How to set scientific notation limits for colorbar?

Question

I am trying to put a scientific notation in all ticks of colorbar (the figure below). I could make it by colorbar(cs, format='%.2e'). The problem is that after I set tick labels using the following lines

cbar.set_ticks(levels)
cbar.set_ticklabels(levels)

the lower limit of notation seems to change to -5 and it shows normal digits for -4 like the figure below (note the tick label 0.00036). I know that one can change scientific notation limits for axes through ticklabel_format. I am looking for the same option for colorbar to change it from -5 to -4.

Any help is appreciated.

Update1:

I have used the following codes but no luck yet.

cbformat = matplotlib.ticker.ScalarFormatter()
cbformat.set_powerlimits((-12,12))
cbformat.set_scientific('%.2e')
cbar = m.colorbar(cs, location='right',format=cbformat, pad="10%")
cbar.set_ticks(levels)
cbar.set_ticklabels(np.round(levels,7))

Update 2:

To reproduce the problem, please use the following codes I copied from this post

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import ticker
from matplotlib.colors import LogNorm

z = np.random.lognormal(mean=10, sigma=3, size=(10,10))
z=z*10**-11
levels = np.linspace(z.min(), np.quantile(z,0.8), 8)
levels = np.append(levels, np.quantile(z,0.85))
levels = np.append(levels, np.quantile(z,0.9))
levels = np.append(levels, np.quantile(z,0.95))
levels = np.append(levels, z.max())   

fig, ax = plt.subplots()
# levels=np.logspace(np.log10(np.min(z)),np.log10(np.max(z)),100)
plot = ax.contourf(z, levels, norm=LogNorm())

cbar = fig.colorbar(plot, location='right',format='%.2e')
cbar.set_ticks(levels)
cbar.set_ticklabels(np.round(levels,9))

plt.show()

Answer 1

you can specify the ticks when creating the colorbar, then the cbformat works. IIUC the following plot is what you want:

fig, ax = plt.subplots()
plot = ax.contourf(z, levels, norm=LogNorm())

cbformat = ticker.ScalarFormatter()
cbformat.set_scientific('%.2e')
cbformat.set_powerlimits((-4,12))
cbformat.set_useMathText(True)
cbar = fig.colorbar(plot, ticks=levels, location='right', format=cbformat)

plt.show()