Spin qubits in quantum dots

Lecture for Quantum Hardware II

TU Delft, 2023

Covering: basic concepts, measurement techniques,

implementations, qubit approaches, current trends

With figures and slides borrowed from

J. Elzerman, R. Hanson, L. Kouwenhoven, M. Veldhorst (TU Delft)
Quantum dot

• A small semiconducting island where electrons are confined

discrete number of electrons

discrete energy level spacing

• Behavior in many ways similar to that of atoms

“artificial atoms”, with properties that can be

engineered and controlled
Energy scales

single electron charging energy: EC = e2/C

(C=8ereoR, disk)
R = 1 nm EC = 0.3 eV => room temp.
R = 100 nm EC = 3 meV => low temp.
R

confinement energy level spacing DE:

R = 1 nm El = 10 eV
R = 100 nm El = 1 meV

thermal energy
T = 300 K kT ~ 26 meV
T = 4.2 K kT ~ 0.35 meV
T = 30 mK kT ~ 2.6 meV
Quantum dots defined in a 2D electron gas

Baart et al, Nature Nano 2016

Quantum dots accumulated electrostatically

Boter et al, IEDM 2019

Veldhorst et al, Nature 2015 Scappucci,

MRS Bulletin 2021
Donors – 31P in Si

Loosely bound 15th

electron of P
Bohr radius ~ 2.5 nm

Coulomb binding
potential

Electron spin ½
Nuclear spin 1/2

B. Kane, Nature 393, 133 (1998) Images taken from A. Morello

Electrically controlled quantum dots

SOURCE DRAIN
ISLAND
e

GATE
Vsd
Vg
I

• Coupled via tunnel barriers to source and drain reservoirs

• Coupled capacitively to gate electrode(s)
Single electron tunneling
Linear response
m(N+1
(N )
GL GR m(N) electrochemical potential,
GL Eadd GR Energy an electron needs to have
m(N)
mS S in order to enter the dot
mD (N) D
m(N) = U(N) - U(N-1)
with U(N) total energy

Vg Vg ’ Eadd extra energy needed to add

one electron
Current (nA)

Eadd = m(N+1) - m(N)

Eadd = Ec + DE
N+1
N+2
N-1
N

Gate voltage (V)

Artificial atoms Kouwenhoven et al., Science 278, 1788 (`97)

Hund’s rule

2p • QDs are artificial atoms

• Change element by tuning Vg
• Full shell N = 2,6,12,20…
1s (noble elements)
• Half-full shell N = 4,9,16…
Two coupled quantum dots

Zero inter-dot capacitance, zero cross-talk

(2,0) (1,0) (0,0)

O hm s cont a ct
naar 2D E G
Al X G a 1 -X As
2D E G G aAs
gat e s
-Vg2

(2,1) (1,1) (0,1)

(2,2) (1,2) (0,2)

Vg1 Vg2
-Vg1
Two coupled quantum dots

Zero inter-dot capacitance, non-zero cross-talk

Left gate also

affects right dot
(2,0) (1,0) (0,0)
O hm s cont a ct
naar 2D E G
Al X G a 1 -X As
2D E G G aAs
gat e s
-Vg2

(2,1) (1,1) (0,1)

(2,2) (1,2) (0,2)

Vg1 Vg2
-Vg1
Two coupled quantum dots –
charge stability diagram

Non-zero inter-dot capacitance and cross-talk

Extra electron in one dot

shifts levels of other dot
(2,0) (1,0) (0,0)
O hm s cont a ct
naar 2D E G
Al X G a 1 -X As
2D E G G aAs
gat e s
-Vg2

(2,1) (1,1) (0,1)

(2,2) (1,2) (0,2)

-Vg1
Charge detection in double dot

Number of charges on dot probed by conductance of external constriction/dot

Tunnel coupled dots

“Bonding/anti-bonding states”
E n erg ies
E+

E1
Em
2W 
12

E2

E-

effect of tunnel coupling:

anti-crossing Oosterkamp et al, Nature 1998
Control signals

• Gate voltage pulses (down to few 100 ps)

• RF/Microwave excitation (up to ~ 50 GHz)

– Electric drive (voltage on a gate)
– Magnetic drive (pass current through wire)
Spin qubits in quantum dots
SL
SR
Loss & DiVincenzo, PRA 1998

Initialization 1-electron, low T, high B0

H0 ~ S wi szi

Read-out convert spin to charge  

then measure charge

ESR pulsed microwave magnetic field 

EZ =
HRF ~ S Ai(t) cos(wi t) sxi gBB

SWAP exchange interaction
HJ ~ S Jij (t) si · sj

Coherence long relaxation time T1
 J(t)
long coherence time T2
Initialization of a single electron spin

Method 1:
relaxation to
ground state

Method 2:
spin-selective
tunneling
Real-time single-electron detection

DRAIN
RESERVOIR

T
IQPC

Q
G
200 nm M P R SOURCE

See single electrons jump on/off the dot in real-time

Vandersypen et al, APL 2004

Pulse-induced tunneling

response
to electron
0.8
tunneling
response
DIQPC (nA)

0.4 to pulse

0.0

-0.4

0 0.5 1.0 1.5

Time (ms)
 Spin read-out principle:
convert spin to charge

charge SPIN UP

-e
N=1 time

charge SPIN DOWN

-e
N=1 N=0 N=1 time
~G-1
 Single-shot spin read-out

q q

t t Main limitations:
detection bandwidth
and spin relaxation

Elzerman, Hanson, Willems v Beveren, Witkamp, LMKV, Kouwenhoven, Nature 2004

 Verification spin read-out

0.3
Spin down fraction

0.2

0.1

0.0 0.5 1.0 1.5 12

Waiting time (ms)
 Spin relaxation in quantum dots –
much slower than orbital relaxation

orbital spin spin

(two-electron) (single electron)

0,3
T1 ~ 1 ms

Spin down fraction

8 Tesla
0,2

0,1
(lower bound)
0,0 0,5 1,0 1,5 12
waiting time (ms)

Fujisawa, Austing, Tokura, Hirayama, Tarucha, Elzerman et al.,

Nature 2002 Nature 2004
Single spin relaxation in quantum dots
timescale: 100 ms up to > 1 s (!)
mechanism: electric field fluctuations (from phonons)
via spin-orbit coupling (Rashba, Dresselhaus) B
B

2
10
100
B-5
1
10

T1 (ms)
10 10
0

-1
10
1
0.0 0.5 1.0 1.5 2.0 4 6 8 101214
Magnetic field [T]
Elzerman et al., Nature 2004
Kroutvar et al., Nature 2004
Amasha et al., cond-mat/0607110

Theory: Khaetskii & Nazarov, PRB 2000, 2001

Coherent rotations of a single electron spin
GaAs

BDC

BAC 

Koppens et al. Nature 2006

W W

28
Si

250 μm

Veldhorst et al, Nature Nano 2014

 Electric-dipole spin resonance DC

Nowack, Koppens, Nazarov, LMKV, Science 2007

AC

Easier local addressing

All-electrical control possible
200
0.2
180

160 use spin-orbit interaction

(pA)
(fA)

140 Golovach, Borhani, Loss PRB 2006

current

120
Dot current

100
0.1
1515 g ~ 0.39
dot

80

RF freq. (GHz)
RF frequency (GHz)
60
1010
40
15.2 GHz 55
0200 200 400 600 800
0 400time (ns)
Burst 800
0 00 0.5 1 1.5 2 2.5 3
RF burst length (ns) 0 1 2
external magnetic field (T) 3
Bext (T)
 Other mechanisms for electrical driving:
magnetic field gradient

H ~ sx,z x

B gradient introduces not only new control mechanism but also new
dephasing mechanism: charge noise now also couples to spin
Pioro-Ladriere, Obata, Tokura, Shin, Kubo, Yoshida, Taniyama, Tarucha,
Nat. Phys. 2008
> 99.9% fidelity single-qubit gates
Yoneda et al Nature Nano 2018
Spin qubits in quantum dots
SL
SR
Loss & DiVincenzo, PRA 1998

Initialization 1-electron, low T, high B0

H0 ~ S wi szi

Read-out convert spin to charge  

then measure charge

ESR pulsed microwave magnetic field 

EZ =
HRF ~ S Ai(t) cos(wi t) sxi gBB

SWAP exchange interaction
HJ ~ S Jij (t) si · sj

Coherence long relaxation time T1
 J(t)
long coherence time T2
Study material
Study material
• Vandersypen and Eriksson, Quantum computing with semiconductor
spins, Physics Today 72, 8, 38 (2019)
• Selected sections (see Brightspace) of Hanson et al, Spins in few-
electron dots, Rev. Mod. Phys. 79, 1217 (2007)

Additional reading:
• Single quantum dots:
– Kouwenhoven et al, Rep. Progr. Phys. 64 (6), 701 (2001)
• Double quantum dots:
– van der Wiel et al., Rev. Mod. Phys. 75, 1 (2003)
• Spin qubits in silicon
– Zwanenburg et al, Rev. Mod. Phys. 85, 961 (2013)
• Scaling spin qubits
– Vandersypen et al, npj Q Info 3, 34 (2017)
Continued on Friday!