

• Quantum confinement (QC) of electrons and holes.
• Formation of quantized and discrete electronic energy levels.
• Energy gap (ΔE) between the successive energy levels increases as the size of the nanomaterial decreases as in a ‘particle in a box’ model.
• The energy level spacing in the nanomaterial is greater than the thermal energy as in atom (ΔE > k_BT).

Important effects of size quantization:

Applications of QDs

Source: Science; August 6; 2021; Semiconductor quantum dots: Technological progress and future challenges

Migration to quantum ‘components’

• Enormous progress have been made in the computer technology.
• The no. of atoms needed to represent a bit in memory has been decreasing exponentially since 1950.
• Similarly, the no. of transistors per chip, clock speed, and energy dissipation per logic operation have also exhibited exponential improving trend.
• However, such rate of improvement is not sustainable any more after the year 2020 since one bit of information will require only few/ one atom (s) to represent it.
• The problem is that the behavior of a computer’s components will be dominated by the principles of quantum physics at this size.

Requirement: QDs in room temperature

Quantum dots should fulfill the following requirements to make them useful for devices at room temperature:

1. Sufficiently deep localizing potential and small QD size.
2. QD ensembles should show high uniformity.
3. Material should be coherent, i.e., defect free.

Size

For a band offset of 0.3 eV, QD diameter ~ 4 nm.

Maximum limit

E₂ and E₁ are the 2^nd and 1^st energy states.

For GaAs/AlGaAs QDs: ~ 12 nm; InAs/AlGaAs QDs: ~ 20 nm.

ΔE_c: conduction band offset.

Materials, Devices and Technology at Quantum scale

Quantum Dot (QD) technology

• Epitaxial growth / deposition:

- CVD

- MOCVD

- MBE

D. Bimberg, M. Grundmann, N. Ledenstov, “Quantum Dot Heterostructures”

Exciton

• When an electron excited from the VB to CB through the absorption of incident photon energy, it creates a hole in the VB. The electron-hole pair (excited electron in CB and hole in the VB) is called the exciton.
• Such electron-hole pair may be considered to constitute a hydrogen- like atom in solid state.
• The effective mass approximation in semiconductor allows to neglect the periodic crystal potential and considers the electrons and holes as free particles.
• The wave functions and their energy values can be obtained by solving the Schrodinger equation analogous to the one describing the electron state in a hydrogen atom.

Excitation of an electron from the VB to CB producing an exciton

(e^- is in CB and h⁺ residing in VB)

Illustration of Exciton in semiconductor

Exciton

The energy values of the exciton are the solutions to the particle in a box with the mass replaced by reduced mass.

R=radius of a spherical particle; m* = corresponding effective mass;

Exciton

The exciton binding energy due to Coulomb attraction between the negatively charged electron and positively charged hole is given by the following relationships (in analogy with those of H-atom where is to be replaced by reduced mass):

For H-atom?

Excitonic Bohr Radius

The equation of Bohr radius of H-atom is given by (in SI unit):

The Bohr radius of exciton in a semiconductor is the natural separation distance between the hole (in VB) and electron (CB) and it is a characteristics property of that semiconductor. It is given by:

Excitonic Bohr Radius

Excitonic Bohr radius of:

GaAs:

CdSe:

Quantum scale materials

• Comparison of Bang gap and Exciton Bohr radius of few semiconductors:

Realization of quantum effects in MOS devices

Distribution of charges with voltage

Holes are attracted

Holes are repelled

Electrons are attracted

Accumulation

Depletion

Inversion

• Charge distributions depend on the applied voltage and frequency of the applied ac signal.

Band diagram in MOS

• For ψ_s<0, accumulation of holes (band bends upward) for p-type substrate.
• For ψ_s > 0, accumulation of electrons for n-type substrate.

When the band bends downward, ψ_s> 0; and when it bends upward, ψ_s< 0.

Band bending and Surface potential

Surface quantization in MOS

• Inversion layer forms within 5 to 7 nm of the interface.

Surface quantization

• Surface quantization effect modifies transport properties in the MOS devices significantly.

• Surface quantization effect can be exploited for developing novel optoelectronic and quantum devices.

• Carriers do not gather exactly @ interface.
• The carriers’ peak appears at a certain depth from the interface.

Voltage Tunable Quantum Dots

(VTQDs)

MOS based QD (VTQD)

Nanowire MOS based VTQD

Approach for analytical modeling

J Appl Phys. 115 (12), 124502 2014); 125 (8), 082506(2019), Sup. Latt. Micro. Struct. 97, 548-555 (2016); Trans. Elect. Dev. 65 (2), 411-418 (2016); J. Comp. Elect. 18 (2), 465 (2019), Phys. Rev. Appl. 15(2), 024055 (2021), 15 (5), 054060 (2021).

Schematic representation of VTQD formation

Fabrication of VTQD array

Optimizing EBL-parameters for fabrication

• Optimization process:

• Optimizing parameters:

PMMA concentration, PMMA-layer thickness, Pre-bake time, Write field, Operating voltage, Operating distance, Magnification, System aperture, Electron dose, Developer and stopper solution concentration and time, Post-bake time.

Fabrication process steps

Fabricated vertical Pt/SiO₂/Ge-nanowire MOS

• FESEM images are taken at 120 kX magnification and 3 kV operating voltage with electron dose of 65k µC/cm².

Fabricated vertical NWMOS device: contd…

Fabricated vertical NWMOS device: contd…

• TEM image of single nanowire MOS device showing different regions.

• Obtained Nanowire-radius ~25 nm & oxide thickness ~20 nm.

• Showing formation of Ge [400]-plane at 2θ = 62.6⁰ on [400]-plane of Si substrate.

• SAED spots arranged in ring like pattern, attributed to [400]-plane of Ge.

Room Temperature Confinement

S. Chattopadhyay et al.; Physical Review Applied, 2021.

• Indicates the formation of voltage tunable Quantum Dot.
• In inversion, as per calculation, three quantum states are created within 2.5 V bias.
• Thus, 6 electrons are confined in the these states.
• The electron states and electron numbers can be engineered!!

A proposed MOSFET based Qubit

Possible MOSFET based Qubit

• A ballistic dual-gate NW-FET (<10 nm) on SOI substrate.
• Gate voltages create VTQDs within Nanowire with quantized/ discrete energy levels.
• Choose materials with low effective mass (e.g., GaAs ~0.06, InP ~0.08, Ge ~0.08).
• Energy spacing of quantum states in VTQDs ~ 500 meV (>> KT_Room/q); the effects of phonon scattering is negligible at Room Temperature.
• Gate voltages manipulate the superposition along with phase.
• Small drain bias is applied for read-out.

Operational at Room Temp.

Superposition and Measurement

S. Chattopadhyay et al., Advanced Quantum Technologies 6, 2200072 (2023).

Qubit Gate Operation

(V_G1, V_G2) = (1.15, 1.30) V

|0〉

ΔV_G2 = 46 mV

|1〉

NOT

ΔV_G2 = 42 mV

Hadamard

Initialization

@ |1〉, V_G1, can be varied to change the phase between states, until the 1^st excited state is occupied.

Bloch Sphere

ΔV_G2 = (30 + 2) mV

Qubit Read-out

Rabi oscillation

Stability diagram

• Pulse repetition time (lifetime of electron pulse entering QD-1) ~336.5 ns.
• Qubit oscillation frequency ~25 MHz.
• Characteristic decay time for qubit dephasing ~73 ns.

(0,0)

(1,0)

(0,1)

(1,1)

• Inter-QD resonance tunneling splits single state into ‘bonding’ and ‘anti-bonding’ states.
• Relevant gate voltages lead to either of the states (0,0), (1,0), (0,1), (1,1).
• Curvature of the ‘bonding’ and ‘anti-bonding’ states depend on strength of inter-dot coupling.

Patterned nanowire FET @CRNN

Patterned nanowire FET @CRNN

Summary

• Quantum Dots can be realized in MOS structures.
• Voltage tunable quantum dots (VTQDs) are appropriate for controlling the carrier confinement level.
• It shows the potential for room temperature electron confinement.
• VTQDs can be engineered to realize quantum superposition, leading to qubit generation near room temperature
• Such VTQDs are solely implementable in the mainstream CMOS platform.

Acknowledgement

• Mr. Debopam Bhattacharya
• Mr. Sandip Bhattacharya
• Mr. Subrata Mondal
• Mr. Susoman Dey
• Ms. Ankita Sengupta
• Mr. Bodhisatwa Roy
• Ms. Aindrila Roy

Laboratories:

• QMat & DigiMF Device Group (Quantum Scale Materials, Devices & Digital Microfluidics Research Group).
• Center for Research in Nanoscience and Nanotechnology (CRNN), CU.
• Center of Excellence (COE): System Biology and Biomedical Engineering, CU.

• Dr. Basudev Nag Chowdhuty
• Dr. Subhrajit Sikdar
• Dr. Anupam Karmakar
• Dr. Chirantan Das
• Dr. Ananya Bhattacharya
• Dr. Rajib Saha
• Dr. Subhadip Chakraborty
• Dr. Jenifer Sultana
• Dr. Somdatta Paul
• Dr. Anindita Das
• Dr. Mainak Palit
• Dr. Abhisek Das
• Dr. Kunal Sinha
• Dr. Sayan Kanungo
• Dr. Sulagna Chatterjee

THANK YOU!