John M. Martinis
Analytical formulas are presented for planar capacitors used in transmon design. Tapering of the wire connecting to the Josephson junction is shown to significantly lower its loss.
# Calculator for coplanar surface loss
# John Martinis, Quantala.com
import numpy as np
from scipy import special
def Sa(a,b,t,c):
return ((np.log(4*a/t)+c+np.log((b-a)/(b+a)))+a/b*(np.log(4*b/t)+c+np.log((b-a)/(b+a))))/2/(1-a*a/b/b)
w = 20 # width of center line
g = w*(1/2) # insulator gap
t = 0.1 # thickness of film
ts = 0.002 # 2 nm surface thickness assumption
tand = 0.005 # tan(delta) = 0.005 typical loss tangent assumption
cm,cs = 5.0,1.6
#es,ems,esa,ema = 10,10,10,10 #Jim
#es,ems,esa,ema = 11.7,11.4,4,10 #Will
es,ems,esa,ema = 11.7,9.8,3.8,9.8 #John Si-1
#es,ems,esa,ema = 11.7,3.8,3.8,9.8 #John Si-2
#es,ems,esa,ema = 10,10,5,9.8 #John Al2O3
a = w/2
b = a+g
epsavg = (es+1)/2
epsMS = es**2/ems/epsavg
epsSA = esa/epsavg
epsMA = 1/ema/epsavg
pm = ts/2/a*Sa(a,b,t,cm)/special.ellipk(a*a/b/b)/special.ellipk(1-a*a/b/b)
ps = ts/2/a*Sa(a,b,t,cs)/special.ellipk(a*a/b/b)/special.ellipk(1-a*a/b/b)
pMS = epsMS*pm
pSA = epsSA*ps
pMA = epsMA*pm
QMS = 1/pMS/tand
QSA = 1/pSA/tand
QMA = 1/pMA/tand
print('pMS, pSA, pMA % = {:6.4f}, {:6.4f}, {:6.4f}'.format(100*pMS,100*pSA,100*pMA))
print('1/QMS, 1/QSA, 1/QMA 1e6 = {:6.4f}, {:6.4f}, {:6.4f}'.format(1e6/QMS,1e6/QSA,1e6/QMA))
