1. I have made a simple single supply comparator, ckt is attached.
2. Mosfet I have used in N-channel , with Vgs = 4V. (part number is different as mentioned in ckt).
Supply voltage range is 5V - 6.5V.
3. I have to cut the output when voltage crosses above 6.5V.
4. I have tried three different comparator with LM311 as attached, OP07 & 741 also. All have same problem.

5. Problem is:

If difference at Vin- & Vin+ is high then opamp/comarator works fine i.e full swing at output, which turn off/on mosfet completely & circuit is switched off/on accordingy.

But when voltage difference at vin- & vin+ is very low or almost equal, output voltage of opamp falls. From 6.5V it remain 4.5V or nearby. Dut to mosfet don't get on/off properly & drop across mosfet occurs.. I don 't understand why this problem.

 

Comparators often oscillate right at the threshold voltage. That may be what is happening to you. You can add a 1 meg resistor from the output to the + input to make it switch clean.

 

Why no decoupling capacitors?
Why no connections on pins 5 & 6?
Are you sure the output is not oscillating?

Edit: ronv beat me to it.
Hysteresis is common on comparators for this very reason.

 

If a NFET has a published Vgs threshold of 4V, it takes a Vgs of about 6 to10V to actually turn it on. You would have to use an NFET with an Vgs(th) of less than 2V to make this work. I would recommend replacing the NFet with an NPN Darlington...

This circuit has no hysteresis, so will rapidly switch on/off as the battery voltage decreases.

You might be interested in following this thread.